Noncatastrophic convolutional codes over a finite ring

Noncatastrophic convolutional codes over a finite ring arXiv:2104.06754v1 [cs.IT] 14 Apr 2021 D. Nappa , R. Pintob , C. Rocha∗,c a Departament de Matemàtiques, Universitat d’Alacant, Spain. of Mathematics, University of Aveiro, Portugal. c Instituto Superior de Contabilidade e Administração de Coimbra, Instituto Politécnico de Coimbra, Portugal. b Department Abstract Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized are being polynomial left prime matrices. In this paper we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in a finite ring. In particular, we need to introduce two different notion of primeness in order to fully characterize noncatastrophic encoders over the finite ring Zpr . The second part of the paper is devoted to investigate the notion of free and column distance in this context when the convolutional code is a free finitely generated Zpr -module. We introduce the notion of b-degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on Zp is. 1. Introduction The notion of primeness plays a central role in the polynomial matrix approach to several areas of pure and applied mathematics, such as systems and control theory or coding theory. In this paper we consider polynomial matrices over the finite ring Zpr , where p is a prime and r an integer greater than 1. Our motivation for considering such a finite ring Zpr stems from applications in the area of error-correcting codes and in particular of convolutional codes over Zpr . Such a ring is useful for the so-called coded modulation scheme where ∗ Corresponding author work is supported by The Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020. Diego Napp is partially supported by Ministerio de Ciencia e Innovación via the grant with ref. PID2019-108668GB-I00. 1 This Preprint submitted to Elsevier 15th April 2021 the codewords in Zpr are mapped onto phase-shift-keying (PSK) modulation signals sets. The mapping is such that distances between modulation points are preserved under additive operations in Zpr , see [26] for more details. In [17] Massey and Mittelholzer observed for the first time that convolutional codes over ZM , are the most appropriate class of codes for phase modulation. Note that even though we will focus on the ring Zpr , by the Chinese Remainder Theorem, results on codes over Zpr can be extended to codes over ZM . An important concept in the theory of convolutional codes is the noncatastrophicity. When a catastrophic convolutional generator matrix is used for encoding, finitely many errors in the estimate of the transmitted codeword can lead to infinitely many errors in the estimate of the information sequence. This is of course a catastrophic situation that has to be avoided when designing the generator matrix. Noncatastrophic generator matrices have been characterized as left prime polynomial matrices and have been studied in several contexts depending on the definition considered in each case, see [1, 11, 21, 22]. In this work we define convolutional codes as finitely generated free Zpr [d]-modules of Zpr [d]n , where Zpr [d] is the polynomial ring with coefficients in Zpr and study noncatastrophicity in this setting [13, 18, 19, 25]. In the case of matrices with entries in Zpr [d] we need to distinguish two types of left primeness, namely, zero left prime and factor left prime, as happens in the case of polynomial matrices in several variables over a field [15, 20, 24, 28]. We provide a characterization of zero left prime polynomial matrices from which it follows that when a convolutional code admit a left zero prime generator matrix, i.e., a noncatastrophic