170 research outputs found
On the Structure of the Linear Codes with a Given Automorphism
The purpose of this paper is to present the structure of the linear codes
over a finite field with q elements that have a permutation automorphism of
order m. These codes can be considered as generalized quasi-cyclic codes.
Quasi-cyclic codes and almost quasi-cyclic codes are discussed in detail,
presenting necessary and sufficient conditions for which linear codes with such
an automorphism are self-orthogonal, self-dual, or linear complementary dual
The extremal codes of length 42 with automorphism of order 7
AbstractAll [42, 21, 8] binary self-dual codes with automorphisms of order 7 are enumerated. Up to equivalence there are 16 such codes. These codes are defined with their generator matrices
2^n Bordered Constructions of Self-Dual codes from Group Rings
Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes.
Various techniques involving circulant matrices and matrices from group rings have been used to construct
such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary
self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual
codes by combining many of the previously used techniques. The purpose of this is to construct self-dual
codes that were missed using classical construction techniques by constructing self-dual codes with different
automorphism groups. We apply the technique to codes over finite commutative Frobenius rings of characteristic
2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct
some extremal self-dual codes length 64 and 68, constructing 30 new extremal self-dual codes of length 68
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