Abstract
In many cases there is a need of exhaustive lists of combinatorial objects of a given type. We consider generation of all inequivalent polynomials from which defining polynomials for constructing quasi-cyclic (QC) codes are to be chosen. Using these defining polynomials we construct 34 new good QC codes over GF(11) and 36 such codes over GF(13). In many cases there is a need of exhaustive lists of combinatorial objects of a given type. We consider generation of all inequivalent polynomials from which defining polynomials for constructing quasi-cyclic (QC) codes are to be chosen. Using these defining polynomials we construct 34 new good QC codes over GF(11) and 36 such codes over GF(13).
Highlights
Let GF(q) denote the Galois field of q elements and let V(n,q) denote the vector space of all ordered n-tuples over GF(q)
In many cases there is a need of exhaustive lists of combinatorial objects of a given type
A k Ă n matrix G having as rows the vectors of a basis of a linear code C is called a generator matrix for C
Summary
Let GF(q) denote the Galois field of q elements and let V(n,q) denote the vector space of all ordered n-tuples over GF(q). A linear code C of length n and dimension k over GF(q) is a k-dimensional subspace of V(n,q) Such a code is called [n, k, d]q code if its minimum Hamming distance is d. One is the construction of codes which optimize minimum distance and the other is proving non-existence of codes of certain parameters ([14], [21]). In the former one often uses computers, but this approach becomes ineffective when the dimension of the codes is large, because, as we know, computing the minimum distance of linear codes is an NP-hard problem [30]. The Chenâs table [9] contains only good and best-known QC and QT codes (q †13) These two databases are updated when new codes are discovered. In this paper we present 34 new QC codes over GF(11) and 36 new QC codes over GF(13)
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