Vardøhus Codes: Polar Codes Based on Castle Curves Kernels

CASTLE CURVES AND POLAR CODES 1 Vardøhus Codes: Polar Codes Based on Castle Curves Kernels arXiv:1901.06923v1 [cs.IT] 21 Jan 2019 Eduardo Camps∗, Edgar Martı́nez-Moro† , and Eliseo Sarmiento‡ Abstract In this paper we show some applications of algebraic curves to the construction of kernels of polar codes over a discrete memoryless channel which is symmetric w.r.t the field operations. We will also study the minimum distance of the polar codes proposed, their duals and the exponents of the matrices used for defining them. All the restrictions that we make to our curves will be accomplished by the so called Castle Curves. Index Terms Castle curves, Polar codes, algebraic kernels, Algebraic Geometry codes I. I NTRODUCTION Given a non-singular square matrix G of size l × l over the finite field Fq with q elements (q a power of a prime) it can be used for designing the kernel of a polar code. In such construction it is interesting to study the exponent of the matrix and the information set. Concerning the information set, itwas proved in [3] that given a binary symmetric channel the construction  1 0  can be analyzed in terms of the elements in the polynomial ring based on the binary matrix GA =  1 1 F2 [x1 , . . . , xn ]/hx21 − x1 , . . . , x2n − xn i. That is, for a fixed given monomial order (thus giving a divisibility on the ring and a weight to the variables xi , i = 1, . . . , n) one can get information for its information set. In the same paper the authors also devise a formula for the minimum distance, derived the dual code and they proved that the permutation group of the code was “large”. In this paper we will show some applications of algebraic curves to polar codes. If we apply some restrictions to a discrete memoryless channel (DMC) W : Fq → O and we will study under which assumptions a matrix G polarizes in terms of the curve used to construct its kernel. Also the information set, the minimum distance of a polar code and its dual based on the curve will be shown generalizing some of the results in [3], in particular we will keep the same notation they used for some properties of polar codes. All the restrictions that we make to our ∗ Departamento de Matemáticas – Instituto Politécnico Nacional – México – Email: [email protected] – Partially supported by a grant ’Beca Mixta’, CONACYT (México) † Institute of Mathematics – University of Valladolid – Castilla, Spain – Email: [email protected] – Partially funded by Spanish-MINECO MTM2015-65764-C3-1-P research grant. ‡ Departamento de Matemáticas – Instituto Politécnico Nacional – México – Email: [email protected] – Partially supported by SNI–SEP. January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 2 curves will be accomplished by Castle Curves [17], this is the reason for the title since Vardøhus (Norway) is the only castle known by the authors in the polar region. The structure of the paper is as follows. In Section II we have compile some basic facts on polar codes and algebraic geometric curves needed to understand the paper. Section III reviews some results in [1] and we adapt them for constructing kernel matrices that arise from algebraic curves for a SOF channel that is a discrete memoryless channel which is symmetric w.r.t the field operations. Section IV deals with the computation of the minimum distance and the dual of codes proposed in the previous section. Finally Section V is devoted to the study of the exponent of such codes and how to reduce them to get a better exponent for a given matrix size. II. P RELIMINARIES A. Polar codes Arıkan introduced in [2] a method to get efficient capacity-achieving binary source and channel codes, generalized later by Şaşoğlu et. al. in [18]. In this section, we review briefly some results used in the rest of the work. Given a non-singular matrix G over Fq (q = pr a power of a prime) of size l × l and a discrete memoryless channel N W : Fq → O, take N = ln and define Wn : FN such that q →O Wn (y1N | uN 1 ) = N Y W (yi | uN 1 Gn ) i=1 ⊗n ⊗n is the Kronecker product of G with itself n times and Bn is the matrix N × N uN 1 Bn yields that if (in , . . . , i1 ) is the l-ary expansion of i, then vi = ui′ , where i′ has with Gn = Bn G , where G such for the vector v1N = (i) the expansion (i1 , . . . , in ). The channel Wn is splitted into N channels Wn : Fq → ON × Fi−1 with q Wn(i) (y1N , ui−1 | ui ) = 1 X Wn (y1N | uN 1 ) N −i uN i+1 ∈Fq These channels are compared via its rate, that is, for a channel W : Fq → O   X X 1 W (y|x) I(W ) = W (y|x) logq q WO (y) y∈O x∈Fq where WO (y) represents the probability of receive y through W . We say that G polarizes W if for each δ ∈ (0, 1) we have: lim n  o  (i) i ∈ {1, . . . , N } | I Wn ∈ (1 − δ, 1] lim n  o  (i) i ∈ {1, . . . , N } | I Wn ∈ [0, δ) n→∞ n→∞ N N = I(W ) = 1 − I(W ) If G polarizes, we say that G the kernel of the polarization. Arıkan’s original construction using GA =  a matrix  1 0   polarizes any binary symmetric channel W . Defining a channel over Fq as symmetric if for each x ∈ Fq 1 1 exists a permutation σx such W (y|x) = W (σx′ −x (y)|x′ ) for each y ∈ O and x′ , x ∈ Fq , Mori and Tanaka [16] showed that source polarization is the same as symmetric channel polarization and January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 3 Theorem 1 ([16], Theorem 14). Let G be a non-singular matrix of size l × l over Fq , V an invertible upper triangular matrix and P a permutation matrix. If G′ = V GP is a lower triangular matrix with units on its diagonal (we call it a standard form of G), then for a G with a non-identity standard form the following statements are equivalent • Any symmetric channel is polarized by G. • Fp (G′ ) = Fq for any standard form G′ of G, where Fp (G′ ) denotes the field generated by the adjunction of the elements in G′ . • Fp (G′ ) = Fq for one standard form G′ of G. (i) When a matrix G polarize some channel W we get a efficient codes choosing the best channels Wn . For this purpose, we use the Bhattacharyya parameter for a channel W : Fq → O defined as follows Z(W ) = 1 q(q − 1) X Xp W (y|x)W (y|x′ ). x,x′ ∈Fq ,x6=x′ y∈O We construct a polar code choosing an information set An ⊂ {1, . . . , N } with the condition that for each i ∈ An and j ∈ / An yields     Z Wn(i) ≤ Z Wn(j) . The polar code CAn is generated by the rows of Gn indexed by An . Due to the polarization of G over W , this code will have a low block error probability. To see this, we use the concept of rate of polarization or exponent of G introduced by Korada et al. in [10] and generalized by Mori and Tanaka in [14]. The exponent of G is defined as