1,436 research outputs found

    The hull of two classical propagation rules and their applications

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    Propagation rules are of great help in constructing good linear codes. Both Euclidean and Hermitian hulls of linear codes perform an important part in coding theory. In this paper, we consider these two aspects together and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the (u,u+v)(\mathbf{u},\mathbf{u+v})-construction. Some new criteria for resulting codes derived from these two propagation rules being self-dual, self-orthogonal or linear complement dual (LCD) codes are given. As applications, we construct some linear codes with prescribed hull dimensions and many new binary, ternary Euclidean formally self-dual (FSD) LCD codes, quaternary Hermitian FSD LCD codes and good quaternary Hermitian LCD codes which are optimal or have best or almost best known parameters according to Datebase at http://www.codetables.dehttp://www.codetables.de. Moreover, our methods contributes positively to improve the lower bounds on the minimum distance of known LCD codes.Comment: 16 pages, 5 table

    Euclidean and Hermitian LCD MDS codes

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    Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) have been of much interest from many researchers due to their theoretical significant and practical implications. However, little work has been done on LCD MDS codes. In particular, determining the existence of qq-ary [n,k][n,k] LCD MDS codes for various lengths nn and dimensions kk is a basic and interesting problem. In this paper, we firstly study the problem of the existence of qq-ary [n,k][n,k] LCD MDS codes and completely solve it for the Euclidean case. More specifically, we show that for q>3q>3 there exists a qq-ary [n,k][n,k] Euclidean LCD MDS code, where 0≤k≤n≤q+10\le k \le n\le q+1, or, q=2mq=2^{m}, n=q+2n=q+2 and k=3orq−1k= 3 \text{or} q-1. Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes

    On MDS Negacyclic LCD Codes

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    Linear codes with complementary duals (LCD) have a great deal of significance amongst linear codes. Maximum distance separable (MDS) codes are also an important class of linear codes since they achieve the greatest error correcting and detecting capabilities for fixed length and dimension. The construction of linear codes that are both LCD and MDS is a hard task in coding theory. In this paper, we study the constructions of LCD codes that are MDS from negacyclic codes over finite fields of odd prime power qq elements. We construct four families of MDS negacyclic LCD codes of length n∣q−12n|\frac{{q-1}}{2}, n∣q+12n|\frac{{q+1}}{2} and a family of negacyclic LCD codes of length n=q−1n=q-1. Furthermore, we obtain five families of q2q^{2}-ary Hermitian MDS negacyclic LCD codes of length n∣(q−1)n|\left( q-1\right) and four families of Hermitian negacyclic LCD codes of length n=q2+1.n=q^{2}+1. For both Euclidean and Hermitian cases the dimensions of these codes are determined and for some classes the minimum distances are settled. For the other cases, by studying qq and q2q^{2}-cyclotomic classes we give lower bounds on the minimum distance
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