Construction of quasi-cyclic self-dual codes

초록

There is a one-to-one correspondence between e-quasi-cyclic codes over a finite field F-q and linear codes over a ring R = F-q[Y]/(Y-m - 1). Using this correspondence, we prove that every l-quasi-cyclic self-dual code of length me over a finite field F-q can be obtained by the building-up construction, provided that char(F-q) = 2 or q equivalent to 1 (mod 4), en is a prime p, and q is a primitive element of F-p. We determine possible weight enumerators of a binary l-quasi-cyclic self-dual code of length pl (with p a prime) in terms of divisibility by p. We improve the result of Bonnecaze et al. (2003) [3] by constructing new binary cubic (i.e., l-quasi-cyclic codes of length 3l) optimal self-dual codes of lengths 30, 36, 42, 48 (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When m = 5, we obtain a new 8-quasi-cyclic self-dual [40, 20, 12] code over F-3 and a new 6-quasi-cyclic self-dual [30, 15, 10] code over F-4. When m = 7, we find a new 4-quasi-cyclic self-dual [28, 14, 9] code over F-4 and a new 6-quasi-cyclic self-dual [42, 21, 12] code over F-4. (C) 2011 Elsevier Inc. All rights reserved.

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