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Reed–Muller code

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IEEE Transactions on Information Theory

MDS codes and self-dual codes are important families of classical codes in coding theory. Therefore, it is of interest to investigate MDS self-dual codes. The existence of MDS selfdual codes over finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> is completely solved for q is even. In the literature, there are many known constructions of MDS self-dual codes for q is odd. In this paper, we present a unified approach on the existence of MDS self-dual codes with concise statements and simplified proof. It is illustrated that some of known results can also be stated in this framework. Furthermore, we can obtain some new MDS self-dual codes, especially for the case when q is not a square.

A Unified Approach to Construct MDS Self-Dual Codes via Reed-Solomon Codes