E(G) = l 1 X ln Di , l ln l i=1 where Di is called the partial distance and it is define as Di = d(Gi , hGi+1 , . . . , Gl i), where Gi is the i-th row of G. The original Arıkan’s matrix GA has exponent 12 . The exponent is the value such that • for any fixed β < E(G) β lim inf P [Zn ≤ 2−N ] = I(W ). n→∞ • For any fixed β > E(G) β lim inf P [Zn ≥ 2−N ] = 1, n→∞ where Zn = Z(Wn′ ) and Wn′ = (Bn ) ′ Wn−1 1 with {Bn }n∈N independent random variables identically distributed over {1, . . . , l}. Anderson and Matthews proved in [1] that this means that for any β < E(G) polar coding using kernel G over a DMC channel W at a fixed rate 0 < R < I(W ) and block length N = ln implies β Pe = O(2−N ), where Pe is the probability of block error. The partial distances Di can be estimated by the sucesion of nested codes hGi , . . . , Gl i and shortening matrices leads to a good exponents in smaller sizes. Compute the exponent and the information set An are two of the main January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 4 problems in polar coding. About the last topic Bardet et al. [3] proved that for GA the structure of the information set can be derived from a monomial order over F2 [x1 , . . . , xn ] and they proved also that minimum distances are computable and duals of polar codes have similar structures using the fact that rational curves over F2 have nice properties. All these conditions leads to consider a special type of algebraic curves, the Castle-like curves that have a nested code structure. They can be described in terms of a finite-generated algebra and satisfied the isometry-dual property. B. Algebraic pointed curves and AG codes Let us remember some facts about algebraic geometry (AG) codes over curves (for an extensive account on AG codes see for example [13]). By a curve we mean a projective, non-singular, geometrically irreducible algebraic curve X over Fq and we denote by X (Fq ) its rational points, by Fq (X ) its function field and by g = g(X ) its Pl genus. We will consider two rational divisors D = i=1 Pi , where the Pi , i = 1, . . . , l are distinct rational points in the curve (rational places) and G such that supp D ∩ supp G = ∅ and 1 ≤ deg(G) ≤ n + 2g − 1. We define the evaluation map evD : L(G) → Flq as evD (f ) = (f (P1 ), . . . , f (Pl )), where L(G) is the vector space of rational functions over the curve such that either f = 0 or div(f ) + G ≥ 0. We define the evaluation code as C(D, G) = evD (L(G)). The kernel of evD is L(G − D) and the length of C(D, G) is deg D, its dimension k = l(G) − l(G − D) and its minimum distance δ(C(D, G)) ≥ deg D − deg G. Given Q ∈ X (Fq ) we called a pointed curve to the pair (X , Q). We denote by H(Q) to the Weierstrass semigroup of Q and given D as before we denote H ∗ (Q) = {m ∈ N0 | C(D, (m − 1)Q) 6= C(D, mQ)}. Clearly, |H ∗ (Q)| = l and we can write H ∗ (Q) = {m1 , . . . , ml }. Most of information of the codes {C(D, mQ)}m∈N0 is contained in H ∗ (Q). If X is a curve of genus g we say that H(Q) is symmetric if h ∈ H(Q) ⇐⇒ 2g − 1 − h ∈ / H(Q). When H(Q) is symmetric and D ≡ lQ we have H ∗ (Q) = H(Q) ∩ {0, 1, . . . , n − 1} ∪ {l1 , . . . , lg } where li are gaps of Q [7]. The isometry-dual condition for a sequence of codes of length l {Ci }li=1 , Ci ( Ci+1 , means that there is x ∈ Flq such that for each i ∈ {1, . . . , l}, Ci⊥ is isometric by x to Cl−i . In [7] they also proved that the following statements are equivalent when l ≥ 2g − 2 • The set {C(D, mQ)}m∈H ∗ (Q) satisfies the isometry-dual condition, • the divisor (l + 2g − 2)Q − D is canonical, • l + 2g − 1 ∈ H ∗ (Q). Then if l ≥ 2g − 2 and D ≡ lQ, the sequence of nested codes {C(D, mi Q)}mi ∈H ∗ (Q) satisfy the isometry dual condition. Observe that a rational curve satisfies these conditions. January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 5 III. A LGEBRAIC C URVES AND K ERNELS From now on G will be a non-singular square matrix G of size l × l over the finite field Fq with q elements (q = pr a power of a prime) and W : Fq → O a DMC channel. Let Gn be the matrix used for constructing a polar code of length N = ln based on G and consider Wn (y1N |uN 1 )= N Y W (yk |ul1 (Gn )∗,k ) k=1 and the partitions given by Wn(i) = (y1N , ui−1 1 |ui ) = X Wn (y1N |uN 1 ). uN i+1 (j)  (i−1)l+j (i) and Wn are the same in the sense that the parameters I(W ) y Z(W ) Note that the channels Wn−1 are equal in both cases. Proposition 2. If W : Fq → O is a DMC given two integers 1 ≤ i ≤ ln−1 and 1 ≤ j ≤ l then (j)  (i) Wn−1 = Wn((i−1)l+j) . 1 Proof: Let l  n−1 n × Fi−1 → Ol f : Ol × Fl(i−1) q q defined as follows n l(i−1) f (y1l , u1 n−1 (i−1)l l kl l ) = (y(k−1)l n−1 +1 , u1 G∗,k , . . . , u (i−2)l+1 G∗,k )k=1 . Then we have that n (i−1)l+j−1 Wn((i−1)l+j) (y1l , u1 |uj ) n l X = 1 q ln −1 n X l Y X l lY Y W yk n ul(i−1)l+j+1 k=1 n−1 = (∗) = = 1 q ln −1 1 q l−1   l Y   k=1 uil (i−1)l+j+1 1 q l−1 W n ul(i−1)l+j+1 k=1 k′ =1 X X l Y k=1 uil (i−1)l+j+1 (i) = Wn−1 (j)  1 n u(Gn )∗,k h=1  ! y(k−1)ln−1 +k′  uhl (h−1)l+1 G∗,k n−1 1 q ln−1 −1 X lY n−1 l vi+1  W y(k−1)ln−1 +k′ k′ =1 ln−1 h=1 (Gn−1 )∗,k′     i ln−1 ((uhl (h−1)l+1 G∗,k )h=1 , vi+1 )(Gn−1 )∗,k′    n l(i−1) (i) G )k uil Wn−1 f (y1l , u1 ∗,k (i−1)l+1 l(i−1) f (y1l , u1 (i−1)l+j−1 ), u(i−1)l+1 u(i−1)l+j  where the equality (∗) follows from the fact that the space generated by the last ln − (i − 1)l − j + 1 rows in the matrix Gn has the same dimension as the space l times cartesian product of the space given by the last ln−1 − i + 1 rows in Gn−1 . Therefore, since there is a bijection between the output alphabets of both channels, their parameters are the same. January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 6 We will be interested in those channels where we can recognize the operations among their elements, more formally Definition 3. Let W : Fq → O be a DMC. We say that W is symmetric w.r.t the field addition if for each x ∈ Fq there is a permutation σx ∈ SG(O) such that W (y|x) = W (σx′ −x (y)|x′ ), ∀x, x′ ∈ FQ , ∀y ∈ O. We say that W is symmetric w.r.t. the field product if for each α ∈ F∗q there is a permutation ψα such that W (y|x) = W (ψα (y)|αx), ∀x ∈ Fq , ∀y ∈ O We say that W is symmetric w.r.t. the field operations in Fq (SOF) if W is symmetric w.r.t. the field addition and product. Remark 4. Note that if the channel W is symmetric w.r.t the field addition for each α ∈ Fq we have that W (y|x) = W (σα (y)|α + x). Example 5. Consider the channel WSq : Fq → Fq with transition probabilities given by WSq (y|x) = (1 − p)χx (y) + p . n Then WSq is a SOF channel. This channel has been studied in [6], [5], [11], [19]. Corollary 6. In the binary case q = 2 we have that any channel symmetric w.r.t the field addition is also a SOF channel. Using Proposition 2 the polarization process can be analyzed inductively using the following result. Proposition 7. If W : Fq → O is a SOF channel and G a non-singular square matrix of size l × l over Fq , then (i) W1 is also a SOF channel. Proof: The symmetry w.r.t the addition is known, see [16]. Therefore we will check the symmetry w.r.t the product. Let i ∈ {1, . . . , N } and α ∈ F∗q , then we have that X Wn(i) (y1l , ui−1 Wn (y1N |uN 1 ) 1 |ui ) = uN i+1 = N XY k=1 uN i+1 = N XY uN i+1 k=1  W yk  N X j=1  uj (Gn )j,k  W ψα (yk ) α N X j=1  uj (Gn )j,k  i−1 = Wn(i) (ψα (yk ))N k=1 , αu1 |αui since (ui+1 , . . . , uN ) 7→ (αui+1 , . . . , αuN ) is a bijection. Hence we define
