Codes and colorings
Through this page, codes and colorings from the
research reported in "T. Etzion and P. R. J. Östergård,
Greedy and heruristic algorithms
for codes and colorings, IEEE Transactions on Information
Theory 44 (1998), 382-388" can be obtained.
Back
to the home page of the article.
Colorings for constant weight codes
We here list the new colorings for constant weight codes
using the same notation as in
[Brower et al., A new table of constant weight
codes, IEEE Trans. Inform. Theory 36 (1990),
1334-1380.]
That is, we first order all the words of length n and
weight w in lexicographic order, and number the
disjoint codes 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, and so on. The
word in position i in this lexicographic order now
belongs to the code whose digit appears in position i of
this notation.
PI(13,5)=(123,123,123,122,113,109,109,106,99,88,72,55,34,11)
PI(13,5)=(123,123,123,121,113,109,109,106,98,89,76,57,34,6)
PI(13,5)=(123,123,123,121,113,108,108,106,96,93,77,56,34,6)
PI(13,6)=(166,166,166,157,150,144,141,134,128,118,98,78,50,19,1)
PI(13,6)=(166,166,165,156,150,143,140,134,130,119,104,79,47,16,1)
PI(14,5)=(169,169,169,169,155,153,151,148,146,139,131,113,88,66,29,7)
PI(14,5)=(169,169,169,169,155,153,151,148,146,138,132,113,92,59,29,10)
PI(14,6)=(278,275,265,257,248,231,229,220,211,201,182,158,127,82,31,7,1)
PI(14,6)=(278,273,265,257,250,231,229,219,211,203,184,156,127,81,35,4)
PI(14,7)=(325,322,307,298,281,265,253,250,240,228,198,174,135,85,50,16,5)
The last of these partitions was earlier (and is still in
the report) in error.
Colorings for asymmetric codes
We here list the new partitions for asymmetric codes.
The notation is analogous to that for constant weight
codes; now, of course,
we order all binary words of length n.
PI(9)= (62,62,62,62,58,54,51,44,34,19,4)
PI(9)= (62,62,62,62,56,56,52,44,30,20,6)
PI(10)=(112,110,110,109,104,98,95,87,75,61,46,14,3)
PI(11)=(198,197,194,189,182,180,177,170,158,140,117,88,46,12)
New asymmetric codes
The new asymmetric codes found are presented here using a
compressed notation similar to that in
[Brower et al., A new table of constant weight
codes, IEEE Trans. Inform. Theory, 36 (1990),
1334-1380.]
The codewords are first sorted into lexicographic order,
yielding the following sequence of codewords:
c1,c2,...,cM.
The compressed notation for these codewords is
a1,a2,...,aM.
To find ai, 1<=i<=M,
let u1,u2,... be the
lexicographically sorted list of
all vectors that have asymmetric distance at least 2 from
the subcode {c1,...,ci-1}
and follow ci-1
(if i=1 we start from the first vector).
If ci=ur, we set
ai=r-1. Informally, given
c1,c2,...,ci-1, we get ci by skipping
ai lexicographic words that can be added to the code.
We also abbreviate a,a,...,a (k times) by
a^k.
Note that the new codes of lengths 10 and 11 are not listed
here but can be obtained from the partitions above.
Aa(12,2) >= 378
Aa(13,2) >= 699
Aa(14,2) >= 1273
Last update: April 14, 1999 by
Patric Östergård.