Abstract
Let F q be a finite field of cardinality q, m 1, m 2, . . . , m l be any positive integers, and \({A_i=F_q[x]/(x^{m_i}-1)}\) for i = 1, . . . , l. A generalized quasi-cyclic (GQC) code of block length type (m 1, m 2, . . . , m l ) over F q is defined as an F q [x]-submodule of the F q [x]-module \({A_1\times A_2\times\cdots\times A_l}\). By the Chinese Remainder Theorem for F q [x] and enumeration results of submodules of modules over finite commutative chain rings, we investigate structural properties of GQC codes and enumeration of all 1-generator GQC codes and 1-generator GQC codes with a fixed parity-check polynomial respectively. Furthermore, we give an algorithm to count numbers of 1-generator GQC codes.
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Communicated by H. van Tilborg.
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Cao, Y. Structural properties and enumeration of 1-generator generalized quasi-cyclic codes. Des. Codes Cryptogr. 60, 67–79 (2011). https://doi.org/10.1007/s10623-010-9417-5
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DOI: https://doi.org/10.1007/s10623-010-9417-5