i−1 N Ψα (y1N , ui−1 1 ) = ((ψα (yk ))k=1 , αu1 ) January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 7 and we get the result. The following result also follows from [16]. Proposition 8. Let G be a non-singular square matrix of size l × l over Fq , V be an upper triangular invertible (i) matrix and P a permutation matrix and consider G′ = V GP . Let W : Fq → O a SOF channel and W1 (i) y W ′1 the channels associated to the polarization processes with the matrices G and G′ respectively. then we have (i) (i) I(W1 ) = I(W ′ 1 ), (i) (i) Z(W1 ) = Z(W ′ 1 ). Corollary 9. If G polarizes a SOF channel W and the matrices G′ and G are given as in the above proposition, then G′ also polarizes W . Moreover, if An and A′n are the information sets generated by G and G′ respectively, then An = A′n . Proof: It follows from Proposition 8 and Proposition 2. As we have seen before when the channel W is symmetric w.r.t. the addition the kernel of the polar code has all the information in the spaces hGl,∗ i ⊂ . . . ⊂ hG1,∗ , . . . , Gl,∗ i. It is natural to associate this structure with the derivative of an algebraic curve. Let X be and algebraic curve and Pl D = i=1 Pi , where Pi 6= Pj if i 6= j, and Pi rational points (places in X of degree 1) and let us suppose that there exist divisors A1 , . . . , Al such that the support of Ai and D are disjoint for each i , Ai ≤ Ai+1 and Fq = C(D, A1 ) ( . . . ( C(D, Al ) = Flq . (1) We consider now f1 , . . . , fl functions such that hf1 , . . . , fi i = C(D, Ai ) and we build the evaluation matrix G given by Gi,∗ = evD (fi ). Pointed algebraic curves satisfy the above construction. If we are given a pointed curve (X , Q) and D = ∗ Pl i=1 Pi where Pi are different rational points, let H (Q) = {m1 , . . . , ml } and Ai = mi Q, we get the desired structure. Example 10. Consider the field with 4 elements F4 and the Hermitian curve x3 = y 2 + y. If we take Q as the P common pole of x and y and the divisor D = Pα,β where Pα,β is the common zero of x − α and y − β, then deg D = 8 and H ∗ (Q) = {0, 2, 3, 4, 5, 6, 7, 9}. It follows that January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 8 00 01 1α 1α2 αα αα2 α2 α α2 α2 x3 y 0 0 α α2 α α2 α α2 x2 y 0 0 α α2 1 α α2 1 3 x 0 0 1 1 1 1 1 1 xy 0 0 α α2 α2 1 1 α α 2 α α α α2 α α2 α α2 1 1 α α α2 α2 1 1 1 1 1 1 2 0 0 1 1 y 0 1 α x 0 0 1 1 1 x 2 When the channel W is symmetric w.r.t. the addition we will call the kernel associated to the pointed curve (X , Q) to any evaluation matrix generated by a basis {f1 , . . . , fl } where each fi ∈ L(mi Q) \ L(mi−1 Q). Note that it is well defined since by Corollary 9 any matrix of this form produces the same set An . In order to study the structure of those matrices associated to curves note that L(∞Q) = finitely generated algebra. S∞ m=0 L(mQ) is a Proposition 11 ([8], Proposition 5.2). Let (X , Q) be a pointed curve and H(Q) = ha1 , . . . , as i, where {a1 , . . . , as } is a minimal generator set of H(Q), then there exists an ideal I ⊂ Fq [t1 , . . . , ts ] such that L(∞Q) = Fq [t1 , . . . , ts ]/I. Proposition 12. Let D = such that Pl i=1 Pi be a divisor of rational places and let us suppose that there exists a z ∈ L(∞Q) (z) = D − lQ. If fz ∈ Fq [t1 , . . . , ts ] is a polynomial such that fz represents z, then evD (L(∞Q)) = Fq [t1 , . . . , ts ]/hI, fz i. Proof: If x ∈ ker(evD ) ∩ L(mQ), then x ∈ L(mQ − D) and since (z) = D − lQ then we have x ∈ zL((m − l)Q). Ie., the image of x in Fq [t1 , . . . , ts ] is in the ideal generated by the equivalence class represented by fz , hence x ∈ Fq [t1 , . . . , ts ]/hI, fz i. Since m has been arbitrary chosen we have ⊂. The other contention follows since evD (yz) = evD (y) ∗ evD (z) = 0. IV. I NFORMATION SETS FOR SOF CHANNELS In this section we analyze the information set An for a SOF channel. The main tool we will use is channel degradation. January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 9 Definition 13. Let W : I → O and W ′ : I → O′ be two DMC channels. We say the W ′ is a degradation of W and we will denote it as W ′  W , if there exists a channel W ′′ : O → O′ such that W ′ (y|x) = X W ′′ (y|z)W (z|x) z∈O ′ for any y ∈ O , x ∈ I. One can think on degradation as a “composition” of channels in the sense that the transition probability of W ′ represents the probability of the event if we send x trough channel W and the received is transmited by channel W ′′ we get y. That is x W W′ z W ′′ y Therefore degradation make the transmission gets worse. Proposition 14. If channels W : I → O and W ′ : I → O′ satisfy W ′  W , then Z(W ) ≤ Z(W ′ ), I(W ) ≥ I(W ′ ). Proof: Let a, b ∈ I, then Za,b (W ′ ) = Xp W ′ (y|a)W ′ (y|b) y∈O ′ = X sX y∈O ′ W ′′ (y|z)W (z|a) z∈O X W ′′ (y|z)W (z|b) z∈O X Xp ≥ W (z|a)W (z|b)W ′′ (y|z) y∈O ′ z∈O = Xp W (z|a)W (z|b) z∈O = Za,b (W ) where the inequality follows from Cauchy-Schwartz. If we take the mean among all the pairs (a, b) ∈ I 2 , a 6= b it follows the desired result. The second inequality follows form the data processing inequality [4]. Moreover, degradation is preserved by the polarization process, more formally Proposition 15. Let W : Fq → O and W ′ : Fq → O′ be two channels such that W ′  W and let G be a non-singular square matrix of size l × l over the finite field Fq , then (i) (i) W ′ 1  W1 . January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 10 Proof: (i) W ′ 1 (y1l , ui−1 1 |ui ) = l XY W ′ (yk |uG∗,k ) uli+1 k=1 = l X XY W ′′ (yk |z)W (z|uG∗,k ) uli+1 k=1 z∈O = l XXY W ′′ (yk |zk )W (zk |uG∗,k ) uli+1 z1l k=1 = l XY W ′′ (yk |zk ) l XY W (zk |uG∗,k ) uli+1 k=1 z1l k=1 = l XY (i) W ′′ (yk |zk )W1 (z1l , ui−1 1 |ui ). z1l k=1 i−1 l If we define W ′′′ (y1l , ui−1 1 |z1 , u1 ) = Ql k=1 W ′′ (yk |zk ) we conclude the proof. Lemma 16. Let (X , Q) be a pointed curve of genus g such that l ≥ 2g and H(Q) = ha1 , . . . , as i is a minimal generator set of H(Q). Let us define fi : H ∗ (Q) → H ∗ (Q) as    m − ai m − ai ∈ H ∗ (Q) fi (m) = ,   l + m − ai m − ai ∈ / H ∗ (Q) then fi is a bijection. Proof: Since l ≥ 2g we know that H ∗ (Q) = H(Q) \ {l + H(Q)} = H(Q) ∩ {0, . . . , l} ∪ {l + l1 , . . . , l + lg }, where li are the gaps of Q. If ai ≤ m < n, then either m−ai ∈ H(Q) (therefore in H ∗ (Q)) or m−ai ∈ / H(Q) and hence l+m−ai ∈ H ∗ (Q), while l + m ∈ / H ∗ (Q) (if not m will be a gap). If m > l and m − ai < l, then m − ai ∈ H ∗ (Q) but m − ai ∈ / fi ({ai , . . . , l − 1} ∩ H(Q)) ⊂ {0, . . . , l − ai − 1}. On the other hand, if m > l y m − ai > l then m − ai ∈ H ∗ (Q). Of course, if m − ai ∈ / H ∗ (Q) then m − ai − l is a non-gap, but m − l is a gap, which contradicts ai ∈ H(Q). Therefore m − ai 6= fi (m′ ) for any ai ≤ m′ < l. Finally, if m < ai then −ai ≤ m − ai < 0 and hence l − ai ≤ l + m − ai < l and since m is a non-gap it is not covered in the previous cases, therefore fi is injective and by cardinality it is bijective. From now on T = (t1 , . . . , ts ) y sea R[T ] = Fq [t1 , . . . , ts ]/I(T ), where the ideal I(T ) is the one given in Proposition 11. Theorem 17. Let W be a SOF channel. Let us consider the pointed curve (X , Q) and a divisor on the curve D = (z)0 with l = deg(z)0 ≥ 2g, and let H ∗ (Q) = {m0 , . . . , ml−1 }. If we consider also an element ml−i ∈ H ∗ (Q) with ml−i < l. If ml−j = ml−i − ar ∈ H ∗ (Q), where ar is one of the generators of H(Q), then (i) (j) W1  W1 . January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 11 Proof: We can choose monomials Mk in R[T ] such that vQ (Mk ) = ml−k and such that if ml−k −ar ∈ H ∗ (Q) then tr |Mk and also if b = max{a ∈ N0 | tar |f, f ∈ L(mr−k Q) \ L((mr−k − 1)Q)} then tbr |Mk . We construct the kernel G evaluating those monomials. Thus if tr |Mk for any k then Mk tr = Mk′ for some k ′ it is clear that ml−k = ml−k′ + ar . Consider the polynomial f = Pl k=1 uk Mk and denote as W1 (y|f ) = Ql k=1 W (yk |f (Pk )). We have that W1 (y|f ) = W1 (y|ul1 ). If we consider A = {k ∈ {1, . . . , l}| ml−k − ar ∈ H ∗ (Q)}, then ! X X uk Mk . uk Mk + W1 (y|f ) = W1 y k∈A k∈A / Let Fr the function Fr (Mk ) = Mk′ ⇔ fr (ml−k ) = ml−k′ . Applying Lemma 16 we have a bijection of the chosen monomials and also W1 (y|f ) = W1 y X uk Fi (Mk )tr + k∈A X uk Mk k∈A / ! , where ui Mi = ui Mj tr . We define y = (yα1 , . . . , yαz ) where supp tr = {α1 < . . . < αz } and if g is a polynomial we define σg (y) = (σg(Pα1 ) (yα1 ), . . . , σg(Pαz ) (yαz )),   −1 −1 ψt−1 (y) ) . ), . . . , ψ (y (y = ψ tr (Pα ) αz tr (Pα ) α1 r z 1 Note that W1 (y|f ) = W1 y X uk Fr (Mk )tr + W X k∈A Y = X uk Mk k∈A / yα uk Mk (Pα ) k∈A / α∈supp / tr ! ! z Y W yαh h=1 X uk Fr (Mk )tr + k∈A uk Mk (Pα ) α∈supp / tr X k∈A / Y X uk Mk (Pα ) = W W yα yα α∈supp / tr k∈A / Y X = α∈supp / tr W yα X ! (Pαh ) ! uk (Fr (Mk )tr − Mk ). Since we k∈A / are in a SOF channel it follows W1 (y|f ) = uk Mk k∈A / Let û be the result of taking only those indexes uk with k ∈ / A and g(û) = Y X uk Mk (Pα ) k∈A / ! ! ! z Y W yαh h=1 z Y X uk Fr (Mk )tr + k∈A W σg(û) (y)h h=1 z Y X tr l X W ψt−1 r ! uk Fi (Mk ) (Pαh ) k=1 (Pαh ) ! ! l
X uk Fr (Mk )(Pαh ) σg(û) (y) h k=1 h=1 uk Mk k∈A / ! ! . Now since Fr is a bijection there is a permutation ϕ : {1, . . . , l} → {1, . . . , l} such that l X k=1 that also satisfies uk Fr (Mk ) = l X uϕ(k) Mk , k=1 ϕ(j) = i ϕ−1 (k) > j ⇐⇒ k ∈ A, k > i. January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 12 This last fact is because if ml−k − ar ∈ / H ∗ (Q) then l + ml−k − ar ≥ l − ar > ml−i − ar . Now let us consider the channel given by Q : Y l × Fj−1 → Y l × Fi−1 as follows q j−1 Q(y, ui−1 ) 1 |z, v1 Y = W yα α∈supp / tr X vϕ−1 (k) Mk (Pα ) k∈A / if uϕ(k) = vk for 1 ≤ k ≤ i−1 and y = σg(v) (ψtr (z)) with g(v) = If v ∈ Flq is a vector where vj = ui , then P ! k∈A / vϕ−1 (k) (Fr (Mk )−Mk ) and 0 elsewhere. (j) j−1 Q(y, ui−1 )W1 (z, v1j−1 |ui ) 1 |z, v1 X z,v1j−1 (∗) = = 1 q l−1 X X z,v1j−1 1 q l−1 z Y l vj+1 z,v1j−1 W = q l−1 = = W yα k∈A / Y X W yα l α∈supp / tr vj+1 ψt−1 r X α∈supp / tr X X h=1 1 Y v1j−1 1 l
X σg(v) (y) h vk Mk (Pαh ) q l−1 1 q l−1 Y W yα Y uli+1 α∈suppt / r X W1 (y|u) W X yα X W1 (z|v) Y W zα X vk Mk (Pα ) k∈A / α∈supp / tr ! ! vϕ−1 (k) Mk (Pα ) k∈A / l α∈supp / tr vj+1 X vϕ−1 (k) Mk (Pα ) ! k∈A / k=1 XX vϕ−1 (k) Mk (Pα ) ! uk Mk (Pα ) k∈A / ! z Y ! z Y W h=1 W ψt−1 r h=1 ψt−1 r l
X vk Mk (Pαh ) σg(v) (y) h k=1 l
X σg(û) (y) h uϕ(k) Mk (Pαh ) k=1 ! ! uli+1 (i) =W1 (y, ui−1 1 |ui ) where from step (∗) on the sum that ranges in z, v1j−1 is only over those indexes that z = ψt−1 (σg(v) (y)) and r uϕ(k) = vk for 1 ≤ k ≤ i − 1. Finally since for any matrix G defining the kernel the information set does not change then the result follows. In order to fix a matrix given a pointed curve (X , Q) so we can describe the polar code in terms of the function field Pl−1 we will take D = (z)0 = i=0 Pi and H ∗ (Q) = {m1 = 0, . . . , ml }. It is known that evD (L(∞Q)) = R[T ]/hfz i, thus a basis is given by evD (L(∞Q)) = ∆(I(T ), fz ) = {M0 , . . . , Ml−1 } where (Mi )∞ = mi+1 Q. Evaluating that basis we will construct the matrix G for the polarization process. From now on we shall consider always the matrix for constructing polar codes from the pointed curve (X , Q). For each n ∈ N and each i ∈ {0, . . . , ln −1} we denote by (in , . . . , i1 ) to the l-ary expansion of i, ie. 0 ≤ ik ≤ l−1 and i= n X ik lk−1 . k=1 Let Pin = (Pin , . . . , Pi1 ), Fq [X1 , . . . , Xn ], Xk = (xk1 , . . . , xks ) and In = hI(X1 ), . . . , I(Xn ), fz (X1 ), . . . , fz (Xn )i. January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 13 In the polynomial ring Fq [X1 , . . . , Xn ] we take the monomial ordering inherit from the weights in the variables xi , ie. the monomial ordering defined by the vectors with entries wn+1−i,il+j = −vQ (xij ), and we will break ties with RevLex if it is needed. As a resume, we get n copies of the original ring R[T ] and order it with weights inherit from the valuation in Q. Proposition 18. Let Mkn (X1 , . . . , Xn ) = Mk1 (X1 ) · · · Mkn (Xn ), where (kn , . . . , k1 ) is l-ary expansion of 0 ≤ k ≤ ln − 1 and Mkn (Pjn ) = Mk1 (Pjn ) · · · Mkn (Pj1 ), then ∆(In ) = {Mkn (X1 , . . . , Xn ) | k ∈ {0, . . . , ln − 1} and the matrix Gn in the polarization process satisfies Gn (i, j) = M(lnn −i) (Pjn ). Proof: The equality in ∆(In ) is clear. The proof of the statements related with Gn and the columns are also clear since it is just an application of the Kronecker product. For checking the property on the rows we will use induction. Case n = 1 is clear so let us suppose it is true in the step n − 1. Due to the bit-reversal in the polarization matrix we know that the row jln−1 + iwhose l-ary expansion is (j, in , . . . , i1 ) is the row in