encoder, then the code can be described by means of a parity-check polynomial matrix. The second part of the paper is devoted to investigating the Hamming distances of these codes as these will determine their error-correcting capabilities. In the context of convolutional codes the column distance is arguably the most important notion of distance [9] and therefore we shall focus on the study of this particular distance. To this end we introduce a novel notion, called the b-degree, and derive bounds on the column distance in terms of the dimension, length and b-degree. In [21] it was proven that the free distance of convolutional codes over Zpr it is determined by its projection over Zp . The authors used this fact and the Hensel lift of a cyclic code in [21] to build optimal convolutional codes with respect to the free distance. Here we show that a convolutional code over Zp is optimal with respect to the column distance if and only if its projection is. This will allow the construction of optimal convolutional codes over Zpr from well-known classes of convolutional codes over Zp . The results of the paper are twofold: we first analyse the primeness of polynomial matrices with entries in Zpr [d] in Section 2 and second we investigate the column distances of free convolutional codes over Zpr in Section 3. In each section we briefly provide some preliminaries: in Section 2 we recall known results of primeness of polynomial matrices over finite fields and in Section 3 the definitions of convolutional codes, free distance and column distances are 2 presented. 2. Primeness of polynomial matrices over Zpr We denote by F[d] the ring of polynomials in the indeterminate d and coefficients in a finite field F and by F(d) the field of rational functions defined in F. Next we will present results that are well-known in the literature, see [5, 6] for more details. 2.1. Primeness of polynomial matrices over a finite field F Definition 2.1. A polynomial matrix U (d) ∈ F[d]k×k is unimodular if it is invertible and U (d)−1 ∈ F[d]k×k . Lemma 2.1. Let U (d) ∈ F[d]k×k . Then U (d) is unimodular if and only if det U (d) ∈ F\{0}. Definition 2.2. A polynomial matrix A(d) ∈ F[d]k×n is left prime if in all factorizations A(d) = ∆(d)Ā(d), with ∆(d) ∈ F[d]k×k , and Ā(d) ∈ F[d]k×n , the left factor ∆(d) is unimodular. Left prime matrices admit several characterizations as stated in the next theorem. Theorem 2.1. Let A(d) ∈ F[d]k×n . The following are equivalent: 1. A(d) is left prime; 2. there exist unimodular matrices U (d) ∈ F[d]k×k and V (d) ∈ F[d]n×n such that U (d)A(d)V (d) = [Ik 0]; 3. there exists a unimodular matrix V (d) ∈ F[d]n×n such that A(d)V (d) = [Ik 0];   A(d) 4. there exists B(d) ∈ F[d](n−k)×n such that is unimodular; B(d) 5. A(d) admits a polynomial right inverse; 6. for all u(d) ∈ F(d)k , u(d)A(d) ∈ F[d]n implies that u(d) ∈ F[d]k ; 7. A(α) has rank k for all α ∈ F̄, where F̄ denotes the algebraic closure of F; 8. the ideal generated by all the k-th order minors of A(d) is F[d]. 3 2.2. Primeness of polynomial matrices over Zpr In this section we study the notion of left prime for polynomial matrices over Zpr . We denote by Zpr [d] the ring of polynomials in the indeterminate d, with coefficients in Zpr and by Zpr (d) the ring of rational functions defined, see [8], as the set   p(d) : p(d), q(d) ∈ Zpr [d] and the coefficient of the smallest power of d in q(d) is a unit . q(d) This condition allows us to treat a rational function as an equivalence class in the relation p1 (D) p(D) ∼ if and only if p(D)q1 (D) = p1 (D)q(D). q(D) q1 (D) Any element a ∈ Zpr has a p-adic expansion [3], i.e., it can be written uniquely as a