Structure and performance of generalized quasi-cyclic codes

Abstract

Abstract. Generalized quasi-cyclic (GQC) codes form a natural generalization of quasi-cyclic (QC) codes. They are viewed here as mixed alphabet codes over a family of ring alphabets. Decomposing these rings into local rings by the Chinese Remainder Theorem yields a decomposition of GQC codes into a sum of concatenated codes. This decomposition leads to a trace formula, a minimum distance bound, and to a criteria for the GQC code to be self-dual or to be linear complementary dual (LCD). Explicit long GQC codes that are LCD, but not QC, are exhibited.

Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains

“…In the case of QC and GQC codes, the results in [2,3] have a similarity with ours in the sense of producing codes of a modulus from those of factored moduli. We compare these results as follows.…”

“…Whereas, in [2,3], the producing methods is the concatenation which is represented by, e.g., Turyn's (x + a, x + b, x + a + b)-method, our producing method is the multiplication G = G 1 G 2 of generator matrices in Theorems 1,2. In [2,3], the self-duality is preserving, i.e., roughly speaking, if codes mod u 1 and mod u 2 are self-dual in a sense, then the produced code mod u = u 1 u 2 is also self-dual.…”

“…In [2,3], the self-duality is preserving, i.e., roughly speaking, if codes mod u 1 and mod u 2 are self-dual in a sense, then the produced code mod u = u 1 u 2 is also self-dual. Unfortunately, our producing method does not have this preserving property of self-duality.…”

“…Unfortunately, our producing method does not have this preserving property of self-duality. From the viewpoint of computational complexity, our method can have an advantage over those of [2,3] because, whereas Turyn-type methods require overall combination of codewords in the worst case, our method requires only multiplying two l-by-l matrices. Consequently, it is important to use different methods according to the desired types of codes.…”

“…The structural properties of quasi-cyclic (QC) [1,2] and generalized quasi-cyclic (GQC) codes [3][4][5] have been reported. On the other hand, cyclic codes can be extended to pseudo-cyclic (PC) and generalized pseudo-cyclic (GPC) codes [6].…”