Abstract
In this paper, rate 1/p q-ary systematic quasi-cyclic codes are constructed based on matroid theory. The relationship between the generator matrix and minimum distance d is derived. Examples and algorithm are presented.
Highlights
In 1967, quasi-cyclic codes were first found by Townsend and Weldon [1]
Quasi-cyclic codes have been extended by the authors ([2],[3])
A subset C of Fqkp is called a quasi-cyclic code of length kp and index p such that (1) C is a subspace of Fqkp ; ( ) (2) If c0,0, c0, p−1, c1,0, c1, p−1, ck −1,0, ck −1, p−1 is a codeword of C
Summary
In 1967, quasi-cyclic codes were first found by Townsend and Weldon [1]. Quasi-cyclic codes have been extended by the authors ([2],[3]). We call C a rate 1 p quasi-cyclic code It follows from [8] that the generator matrix of a quasicyclic code of length kp and index p is equivalent to a k p generator matrix of the form. To construct systematic quasi-cyclic codes, let G0 = Ik , where Ik is a k k identity matrix. In [9], by a use of the connections between block codes and their matroids, a matroid search algorithm is developed to construct the binary linear quasi-cyclic codes with a large minimum distance. We generalize the method of [9] to q -ary quasi-cyclic codes and construct the q -ary linear quasi-cyclic codes with a large minimum distance.
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