linear combination of 1, p, p2 , . . . . . . , pr−1 , with coefficients in Ap = {0, 1, . . . , p − 1} ⊂ Zpr , a = α0 + α1 p + · · · + αr−1 pr−1 , αi ∈ Ap , i = 0, 1, . . . , r − 1. Note that all elements in Ap \{0} are units. Given a matrix A(d) ∈ Zpr [d]k×n , denote by [A(d)]p its (componentwise) projection into Zp . Definition 2.3. A polynomial matrix U (d) ∈ Zpr [d]k×k is unimodular if it is invertible and U (d)−1 ∈ Zpr [d]k×k . Next lemma characterizes the polynomial matrices over Zpr [d] which admit a right polynomial inverse. Lemma 2.2. A polynomial matrix A(d) ∈ Zpr [d]k×n , with n ≥ k, admits a polynomial right inverse if and only if [A(d)]p ∈ Zp [d]k×n also admits a polynomial right inverse over Zp [d]. Proof 1. If [A(d)]p admits a polynomial right inverse over Zp [d] then there exists B(d) ∈ Zp [d]n×k such that [A(d)]p B(d) = Ik mod p. Considering B(d) as a matrix over Zpr [d], we have that A(d)B(d) = Ik − pX(d), for some X(d) ∈ Zpr [d]k×k and therefore B(d)(Ik + pX(d) + p2 X 2 (d) + · · · + pr−1 X r−1 (d)) is a right inverse of A(d). The converse is obvious. The following theorem is immediate. 4 Theorem 2.2. Let U (d) ∈ Zpr [d]k×k . The following are equivalent: 1. U (d) is unimodular; 2. det U (d) is a unit; 3. [U (d)]p is unimodular in Zp [d]k×k . Left primeness is a property of polynomial matrices which plays a fundamental role when we consider convolutional codes over a finite field F. As mentioned before, a polynomial matrix A(d) ∈ F[d]k×n is left prime if in all factorizations A(d) = ∆(d)Ã(d), with ∆(d) ∈ F[d]k×k , and Ã(d) ∈ F[d]k×n , the left factor ∆(d) is unimodular, or equivalently if the ideal generated by all the k-th order minors of A(d) is F[d] (see Theorem 2.1). However, this equivalence does not hold over Zpr . There are polynomial matrices over Zpr [d] that satisfy the former condition but do not satisfy the later, as it is illustrated in the following example. Example 2.1. The matrix   A(d) = 1 + 3d 1 + d ∈ Z4 [d]2 does not have a nonunimodular left factor, but the ideal generated by its full size minors is {(1 + d)p(d) : p(d) ∈ Zpr [d]}. Therefore, we need to introduce two different notions of primeness when dealing with polynomial matrices over Zpr . Definition 2.4. A polynomial matrix A(d) ∈ Zpr [d]k×n is left factor-prime (ℓF P ) if in all factorizations A(d) = ∆(d)Ā(d) with ∆(d) ∈ Zpr [d]k×k and Ā(d) ∈ Zpr [d]k×n , the left factor ∆(d) is unimodular. Definition 2.5. A polynomial matrix A(d) ∈ Zpr [d]k×n is left zero-prime (ℓZP ) if the ideal generated by all the k-th order minors of A(d) is Zpr [d]. Right factor-prime (rFP) and right zero-prime (rZP) matrices are defined in the same way, upon taking transposes. Remark 2.1. Note that the fact that the conditions of Theorem 2.1 are not anymore equivalent when considering rings instead of fields also occurs when considering the polynomial ring F[d1 , . . . , dn ] in several variables instead of F[d], see [15, 16, 20, 24, 28] for more details. As shown in Example 2.1 factor-primeness does not imply zero-primeness, however the converse is true as stated in the following lemma. 5 Lemma 2.3. Let A(d) ∈ Zpr [d]k×n . If A(d) is left zero-prime then it is also left factor-prime. Proof 2. Let us assume that A(d) is not left factor prime. Then A(d) = X(d)Ã(d) for some Ã(d) ∈ Zpr [d]k×n and X(d) ∈ Zpr [d]k×k nonunimodular. Then by Theorem 2.2 x(d) = det X(d) is not a unit and the ideal generated by all the k-th order minors of A(d) is contained in {x(d)p(d) : p(d) ∈ Zpr [d]}. Consequently, A(d) is not left zero-prime. It is easy to see that an ideal I of Zpr [d] is equal to Zpr [d] if and only if [I]p = {[u]p : u ∈ I} is equal to Zp [d] and, therefore, the next lemma follows immediately. Lemma 2.4. A(d) ∈ Zpr [d]k×n is left zero-prime over Zpr [d] if and only if [A(d)]p ∈ Zp [d]k×n is left prime over Zp [d]. Now, we are in position to prove the following characterizations of left zeroprime matrices with entries in Zpr which can be considered as an extension of Theorem 2.1 to the finite ring case. Theorem 2.3. Let A(d) ∈ Zpr [d]k×n . The following are equivalent: 1. A(d) is left zero-prime; 2. there exist unimodular matrices U (d) ∈ Zpr [d]k×k and V (d) ∈ Zpr [d]n×n such that U (d)A(d)V (d) = [Ik 0]; 3. there exists a unimodular matrix V (d) ∈ Zpr [d]n×n such that A(d)V (d) = [Ik 0];   A(d) 4. there exists B(d) ∈ Zpr [d](n−k)×n such that is unimodular; B(d) 5. A(d) admits a polynomial right inverse; 6. for all u(d) ∈ Zpr (d)k , u(d)A(d) ∈ Zpr [d]n implies that u(d) ∈ Zpr [d]k ; 7. Ā(α) has rank k, mod p, for all α ∈ Z̄p , where Z̄p denotes the algebraic closure of Zp and Ā(d) = [A(d)]p . Proof 3. From Theorems 2.1 and 2.2 and Lemma 2.4 we immediately conclude that 2) ⇒ 1), 3) ⇒ 1), 4) ⇒ 1), 5) ⇒ 1), 7) ⇒ 1) and 1) ⇒ 7). Next we prove the implications 1) ⇒ 2), 2) ⇒ 3), 3) ⇒ 4), 4) ⇒ 5), 5) ⇒ 6) and 6) ⇒ 1). 1) ⇒ 2): Since A(d) is ℓZP , [A(d)]p is left prime over Zp [d] and therefore there exist unimodular matrices U (d) ∈ Zp [d]k×k and V (d) ∈ Zp [d]n×n such that U (d)[A(d)]p V (d) = [Ik 0] mod p. Considering U (d) and V (d) as matrices over Zpr [d] we have that U (d)A(d)V (d) = [X1 (d) X2 (d)], 6 with X1 (d) ∈ Zpr [d]k×k and X2 (d) ∈ Zpr [d]k×(n−k) . Note that, X1 (d) is unimodular because [X1 (d)]p = Ik and that U1 (d) = X1 (d)−1 U (d) and   Ik −X1 (d)−1 X2 (d) V1 (d) = V (d) 0 In−k are polynomial matrices. It is easy too see that U1 (d) and V (d) are unimodular matrices, and that U1 (d)A(d)V1 (d) = [Ik 0]. 2) ⇒ 3): Let U (d) ∈ Zpr [d]k×k and V (d) ∈ Zpr [d]n×n be unimodular matrices such that U (d)A(d)V (d) = [Ik 0]. Then A(d)V (d) = [U (d)−1 0] and, therefore   U (d) 0 V1 (d) = V (d) 0 In−k is a unimodular matrix such that A(d)V1 (d) = [Ik 0]. 3) ⇒ 4): From the assumption A(d) = [Ik 0]Ṽ (d) for some unimodular matrix Ṽ (d) ∈ Zpr [d]n×n , i.e., A(d) is the submatrix of Ṽ (d) constituted by its first k rows. 4) ⇒ 5): Let [X(d) Y (d)] with X(d) ∈ Zpr [d]n×k and Y (d) ∈ Zpr [d]n×(n−k) be such that    A(d)  X(d) Y (d) = In . B(d) Then A(d)X(d) = Ik . 5) ⇒ 6): Let u(d) ∈ Zpr (d)k be such that u(d)A(d) = w(d) ∈ Zpr [d]n and let X(d) ∈ Zpr [d]n×k be a right inverse of A(d). Then u(d) = w(d)X(d), which is a polynomial vector. 6) ⇒ 1): Let us assume that A(d) is not ℓZP . Then [A(d)]p is not left prime over Zp [d], and therefore there exists a nonpolynomial vector u(d) ∈ Zp (d)k such that u(d)[A(d)]p ∈ Zp [d]n mod p. Considering u(d) as a vector over Zpr [d], it follows that pr−1 u(d) ∈ Zpr [d]k is also nonpolynomial and pr−1 u(d)A(d) ∈ Zpr [d]n . 3. Distance properties of free convolutional codes over Zpr In this section we first recall the basic definitions of convolutional codes over Zpr . We consider convolutional codes as free Zpr [d]-submodules of Zpr [d]n , for some n ∈ N, see [13, 18, 19, 25]. We require the encoding map to be injective and therefore focus on free submodules of Zpr [d]n . We note that different definitions have been considered in the literature, see for instance [4, 10, 11, 21]. The nonfree case lies beyond the scope of this work but it can also be treated using the theory of p-basis and p-generating sequences, see for instance [11, 12, 18, 21]. 