the Kronecker product’s matrix with l-ary expansion (i1 , . . . , in , j). That row is in correspondence with the product of the monomials Mj (Xn ) and Mlnn−1 −i (X1 , . . . , Xn−1 ) by the induction hypothesis, so the result follows. As a corollary of Theorem 17 we have Corollary 19. Let G be the matrix associated to the pointed curve (X , Q) and Mi ∈ ∆(I(X1 ), fz ) with deg Mi ≤ l. If Mi 6= ∆(I, Mj ) with i > j, then (l−i) W1 (l−j)  W1 In particular the result follows if Mj |Mi (where the division is in the ring R[T ]). Consider the set An ⊂ ∆(In ) and the code given by n −1 | Mkn ∈ An }i, CAn = h{(Mkn (Pjn ))lj=0 / An we have that we say that CAn is a polar code if for each Mkn ∈ An and for each Mjn ∈ Z(Mkn ) := Z(Wn(l n −k) ) ≤ Z(Mjn ). Proposition 20. Let CAn be a polar code constructed from a pointed curve (X , Q). If Min ∈ An satisfies for all ik < jk , deg Mik ≤ l and Mik ∈ / ∆(I(Xk ), Mjk ), 1 ≤ k ≤ n, then Mjn ∈ An . In particular, if Mjn |Min then Mjn ∈ An . Proof: It follows from induction taking into account that (k) W  W ′ =⇒ W1 January 23, 2019  W1′ (k) DRAFT CASTLE CURVES AND POLAR CODES 14 and  (k) Wn−1 (k′ ) ′ = Wn((k−1)l+k ) . Thus if i1 < j1 , Min = Mi1 (X1 )Min−1 and Mjn = Mj1 (X1 )Mjn−1 then ′ ′ Wn(l n −i) = Wn((l n−1 −i′ −1)l+l−i1 ) (ln−1 −i′ ) (l−i1 ) = (Wn−1 (l n−1  (Wn−1 ) ′ −i ) (l−j1 ) ) (ln−1 −j ′ ) (l−j1 )  (Wn−1 = Wn(l n ) −j) The induction step follows from Corollary 19 above. Definition 21. We say that the code CAn is weakly decreasing if for all Mkn ∈ An we have that if j satisfies ki ≥ ji , (jn , . . . , j1 ) (the l-ary expansion of j) then Mjn ∈ An . Remark 22. The name weakly decreasing is recovered from the one in [3]. We do not have a way of ensuring that a code is weakly decreasing, but from the fact that any polar code is the shortening of a weakly decreasing code, using the proposition above we will check that for some cases the difference between a polar code and weakly decreasing code is not so big (measured as the number of rows that one has to remove). Example 23. Consider the hermitian curve x3 = y 2 + y over F4 pointed in Q the common pole of x and y. In this case ∆(I, x4 − x) = {x3 y, x2 y, x3 , xy, x2 , y, x, 1} that correspond with the values H ∗ (Q) = {9, 7, 6, 5, 4, 3, 2, 0}. If we choose x2 y ∈ A1 then xy, x2 , y, x, 1 ∈ A1 ; if x3 ∈ A1 then we have a weakly decreasing code. On the other hand if x3 ∈ A1 then x2 , y, x, 1 ∈ A1 and for getting a weakly decreasing code is enough to see that xy ∈ An . Corollary 24. Rational curves provide kernels for polar codes that are weakly decreasing. Proof: Those curves there are no gaps and H ∗ (Q) = {0, 1, . . . , q − 1} therefore for all m ∈ H ∗ (Q), m < n and m − 1 ∈ H ∗ (Q). Remark 25. The result stated in the corollary above generalizes the same statement [3] for rational curves over F2 . Definition 26. We say that a code CAn is decreasing if it is weakly decreasing and there are h1 , . . . , hk ∈ {0, . . . , l − 1} (maybe not distinct) such that Mh1 (Xi1 ) · · · Mhk (Xik ) ∈ An , January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 15 then for any jv ≤ iv , v ∈ {1, . . . , k} we have Mh1 (Xj1 ) · · · Mhk (Xjk ) ∈ An . This extra property for being decreasing will be called degrading property. Indeed it makes sense since in each step in the polarization process the new elements are worse than the previous ones. Proposition 27. If CAn is a polar code and Mh1 (Xi1 ) · · · Mhk (Xik ) ∈ An then Mh1 (Xj1 ) · · · Mhk (Xjk ) ∈ An for all jv ≤ iv , 1 ≤ v ≤ k. Proof: We will prove it by induction on n. For n = 2 remember that G2 = B2 G⊗2 where B2 interchanges the i-th row with l-ary expansion (i2 , i1 ) with the one with expansion (i1 , i2 ) (rows are indexed from 0 to l2 − 1) and therefore (B2 )−1 = B2 . Moreover G2 B2 = G⊗2 . We have that G2i,j = Mi1 (Pj2 )Mi2 (Pj1 ) that multiplied by B2 returns G2 B2 i,j = Mi1 (Pj1 )Mi2 (Pj2 ) = G⊗2 i,j . 2 Hence if u ∈ Flq , then (uB2 )(G2 B2 ) = u(G⊗2 B2 ) = uG2 . (∗) Moreover note that the last l entries in uB2 correspond with (ul−1 , u2l−1 , . . . , ul2 −1 ). Whence suppose that Mh (X2 ) ∈ A2 ; that monomial is associated with the row l2 − lh = l(l − h) while the monomial Mh (X1 ) is 2 2 2 2 associated with l2 − h. If we define Q : Y l × Flq −h−1 → Y l × Flq −lh−1 with probabilities 2 2 2 2 Q(y1l , u1l −lh−1 |z1l , v1l −h−1 ) = 1 2 2 2 2 if B2 y1l = z1l and (vB2 )1l −lh−1 = ul1 −lh−1 .Then by (∗), it follows that 2 2 X 2 2 2 2 Q(y1l , u1l −lh−1 |z1l v1l −h−1 )W2l −h 2 2 (z1l , v1l −h−1 |ul2 −lh ) 2 −h−1 z1l ,v1l = 2 X 2 −h−1 z1l ,v1l = X 2 v l2 l −h+1 1 X q l2 −1 2 ul2 2 2 2 2 1 Q(y1l , ul1 −lh−1 |z1l v1l −h−1 )W2 (z|v) q l2 −1 W2 (B2 y|uB2 ) l −lh+1 = X 1 W2 (y|u) q l2 −1 2 ul2 l −lh+1 (l2 −lh) =W2 2 In other words, W2l January 23, 2019 −lh 2  W2l 2 2 (y1l , u1l −lh−1 |ul2 −lh ). −h and therefore Mh (X1 ) ∈ A2 . DRAFT CASTLE CURVES AND POLAR CODES 16 (l)  (li) (i) = Wn , hence if Mh1 (Xi1 ) · · · Mhk (Xik ) ∈ An Let us suppose that it is true for n − 1. Note that Wn−1 and i1 ≥ 2 we have that Mh1 (Xj1 ) · · · Mhk (Xjk ) ∈ An for all jv ≤ iv , v ∈ {1, . . . , k} such that 2 ≥ j1 ≤ i1 . Pk Pk If 2 ≥ j1 < i1 then we have that if i = v=1 hv liv −1 and j = v=1 hv ljv −1 then l|i and l|j by the induction hypothesis in n − 1, therefore Wn(l n −i)  (l) n ( l l−i ) = Wn−1   ln −j l Wn−1  !(l) = Wn(l n −j) , and the result follows. Same reasoning guaranties the result if i1 = j1 = 1. It only remains the case 1 = j1 < i1 . Taking into account that degradation is a transitive relation we can suppose that i1 = 2 and choosing i′ = i − h1 l an j ′ = j − h1 , where i and j are as before. Thus applying the induction step to the case n − 2, we have that n ′ ln −i′ l2 Wn−2 l −j 2  Wn−2l . Applying induction for the case 2 we have  n ′  !(l2 −lh1 ) Wn(l n −i) =  l −i 2 Wn−2l  ln −i′ l2 Wn−2  ln −j 2  Wn−2l = n Wn(l −j) 2  !(l2 −h1 ) 2 !  (l2 −h1 ) ′ 2 and we conclude the proof. Remark 28. The previous result does not make use of the property SOF. Thus a polar code weakly decreasing is decreasing. Example 29. Take the hermitian curve from previous examples and n = 2. If y1 x2 ∈ A2 then, using Proposition 20 we get x2 , y1 , 1 ∈ A2 and applying the proposition above x1 ∈ A2 . If x1 x2 ∈ A2 , with the previous elements we have a descending polar code. V. M INIMUM D ISTANCE AND D UAL OF P OLAR C ODES Let us check some properties of the structure of a polar code constructed from a pointed curve (X , Q). Proposition 30. Let CAn be a decreasing code and let (Kn , . . . , K1 ) be a tuple such for each Mkn ∈ An the l-ary expansion of k, (kn , . . . , k1 ), satisfies ki ≤ Ki for each i ∈ {1, . . . , n}. Take H ∗ (Q) = {m1 , . . . , ml } and di = δ(C(X , D, mKi +1 Q)) and let k ′ be such that Mkn ∈ An for each k ′ ≥ k and d′i = δ(C(X , D, mki′ +1 Q)), then we have n Y i=1 January 23, 2019 d′i ≥ δ(CAn ) ≥ n Y di . i=1 DRAFT CASTLE CURVES AND POLAR CODES 17 Proof: We will proceed by induction. It is clear for n = 1. Let us suppose it is true for n and get the result for n + 1. First note that K1 ≥ K2 ≥ . . . ≥ Kn+1 since if Mkn ∈ An and MKj (Xj )|Mkn (from the hypothesis in the proposition), then MKj (Xj ) ∈ An and MKj (Xj−1 ) ∈ An , since the code is decreasing and therefore Kj ≤ Kj−1 . Let C1 be the generator matrix of C(X , D, mK1 +1 Q) and let A be the matrix with rows the evaluations of Mkn+1 ∈ An+1 with k > l − 1. Then CAn is contained in the code generated by A ⊗ C1 , which is the generator matrix for the matrix product code [C1 · · · C1 ]A and then, by [9, Theorem 2.2] we have the result. The other inequality follows in a similar way. Example 31. Consider Example 29 again taking A2 = {y1 x2 , y1 , x2 , x1 , 1}. This code is contained in the decreasing code generated by A2 ∪ {x2 x1 }, then K1 = 2 > K2 = 1. We know that m3 = 3 and m2 = 2 and the minimum distances for these hermitian codes are 3 and 2 respectively [20], therefore δ(CA2 ) ≥ 6. Remember that isometry-dual condition for a sequence of codes {Ci }li=1 means that exists x ∈ Flq such for each i ∈ {1, . . . , l}, Ci⊥ and Cl−i are isometric according to x. Codes constructed from pointed curves (X , Q) and D ∼ lQ satisfy this condition and we say that the curve satisfy the isometry-dual condition ([7]). We will see that polar codes constructed from these curves preserves a similar condition. Proposition 32. Let G be the kernel for a isometric-dual curve (X , Q) of size l × l. Let CAn be a decreasing code and define n n c A⊥ n = {Mj ∈ ∆(In ) | ji = l − 1 − ki , 1 ≤ i ≤ n, Mk ∈ An } , and this code is also decreasing. Then (CAn )⊥ is isometric to CA⊥ n Proof: If we compare the sizes of the sets we just have to check that CA⊥ ⊂ (CAn )⊥ . It is also clear n that CA⊥ is also a decreasing code. Let f (T ) ∈ L(∞Q) be the element which establish the isometry between n Pl−1 C(X , D, mi Q)⊥ and C(X , D, ml−i Q) for each i ∈ {1, . . . , l}. Then we have that i=0 f (Pi )Mj (Pi )Mk (Pi ) = 0 for each j ∈ {0, . . . , l − 1} and for every k ∈ {0, . . . , l − 1 − j}. Take F = f (X1 ) · · · f (Xn ) and Mkn ∈ An and Mkn′ ∈ CA⊥ . Then we have n n lX −1 F (Pin )Mkn (Pin )Mkn′ (Pin ) = n Y j=1 i=0 l−1 X ! f (Pi )Mkj (Pi )Mkj′ (Pi ) . i=0 We claim that there exists j ∈ {1, . . . , n} such that kj′ ≤ l − 1 − kj . If this does not happen we would have kj′ > l − 1 − kj , ∀j ∈ {1, . . . , n} l − 1 − kj′ < kj , ∀j ∈ {1, . . . , n} but CAn is decreasing, then for k ′ = Pn j=1 (l − 1 − kj′ )lj−1 , Mkn′ ∈ An \ A⊥ n , which is a contradiction. Therefore it exists such j and the sum over it is 0 and we have the result. January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 18 Corollary 33. If in the proof of Proposition 32 we have that the function f evaluates to ev(f ) = (1, 1, . . . , 1), then ⊥ = CA⊥ CA n n Codes with kernel Gq satisfies this condition. Corollary 34. Let CAn be a decreasing code from a isometric-dual curve. Let Bn and Cn be decreasing sets such that Cn ⊂ An ⊂ Bn then ⊥ ⊂ CCn⊥ . CBn⊥ ⊂ CA n Example 35. We have already mentioned that each polar code can be seen as a shortened code obtained from a decreasing code, then we can complete its dual from the dual of the decreasing one. Let us take again A2 = {y1 x2 , x2 , y1 , x1 , 1} and A′2 = A2 ∪ {x1 x2 }. This is a decreasing set and we have ⊥ A′ 2 ={x1 y1 x32 y2 , x21 x32 y2 , y1 x32 y2 , x1 x32 y2 , x32 y2 , x1 y1 x22 y2 , x21 x22 y2 , y1 x22 y2 , x1 x22 y2 , x22 y2 , x31 y1 x32 , x21 y1 x32 , x31 x32 , x1 y1 x32 , x21 x32 , y1 x32 , x1 x32 , x32 , x31 y1 x2 y2 , x21 y1 x2 y2 , x31 x2 y2 , x1 y1 x2 y2 , x21 x2 y2 , y1 x2 y2 , x1 x2 y2 , x2 y2 , x31 y1 x22 , x21 y1 x22 , x31 x22 , x1 y1 x22 , x21 x22 , y1 x22 , x1 x22 , x22 , x31 y1 y2 , x21 y1 y2 , x31 y2 , x1 y1 y2 , x21 y2 , y1 y2 , x1 y2 , y2 , x31 y1 x2 , x21 y1 x2 , x31 x2 , x1 y1 x2 , x21 x2 , y1 x2 , x1 x2 , x2 , x31 y1 , x21 y1 , x31 , x1 y1 , x21 , y1 , x1 , 1} In this case the isometry is given by (1, . . . , 1) and if we add an orthogonal vector to the evaluations of A2 ⊥ but not to the one of x1 x2 , we would have a generator set for CA . One of these vectors is the evaluation of 2 ⊥ ⊥ g = x2 x1 (x1 y1 + 1), then A′ 2 ∪ {g} generates CA . 2 All the conditions asked for the pointed curves (X , Q) are satisfied by weak Castle and Castle curves [17]. We say that a pointed curve (X , Q) over Fq is weak Castle if H(Q) is symmetric and there is a morphism φ : P → P with (φ)∞ = hQ and α1 , . . . , αa ∈ Fq such that φ−1 (αi ) ∩ X (Fq ) = h. (X , Q) be a pointed Castle curve if it is weak Castle and h is the multiplicity of H(Q) and r = q. January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 19 VI. M ODIFYING KERNELS FROM ALGEBRAIC CURVES Remember that given a square-matrix G of size l × l over Fq with rows G1 , . . . , Gl , the exponent E(G) of the matrix G is defined as E(G) = l 1 X ln Di , l ln l i=1 where Di is the called partial distance and it is the minimum of the Hamming distances d(Gi , v), with v ∈ hGi+1 , . . . , Gl i. Suppose G is non-singular over Fq of size l × l and G′ is as G. If G′ G−1 is a upper-triangular invertible matrix, then E(G) = E(G′ ); therefore, each matrix coming from a pointed curve has the same exponent. Looking for the best matrices over a given size, shortening codes is a good way to find them, for example this was the way to find the best matrix over F2 of size 16 (see [10]). Next theorem was proved by Anderson and Matthews in [1]. It says that shortening kernels from algebraic curves does not change the final structure of the code. Theorem 36. Let G be a kernel from a pointed curve (X , Q) with D = ′ ′ Pl i=1 Pi . Taking the j-th column, we can shorten G to obtain the matrix G . Then we have that G is the kernel arising from the codes {C(D − Pj , mQ − Pj )}m∈H ∗ (Q) . We can repeat this process to obtain polar codes from kernel associated to divisors of the form mQ − P P. However, if we take points coming from zero divisor of elements in L(∞Q) we will have a matrix with the same structure. Pl Proposition 37. Let (X , Q) a pointed curve and z ∈ Fq (X ) with (z) = D − lQ, D = i=1 Pi . Let’s suppose Ps there is z ′ ∈ Fq (X ) such that (z ′ ) = i=1 Pki − sQ with ki 6= kj if i 6= j; define D′ = (z ′ )0 . Let ϕ : Flq → Fsq be the mapping such ϕ(c) is the same word c but erasing the entries indexed by {i ∈ {1, . . . , n} | Pi ∈ / supp z ′ }. Let ψ : R/hI, fz i → R/hI, fz′ i the natural mapping between both rings. Then