Abstract
Let $d_{q}(n,k)$ be the maximum possible minimum Hamming distance of a linear [$n,k$] code over $\mathbb{F}_{q}$. Tables of best known linear codes exist for small fields and some results are known for larger fields. Quasi-twisted codes are constructed using $m \times m$ twistulant matrices and many of these are the best known codes. In this paper, the number of $m \times m$ twistulant matrices over $\mathbb{F}_q$ is enumerated and linear codes over $\mathbb{F}_{17}$ and $\mathbb{F}_{19}$ are constructed for $k$ up to $5$.
Highlights
Tables of best known linear codes exist for small fields and some results are known for larger fields
The number of m × m twistulant matrices over Fq is enumerated and linear codes over F17 and F19 are constructed for k up to 5
Let Fq denote the finite field of q elements, and V (n, q) the vector space of n-tuples over Fq
Summary
Let Fq denote the finite field of q elements, and V (n, q) the vector space of n-tuples over Fq. The Griesmer bound is a well-known lower bound on nq(k, d) k−1 d nq(k, d) ≥ gq(k, d) =. The Singleton bound [12] is a lower bound on nq(k, d) and is given by nq(k, d) ≥ d + k − 1. Codes that meet this bound are called maximum distance separable (MDS). Codes over F17 and F19 for k up to 5 are presented These codes establish lower bounds on the minimum distance. Many of these meet the Singleton and/or Griesmer bounds, and so are optimal
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