7 3.1. Convolutional codes Definition 3.1. A convolutional code C of rate k/n is a free submodule of Zpr [d]n of rank k. A matrix G(d) ∈ Zpr [d]k×n whose rows form a basis of C is called an encoder of C. Thus, an encoder of C is a full row rank matrix G(d) ∈ Zpr [d]k×n such that C = Im Zpr [d] G(d) = {u(d)G(d) : u(d) ∈ Zpr [d]k }. Equivalent encoders are full row rank matrices that are encoders of the same code. Then two equivalent encoders G1 (d), G2 (d) ∈ Zpr [d]k×n are such that G2 (d) = U (d)G1 (d) for some unimodular matrix U (d) ∈ Zpr [d]k×k . Thus, it follows that if a convolutional code admits a left zero-prime encoder then all its encoders are also left zero-prime. We call such codes noncatastrophic codes and they are the ones that admit a kernel representation as stated in the following theorem. Theorem 3.1. Let C be a convolutional code of rate k/n. Then, there exists a full column rank polynomial matrix H(d) ∈ Zpr [d]n×(n−k) such that C = kerZpr [d] H(d) = {w(d) ∈ Zpr [d]n : w(d)H(d) = 0} if and only if C is noncatastrophic. Proof 4. Let us assume first that C is noncatastrophic and let G(d) ∈ Zpr [d]k×n be an encoder of C. Then, G(d) is ℓZP and therefore, by Theorem 2.3, there exist polynomial matrices B(d) ∈ F[d](n−k)×n , X(d) ∈ F[d]n×k and H(d) ∈ F[d]n×(n−k) such that        G(d) G(d)  X(d) H(d) = X(d) H(d) = In B(d) B(d) This means that H(d) is a full column rank matrix such that G(d)H(d) = 0, and therefore C ⊂ kerZpr [d] H(d). On the other hand, if w(d) ∈ kerZpr [d] H(d) we have that     G(d) w(d) = w(d) X(d) H(d) B(d)     G(d) = w(d)X(d) 0 B(d) = u(d)G(d), where u(d) = w(d)X(d) ∈ Zpr [d]k , i.e. w(d) ∈ C. For the converse let us assume that C is not a noncatastrophic convolutional code and that C = kerZpr [d] H(d) for some full column rank matrix H(d) ∈ Zpr [d]n×(n−k) . Let G(d) be an encoder of C. Then, since G(d) is not left zeroprime, [G(d)]p = X(d)G̃(d) mod p 8 for some invertible but nonunimodular matrix X(d) ∈ Zp [d]k×k and G̃(d) ∈ Zp [d]k×n . Considering X(d) and G̃(d) as matrices over Zpr [d], we have that pr−1 G(d) = pr−1 X(d)G̃(d) and therefore pr−1 X(d)−1 G(d) = pr−1 G̃(d). Since X(d) is not unimodular, there exists an i ∈ {1, . . . , k} such that the ith row of pr−1 X(d)−1 is not polynomial. Let us represent such row by ℓi (d), i.e., ℓi (d) = ei pr−1 X(d), where ei is the i-th vector of the canonical basis. Then ℓi (d)G(d) does not belong to C because ℓi (d) is not polynomial, but since ℓi (d)G(d) = ei pr−1 G̃(d) is a polynomial vector, it follows that ℓi (d)G(d) belongs to kerZpr [d] H(d), which is a contradiction. If C is a noncatastrophic convolutional code, then a full column rank polynomial matrix H(d) such that C = kerZpr [d] H(d) is called a parity-check matrix of C. We conclude this section by giving a result on the relation between the order of an information sequence u(d) and the corresponding codeword w(d) = u(d)G(d) where G(d) is an encoder. This relation will be useful later on the paper. Let a ∈ Zpr . We define the order of a to be ℓ, and we write ord(a) = ℓ, if the set aZpr has pℓ elements. Then, ord(a) = ℓ if and only if pℓ−1 a is a nonzero element of pr−1 Zpr and pℓ a = 0. In the same way we define the order of a polynomial vector w(d) ∈ Zpr [d]m to be ℓ, and we write ord(w) = ℓ, if pℓ−1 w(d) 6= 0 and pℓ w(d) = 0. This means that pℓ−1 w(d) is a nonzero element of pr−1 Zpr [d]m . The following lemma relates the orders of an information sequence and the corresponding codeword. We omit the simple proof. Lemma 3.1. Let C be a convolutional code of rate k/n, G(d) an encoder of C and w(d) = u(d)G(d), with u(d) ∈ Zpr [d]k , a codeword of C. Then ord(w) = ord(u). 