Abstract
A well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ringR. A sufficient condition is given, as well, for such a bound to be sharp. It is also shown that an MPC is free when its input codes are all free, in which case a generator matrix is given. IfRis finite, a sufficient condition is provided for the dual of an MPC to be an MPC, a generator matrix for such a dual is given, and characterizations of LCD, self-dual, and self-orthogonal MPCs are presented. Finally, the results of this paper are used along with previous results of the authors to construct novel MPCs arising fromσ,δ-codes. Some properties of such constructions are also studied.
Highlights
Over the past two decades, studying codes over commutative rings and their properties has been attracting a great deal of attention
As far as engineering applications are concerned, it is understood that codes over special types of commutative rings are more relevant; namely, finite Frobenius rings
If C is an R-submodule of Rn, C is called a linear code over R. e R-submodule of Rn generated by a code in Rn is obviously a linear code over R, so all codes considered in this paper are linear
Summary
Over the past two decades, studying codes over commutative rings (especially finite ones) and their properties has been attracting a great deal of attention. We give in Proposition 2 sufficient conditions for a matrixproduct code over a commutative ring to be free, and we give its generator matrix in Corollary 1. We generalize in Proposition 3 a well-know fact that tells when the dual of a matrixproduct code is a matrix-product code.
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