Linear complementary dual, maximum distance separable codes

Abstract

Linear complementary dual (LCD) maximum distance separable (MDS) codes are constructed to given specifications. For given $n$ and $r<n$, with $n$ or $r$ (or both) odd, MDS LCD $(n,r)$ codes are constructed over finite fields whose characteristic does not divide $n$. Series of LCD MDS codes are constructed to required rate and required error-correcting capability. Given the field $GF(q)$ and $n/(q-1)$, LCD MDS codes of length $n$ and dimension $r$ are explicitly constructed over $GF(q)$ for all $r<n$ when $n$ is odd and for all odd $r<n$ when $n$ is even. For given dimension and given error-correcting capability LCD MDS codes are constructed to these specifications with smallest possible length. Series of asymptotically good LCD MDS codes are explicitly constructed. Efficient encoding and decoding algorithms exist for all the constructed codes. Linear complementary dual codes have importance in data storage, communications' systems and security.