3.2. Distance properties Next we study the free distance and column distances of a convolutional code over Zpr . Such distances were also investigated in [18, 19, 21] for not necessarily free convolutional codes using the notion of p-basis, see [19, 21]. For the free case addressed in this work we introduce the notion of b-degree of a code and derive new bounds on the free and column distance in terms of the length, dimension and b-degree of the code. First, we formally present the definitions of free distance and column distance. The free distance of a convolutional code is defined as dfree (C) = min{wt(w(d)) : w(d) ∈ C, w(d) 6= 0} 9 P P where for w(d) = i∈N0 wi di , wt(w(d)) = i∈N0 wt(wi ), with wt(wi ) to be the number of nonzero entries of wi . Let [C]p = {[w(d)]p : w(d) ∈ C} and define dfree ([C]p ) = min{wt(v(d)) : v(d) ∈ [C]p , v(d) 6= 0}. In [21, Theorem 5.3] it was shown that dfree (C) ≥ dfree ([C]p ). (1) This can be alternatively shown as follows. Note that [C]p ≃ pr−1 C and that dfree ([C]p ) = dfree (pr−1 C) with dfree (pr−1 C) = min{wt(pr−1 w(d)) : w(d) ∈ C, [w(d)]p 6= 0}. Let w(d) be a nonzero codeword of C of order ℓ. Lemma 3.1 implies that pℓ−1 w(d) is a nonzero vector of pr−1 C, and since wt(w(d)) ≥ wt(pℓ−1 w(d)) the inequality (1) follows. Next theorem shows that inequality (1) is in fact an equality. Theorem 3.2. Let C be a convolutional code. Then dfree (C) = dfree ([C]p ). Proof 5. We only need to prove that dfree (C) ≤ dfree ([C]p ). For that let w(d) be a nonzero codeword of [C]p . Then there exists w̃(d) ∈ C such that [w̃(d)]p = w(d). Thus, pr−1 w̃(d) ∈ C with wt(pr−1 w̃(d)) = wt(w(d)), which implies that dfree (C) ≤ dfree ([C]p ). The maximum value that the free distance of a convolutional code over a finite field of rate k/n can attain depends also of the degree of the code which is defined as the maximum of the degrees of the determinants of the submatrices of one and hence any generator matrix of C. If C is a convolutional code over a finite field of rate k/n and degree δ, then   δ dfree (C) ≤ (n − k) ⌊ ⌋ + 1 + δ + 1. (2) k This upper bound is called the Generalized Singleton bound and was found first in [23] in the field case and then extended in [19, 21] for the ring case using the notion of p-degree that is not used in this work. For the case of free modules of Znpr that is considered here we will obtain a new expression for the bound on the free distance of a free convolutional code C. For that we need to introduce the novel concept of b-degree of C. Definition 3.2. Let C be a convolutional code over Zpr [d]. The b-degree of C is equal to the degree of [C]p . 10 The b-degree of a convolutional code C can be easily obtained by calculating the maximum degree of the full size minors of [G(d)]p mod p, for any encoder G(d) of C. The next result follows immediately from Theorem 3.2, Definition 3.2 and from the expression (2). Theorem 3.3. Let C be a convolutional code of rate k/n and b-degree δ. Then   δ dfree (C) ≤ (n − k) ⌊ ⌋ + 1 + δ + 1. k A convolutional code of rate k/n and b-degree δ with free distance (n −
k) ⌊ kδ ⌋ + 1 + δ + 1 is said to be a Maximum Distance Separable (MDS) code. It follows immediately from Theorem 3.2 that a convolutional code C is MDS if and only if [C]p is also MDS over Zp [d]. Another type of distances of a convolutional code which can be very useful in sequential decoding and have showed to have a potential use in streaming applications are the column distances [14]. Pν Let G(d) be an encoder of C and let us write G(d) = i=0 Gi di , Gi ∈ P Zk×n . The codeword w(d) = i∈N0 wi di , wi ∈ Znpr , corresponding to u(d) = r Pp i k i∈N0 ui d , ui ∈ Zpr , is such that [w0 w1 · · · wj ] = [u0 u1 · · · uj ]Gcj where    Gcj =   G0 G1 G0 ··· ... .. .  