R/hI, fz i evD ψ R/hI, fz′ i Fnq ϕ evD′ Fsq is commutative. The kernel G constructed from D and Q has as submatrix G′ , the kernel from D′ and Q. Proof: Take f, f ′ ∈ R/hI, fz i such that ϕ(evD (f )) = ϕ(evD (f ′ )). This occurs if and only if evD (f )j = evD (f ′ )j ∀j ∈ Pj ∈ supp z ′ . This is evD′ (f ) = evD′ (f ′ ), then we have f − f ′ ∈ hI, fz′ i, implying ψ(f ) = ψ(f ′ ) as we wanted it. January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 20 Corollary 38. The matrix G′ of the proposition above is isometric to the one obtained after shortening G with the process described in Theorem 36. Qa Corollary 39. A Castle-like curve with D = (z)0 = ( i=1 (φ − αi ))0 produces a sequence of a kernels (each one Q submatrix of the next) coming from the divisors of ji=1 (φ − αi ), j ∈ {1, . . . , a}. Example 40. Take again the hermitian curve over F4 where α is a primitive element, x3 = y 2 + y. This is a Castle curve with kernel 00 01 1α 1α2 αα αα2 α2 α α2 α2 x3 y 0 0 α α2 α α2 α α2 x2 y 0 0 α α2 1 α α2 1 x3 0 0 1 1 1 1 1 1 xy 0 0 α α2 α2 1 1 α x2 0 0 1 1 α2 α2 α α y 0 1 α α2 α α2 α α2 x 0 0 1 1 α α α2 α2 1 1 1 1 1 1 1 1 1 If we shorten this kernel taking the points with x = 0 like in Theorem 36, starting with 00. 1α 1α2 αα αα2 α2 α α2 α2 x3 y α α2 α α2 α α2 x2 y α α2 1 α α2 1 x3 1 1 1 1 1 1 xy α α2 α2 1 1 α x2 1 1 α2 α2 α α x 1 1 α α α2 α2 . This matrix comes from the codes with divisor (x3 − 1)0 and P∞ − P00 − P01 . From the original kernel, if we remove the columns of that points and the rows products of x3 . 1α 1α2 x2 y January 23, 2019 αα αα2 α2 α α2 α2 α α2 1 α α2 xy α 2 α α 1 1 α x2 1 1 α2 α2 α α 2 2 1 2 y α α α α α α2 x 1 1 α α α2 α2 1 1 1 1 1 1 1 DRAFT CASTLE CURVES AND POLAR CODES 21 This matrix comes from the divisor (x3 − 1)0 and P∞ . The isometry between both matrices is clear and this second matrix has the same structure as the original one, so we can apply the analysis of information set, minimum distance and its dual like before. As an example of the last corollary we can give the next matrix sequence from the hermitian curve 00 01 1α 1α2 00 01 1α 1α2 00 01 y 0 1 1 1 1 2 x2 y 0 0 α α2 2 αα αα2 1 α 2 xy 0 0 α α xy 0 0 α α α 1 y 0 1 α α2 x2 0 0 1 α2 α2 α 2 x 0 0 1 1 y 0 1 α α α α2 1 1 1 1 1 x 0 0 1 1 α α 1 1 1 1 1 1 1 00 01 1α 1α2 αα αα2 α2 α α2 α2 x3 y 0 0 α α2 α α2 α α2 x2 y 0 0 α α2 1 α α2 1 x3 0 0 1 1 1 1 1 1 2 2 xy 0 0 α α α 1 1 α x2 0 0 1 1 α2 α2 α α 2 2 y 0 1 α α α α α α2 x 0 0 1 1 α α α2 α2 1 1 1 1 1 1 1 1 1 . Their exponents are, respectively 1 1 ln(6 · 4 · 3 · 2 · 2 · 1) ln(8 · 6 · 5 · 4 · 3 · 2 · 2 · 1) , , ≈ 0.5268, ≈ 0.5622 2 2 6 ln(6) 8 ln(8) Now we will check another resource to search matrices with good exponents arising from AG codes. We will need the next result. Proposition 41. Let G and G′ be two matrices over Fq of size l × l and l′ × l′ respectively, non-singular and with ′ partial distances {Di (G)}li=1 and {Di (G′ )}li=1 . Then for the matrix G′ ⊗ G we have Dk (G′ ⊗ G) = Di′ (G′ ) · Di (G) where k = (i′ − 1)l + i. Proof: For the first l rows is clear since they are just copies of the original G. Let us suppose the result for the first l′ l − (kl + l) + 1 rows (0 ≤ k ≤ l′ − 1) and let’s prove it for the rows l′ l − (k + 1)l − j, 0 ≤ j ≤ l − 1. If we begin with h = l′ l − (k + 1)l we observe that the partial distance Dh (G′ ⊗ G) is the same as the one of   {G′i }ki=1 ⊗ G   (∗) ′ l′ {Gi }i=k+1 ⊗ Il January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 22 where Il is the identity matrix of size l and G′i is the i-th row of G′ . Since the matrix G is non-singular and the h-th partial distance this is the distance d(G′ ⊗ Gh , hG′ ⊗ Gh+1 , . . . , G′ ⊗ Gl′ l i), then the last vector space is generated by the tensor product of the last l′ − k rows of G′ with Il . Also we know that G′l′ −k ⊗ Gl = l X G′l′ −k ⊗ (Gl,j ej ), j=1 where ej is the j-th vector of the canonical basis for Flq . Notice that if u and v are two vectors with disjoint supports, then the Hamming weight w(v + u) = w(v) + w(u); also, w(v ⊗ u) = w(v) · w(u). Then if we take some elements αi ∈ Fq , l′ l − (k + 1)l + 1 ≤ i ≤ l′ l we have   l′ ,l X w G′l′ −k ⊗ Gl + , α(i−1)l+j G′i ⊗ ej  i=l′ −k+1,j=1  =w  l X ′ l ,l X G′l′ −k ⊗ Gl,j ej + j=1 i=l′ −k+1,j=1  α(i−1)l+j G′i ⊗ ej      l′ X X α (i−1)l+j ′   G′l′ −k + α(i−1)l+j G′i  ⊗ ej  Gi ⊗ ej + =w  G l,j i=l′ −k+1 j∈supp Gl i=l′ −k+1 j ∈supp / Gl       ′ l′ l X X X X α ◦ (i−1)l+j ′  Gi ⊗ ej  + = w G′l′ −k + w  α(i−1)l+j G′i  ⊗ ej  G l,j j∈supp Gl i=l′ −k+1 i=l′ −k+1 j ∈supp / Gl    ′ l X X α(i−1)l+j ′ Gi  ⊗ ej  w G′l′ −k + ≥ Gl,j j∈supp Gl i=l′ −k+1   l′ X X α (i−1)l+j ′ = w G′l′ −k + Gi  w(ej ) Gl,j ′  X  j∈supp Gl ≥ X ′ l X i=l −k+1 Dl′ −k (G′ ) j∈supp Gl =Dl (G) · Dl′ −k (G′ ), where we have that ◦ sin v ⊗ ej has disjoints support for different j. The results follows if we change Gl for any vector of weight Dj (G). Next corollary is a general version of the one in [12]. Corollary 42. Let G1 and G2 be two non-singular matrices over Fq of sizes l1 and l2 respectively. Then E(G1 ⊗ G2 ) = E(G1 ) E(G2 ) + . logl1 (l1 l2 ) logl2 (l1 l2 ) Proof: We know that G1 ⊗ G2 has size l1 l2 . For each k ∈ {1, . . . , l1 l2 } we can rewrite k as k = (j − 1)l2 + s January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 23 where 1 ≤ j ≤ l1 y 1 ≤ s ≤ l2 , therefore l l E(G1 ⊗ G2 ) = 1 2 X 1 ln(Dk (G1 ⊗ G2 )) l1 l2 ln(l1 l2 ) k=1 (a) = l l 2 1 X X 1 ln(Dj (G1 )Ds (G2 )) l1 l2 ln(l1 l2 ) j=1 s=1 l = = l 1 2 X X 1 1 ln(Dj (G1 )) + ln(Ds (G2 )) l1 ln(l1 l2 ) j=1 l2 ln(l1 l2 ) s=1 E(G2 ) ln l2 E(G1 ) ln l1 + ln l1 l2 ln l1 l2 where equality (a) follows from the previous proposition. We can extend the analysis done for the kernel defined by one curve to the product of two kernels arising    Pl1 −1 Pl2 −1 ′ ′  from two curves defined over the same field. Let X , (z)0 = i=0 Pi , Q and Y, (z ′ )0 = j=0 Pi , Q be two pointed curves over Fq and S[X] = Fq [x1 , . . . , xs ] and S[Y ] = Fq [y1 , . . . , ys′ ] the polynomial rings where there exist I ⊂ S[X] and I ′ ⊂ S[Y ] such that S[X]/hI, fz i and S[Y ]/hI ′ , fz′ i are isomorphic to the codes associated to the respective curves.We will denote as GX and GY their respective kernels. We denote by S[X, Y ] = Fq [x1 , . . . , xs , y1 , . . . , ys′ ] and by IXY = hI, I ′ , fz , fz′ i. We will endow R[X, Y ] = S[X, Y ]/hI, I ′ i with the weight generated by the inherit vectors w1 = (w(x1 ), . . . , w(xs ), 0, . . . , 0) and w2 = (0, . . . , 0, w(y1 ), . . . , w(ys′ )). Proposition 43. If ∆(I) = {M0 , . . . , Ml1 −1 } and ∆(I ′ ) = {M0′ , . . . , Ml′2 −1 }, then ∆(IXY ) = {M0 M0′ , M0 M1′ , . . . , Ml1 −1 Ml2 −1 } and the rows of the matrix GX ⊗ GY are evaluations of the elements in ∆(IXY ) M · M ′ (Qi ) = M (Pj )M ′ (Pk′ ) i = jl2 + k in decreasing order w.r.t. the induced ordering. Proof: The equality for ∆(IXY ) is clear and the equality on the rows follows from the definition of the Kronecker product since (GX ⊗ GY )i,j = (GX )⌊i/l2 ⌋,⌊j/l2 ⌋ (GY )i mod l2 ,j mod l2 = Mln −⌊i/l2 ⌋ (P⌊j/l2 ⌋ )Ml′n −i ′ Thus we have a set of monomials M̃i = M⌊i/l2 ⌋ Mimod l2 ′ mod l2 (Pj mod l2 ). and Qi as before. Let l = l1 l2 and consider the polar code constructed from the kernel GXY . Now we will work on the polynomial ring R[X1 , Y1 , X2 , Y2 , . . . , Xn , Yn ]. We will define an ordering on Z as follows i ⊳ j if and only if when i = hl2 + k and j = h′ l2 + k ′ we have h < h′ y j < j ′ . With this new ordering our previous definitions are translated easily. January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES • 24 A code CAn with kernel GXY is called weakly decreasing if for M̃kn ∈ An , M̃kn′ |M̃kn it follows that M̃kn′ ∈ An . As a corollary a polar code over a SOF channel is weakly decreasing. • A code is decreasing if it is weakly decreasing and also M̃i1 (Xj1 Yj1 ) · · · M̃ik (Xjk Yjk ) ∈ An implies M̃i1 (Xj1′ Yj1′ ) · · · M̃ik (Xjk′ Yjk′ ) ∈ An for jv′ ≤ jv , v ∈ {1, . . . , k}. As before this last property will be call degrading property. Example 44. From Corollary 42 we can see that a matrix G⊗n has the same exponent as the original matrix G. Let us consider the field F4 and the field of rational functions with variable t and the hermitian curve over F4 [x, y]. We compute the kernel using Q = P∞ (the common pole of x and y) and we construct the matrices GH and GR . The monomial basis we have to consider are LH = {x3 y, x2 y, x3 , xy, x2 , x, y, 1}, LR = {t3 , t2 , t, 1}. Therefore GH ⊗ GR is the evaluation of the monomials LH⊗R ={x3 yt3 , x3 yt2 , x3 yt, x3 y, x2 yt3 , x2 yt2 , x2 yt, x2 y, x3 t3 , x3 t2 , x3 t, x3 , xyt3 , xyt2 , xyt, xy, x2 t3 , x2 t2 , x2 t, x2 , yt3 , yt2 , yt, y, xt3 , xt2 , xt, x, t3 , t2 , t, 1}. The rational curve has exponent ln 4! 4 ln 4 and the hermitian one has exponent ln(8·2·6!) 8 ln 8 thus the kernel GH ⊗ GR has exponent E(GH ⊗ GR ) ≈ 0.5665 . If we construct a polar code from this kernel over a SOF channel and n = 1 we have that if xyt2 , x2 t2 ∈ An then xyt, xy, x2 t, x2 , yt2 , yt, y, xt2 , xt, x, t2 , t, 1 ∈ An . If those are the only elements in the code then we have a decreasing code with minimum distance 6 (since the code associated to xy has minimum distance 3 and the one associated to t2 has minimum distance 2). VII. C ONCLUSION In this paper we have stablished a construction of polar codes from pointed algebraic curves for a discrete memoryless channel which is symmetric w.r.t the field operations. This results extend some results in [3] for a binary symmetric channel. Note that both the families of weak Castle and Castle curves provide good candidates for designing the proposed polar codes since they satisfy the conditions needed in the construction. Even if the nature of the results is mainly theoretical, we believe that it can contribute to a deeper understanding to polar codes over non-binary alphabeths. January 23, 2019 DRAFT CASTLE CURVES AND POLAR CODES 25 R EFERENCES [1] Anderson, S. E., & Matthews, G. L. (2014). Exponents of polar codes using algebraic geometric code kernels. Designs, codes and cryptography, 73(2), 699-717. [2] Arıkan, E. (2009). Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE Transactions on Information Theory, 55(7), 3051-3073. [3] Bardet, M., Dragoi, V., Otmani, A., & Tillich, J. P. (2016). Algebraic properties of polar codes from a new polynomial formalism. In International Symposium on Information Theory ISIT 2016 (pp. 230-234). [4] Cover, T. M., & Thomas, J. A. (2012). Elements of information theory. John Wiley & Sons. [5] Bleichenbacher, D., Kiayias, A., & Yung, M. (2003, June). Decoding of interleaved Reed Solomon codes over noisy data. In International Colloquium on Automata, Languages, and Programming (pp. 97-108). Springer, Berlin, Heidelberg. [6] Dodunekova, R., Dodunekov, S. M., & Nikolova, E. (2008). A survey on proper codes. Discrete Applied Mathematics, 156(9), 1499-1509. [7] Geil, O., Munuera, C., Ruano, D., & Torres, F. (2010). On the order bounds for one-point AG codes. arXiv preprint arXiv:1002.4759. [8] Geil, O., & Pellikaan, R. (2002). On the structure of order domains. Finite Fields and Their Applications, 8(3), 369-396. [9] Hernando, F., Lally, K., & Ruano, D. (2009). Construction and decoding of matrix-product codes from nested codes. Applicable Algebra in Engineering, Communication and Computing, 20(5-6), 497. [10] Korada, S. B., Şaşoğlu, E., & Urbanke, R. (2010). Polar codes: Characterization of exponent, bounds, and constructions. IEEE Transactions on Information Theory, 56(12), 6253-6264. [11] Lechner, G., & Weidmann, C. (2008, September). Optimization of binary LDPC codes for the q-ary symmetric channel with moderate q. In Turbo Codes and Related Topics, 2008 5th International Symposium on (pp. 221-224). IEEE. [12] Lee, M. K., & Yang, K. (2014). The exponent of a polarizing matrix constructed from the Kronecker product. Designs, codes and cryptography, 70(3), 313-322. [13] Martı́nez-Moro, Edgar and Munuera, Carlos and Ruano, Diego (Eds.) (2008). Advances in algebraic geometry codes. Series on Coding Theory and Cryptology Vol. 5. World Scientific Publishing Co. Pte. Ltd. [14] Mori, R., & Tanaka, T. (2010). Channel polarization on q-ary discrete memoryless channels by arbitrary kernels. In Information Theory Proceedings (ISIT), 2010 IEEE International Symposium on (pp. 894-898). IEEE. [15] Mori, R., & Tanaka, T. (2010). Non-binary polar codes using Reed-Solomon codes and algebraic geometry codes. In Information Theory Workshop (ITW), 2010 IEEE (pp. 1-5). IEEE. [16] Mori, R., & Tanaka, T. (2014). Source and channel polarization over finite fields and ReedSolomon matrices. IEEE Transactions on Information Theory, 60(5), 2720-2736. [17] Munuera, C., Sepúlveda, A., & Torres, F. (2008). Algebraic Geometry codes from Castle curves. In Coding Theory and Applications (pp. 117-127). Springer, Berlin, Heidelberg. [18] Şaşoğlu, E., Telatar, E., & Arıkan, E. (2009, October). Polarization for arbitrary discrete memoryless channels. In Information Theory Workshop, 2009. ITW 2009. IEEE (pp. 144-148) [19] Shokrollahi, A. (2004, October). Capacity-approaching codes on the q-ary symmetric channel for large q. In Information Theory Workshop, 2004. IEEE (pp. 204-208). IEEE. [20] Yang, K., & Kumar, P. V. (1992). On the true minimum distance of Hermitian codes. In Coding theory and algebraic geometry (pp. 99-107). Springer, Berlin, Heidelberg. January 23, 2019 DRAFT