Abstract
The class of 1-generator quasi-abelian codes over finite fields is revisited. Alternative and explicit characterization and enumeration of such codes are given. An algorithm to find all 1-generator quasi-abelian codes is provided. Two 1-generator quasi-abelian codes whose minimum distances are improved from Grassl’s online table are presented.
Highlights
As a family of codes with good parameters, rich algebraic structures, and wide ranges of applications, quasi-cyclic codes have been studied for a halfcentury
Given finite abelian groups H ≤ G and a finite field Fq, an H-quasi-abelian code is defined to be an Fq[H]-submodule of Fq[G]
Examples of new codes derived from 1-generator quasi-abelian codes are presented
Summary
As a family of codes with good parameters, rich algebraic structures, and wide ranges of applications (see [8], [9], [11], [10], [13], [14] , and references therein), quasi-cyclic codes have been studied for a halfcentury. An H-quasi-abelian code C is said to be of 1-generator if C is a cyclic Fq[H]-module. Analogous to the case of 1-generator quasi-cyclic codes, the number of 1-generator quasi-abelian codes has been determined in [7]. An explicit construction and an algorithm to determine all 1-generator quasi-abelian codes have not been well studied. We give an alternative discussion on the algebraic structure of 1-generator quasi-abelian codes and an algorithm to find all 1-generator quasi-abelian codes. Examples of new codes derived from 1-generator quasi-abelian codes are presented. An alternative discussion on the algebraic structure of 1-generator quasi-abelian codes is given in Section 3 together with an algorithm to find all 1-generator quasi-abelian codes and the number of such codes.
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