Gj Gj−1     G0 is called the truncated sliding matrix corresponding to G(d) (we consider Gj = 0, if j > ν), [7, 18]. Definition 3.3. [18, 27] Given an encoder G(d) of a convolutional code C, we define the j-th column distance of G(d) as dcj (G) = min{wt([u0 u1 · · · uj ]Gcj ) : ui ∈ Zkpr , u0 6= 0}. Parity-check matrices are very useful in the analysis of such distances. For this reason we restrict the study of such distances to noncatastrophic convolutional codes. Note that if G(d) is an encoder of a noncatastrophic convolutional code C, then G(d) has a right polynomial inverse and therefore G(0) is full row rank and this means that the j-th column distance of C is an invariant of the code and can be obtained as dcj (C) = min{wt([w0 w1 · · · wj ]) : [w0 w1 · · · wj ] ∈ Im Gcj , w0 6= 0}. 11 If H(d) = ℓ ∈ N, then Pℓ i=0 (n−k)×n Hi di , Hi ∈ Zpr is a parity-check of C and H(ℓ) 6= 0, dcj (C) = min{wt([w0 w1 · · · wj ]) : [w0 w1 · · · wj ](Hjc ) = 0, w0 6= 0}, where    Hjc =   H0 H1 H0 ··· ··· .. . Hj Hj−1 .. . H0    ,  with Hj = 0 for j > ℓ. The following theorem is immediate and its proof is analogous to the field case. Theorem 3.4. Let C be a noncatastrophic convolutional code of rate k/n. Then, for any j, d ∈ N, dcj (C) = d if and only if the following conditions are satisfied: 1. there exist d rows of Hjc linearly dependent over Zpr [d] such that one of these rows belongs to the first n rows of Hjc ; 2. all d − 1 rows, in which one of the rows belongs to the first n rows of Hjc , are linearly independent over Zpr [d]. Remark 3.1. Since the lines of a matrix A are linearly independent if and only if [A]p is a full row rank, the conditions 1 and 2 of the Theorem 3.4 can be expressed, respectively, and in an equivalent way, as follows: 1. there exist d rows of [Hjc ]p linearly dependent over Zp [d] such that one of these rows belongs to the first n rows of [Hjc ]p ; 2. all d− 1 rows, in which one of the rows belongs to the first n rows of [Hjc ]p , are linearly independent over Zp [d]. Next theorem provides upper bounds on the column distances of a noncatastrophic convolutional code. These upper bounds were found in [18] for the more general case in which the convolutional codes are not necessarily noncatastrophic free convolutional codes. Although the result is not new, we opted to present the proof, because it is much simpler than the one in [18]. Theorem 3.5. Let C be a noncatastrophic convolutional code of rate k/n. Then dcj (C) ≤ (n − k)(j + 1) + 1, for all j ∈ N0 . Proof 6. Since C is a noncatastrophic convolutional code over Zpr , then[C]p is a noncatastrophic convolutional code code over Zp of rate k/n, and therefore dcj ([C]p ) ≤ (n − k)(j + 1) + 1, P for all j ∈ N0 (see [7]). Let w(d) = i∈N0 wi di ∈ [C]p with w0 6= 0, then wt([w0 w1 · · · wj ]) ≤ (n − k)(j + 1) + 1. 12 n(j+1) . Then, w̃(d) = pr−1 w(d) ∈ Let us consider [w0 w1 · · · wj ] as a vector of Zpr r−1 p C is such that w̃(0) 6= 0 and wt([w̃0 w̃1 · · · w̃j ]) = wt([w0 w1 · · · wj ]). Then dcj (C) ≤ (n − k)(j + 1) + 1. The next result readily follows from [7, Theorem 2.4] and Remark 3.1. Theorem 3.6. Let G(d) be an encoder of a noncatastrophic convolutional code over Zpr [d], C, of rate k/n, k ≤ n, and H(d) be a parity-check matrix of C. The following are equivalent: 1. dcj (C) = (n − k)(j + 1) + 1. 2. every (j + 1)k × (j + 1)k full-size minor of [Gcj ]p formed from the columns with indices 1 ≤ t1 < . . . < t(j+1)k , where tsk+1 > sn, s = 1, . . . , j,is nonzero. 3. every (j + 1)(n − k) × (j + 1)(n − k) full-size minor of [Hjc ]p formed from the columns with indices 1 ≤ r1 < . . . < r(j+1)(n−k) , where rs(n−k) ≤ sn, s = 1, . . . , j, is nonzero. The column distances of a noncatastrophic C do not grow indefinitely, since they are naturally upper bounded by the free distance of C. If C is a noncatastrophic convolutional code over a finite field of rate k/n and degree δ, then C can have   column distances up to the L-th column distance, where L =   maximum δ δ + . L is the largest integer for which k n−k    δ (n − k)(L + 1) + 1 ≤ (n − k) + 1 + δ + 1. k Definition 3.4. [7] Let C be a noncatastrophic code over a finite field F, of rate   δ δ + . C is a Maximum Distance Profile k/n and degree δ. Let L = k n−k (MDP) convolutional code if dcj (C) = (n − k)(j + 1) + 1, for all j ≤ L. Since the upper bounds of the column distances and the generalized Singleton bound coincide with the counterpart notions of [C]p , if C is a noncatastrophic convolutional code over Zpr [d] of rate k/n and b-degree δ then C also can achieve the upper bound for column distance only up to the instant L-th. This leads to the following definition. Definition 3.5. Let C be a noncatastrophic code over Zpr [d], of rate k/n and  δ δ b-degree δ. Let L = + . C is a Maximum Distance Profile (MDP) k n−k convolutional code if dcj (C) = (n − k)(j + 1) + 1, for all j ≤ L. 13 Theorem 3.7. Let C be a noncatastrophic code over Zpr [d], of rate k/n and b-degree δ. C is an MDP code if and only if [C]p is an MDP code over Zp [d].     δ δ + and Proof 7. Let C be an MDP convolutional code, j ≤ L = k n−k X w̃i di ∈ [C]p , w̃0 6= 0 w̃(d) = i∈N0 Let us consider w(d) as a vector of Zpr [n]n , and let w(d) = X wi di ; pr−1 w̃(d) ∈ i∈N0 C, w0 = pr−1 w̃0 6= 0. Since wt([ w̃0 w̃1 . . . w̃j ]) = wt([ w0 wi . . . wj ]) and C is an MDP convolutional code and dcj (C) ≤ wt([ w0 wi . . . wj ]), it follows that (n − k)(j + 1) + 1 ≤ wt([ w̃0 w̃1 . . . w̃j ]), j ≤ L and therefore [C]p is a MDP code. X wi di ∈ C, w0 6= Let us now consider [C]p a MDP code and w(d) = i∈N0     δ δ 0, j ≤ L = + , ℓ the order of [ w0 w1 . . . wj ] and s1 the k n−k smallest integer less than or equal to j, such that ws1 has order ℓ1 . It follows that pr−l [ w0 w1 . . . wj ] = pr−1X [ 0 0 w̃s1 . . . w̃j ], with [w̃s1 ]p 6= 0. Let pr−l w(d) = ds (pr−1 w̃(d)), w̃(d) = w̃s1 +i di , be a codeword. Since G(d) is a i∈N0 left-zero prime matrix, G(0) is full row rank. So ds (pr−1 w̃(d)) = (ds u(d))G(d) and, therefore, pr−1 w̃(d) = u(d)G(d), which implies that pr−1 w̃(d) ∈ C. That way, [w̃(d)]p ∈ [C]p and
wt [ pr−1 w̃s1 . . . pr−1 w̃j−s1 ] = wt ([ w̃s1 . . . w̃j−s1 ]p ) , [w̃s1 ]p 6= 0. Furthermore, [C]p is MDP, so
wt [ pr−1 w̃s1 . . . pr−1 w̃j−s1 + 1 ] ≥ (n − k)(j − s1 + 1) + 1. Let us now consider [ w0 w1 . . . ws1 −1 ] with order ℓ2 ≤ ℓ, such that ˜s1 −1 ], ˜s2 . . . w̃ pr−l2 [ w0 w1 . . . ws1 −1 ] = pr−1 [ 0 0 w̃ ˜s2 (d)]p 6= 0. Repeating the previous reasoning, we have with s2 ≤ s1 − 1 and [w̃ to
wt [ pr−1 w̃s2 . . . pr−1 w̃s1 −1 ] ≥ (n−k)(s1 −1+s2 +1)+1 = (n−k)(s1 +s2 )+1 Successively applying the previous process, we obtain wt ([ w0 w1 . . . wj ]) ≥ (n − k)(j + 1) + 1. According to this theorem, we can easily get an MDP code over Zpr [d] of rate k/n and b-degree δ, from an MDP code over Zp [d], of rate k/n and degree δ. 14 4. Conclusions and future work In this paper we have investigated the central notion of primeness of polynomial matrices over Zpr . We showed that zero left prime encoders define noncatastrophic convolutional codes over Zpr and allow a representation of the convolutional code by means of a polynomial parity-check matrix. We have studied free and columns distances of these codes and show that these are determined by the projection of the code over Zp . A natural and interesting avenue for future investigation is to generalized these results to wider classes of rings such as finite chain rings [2]. 5. 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