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Optimal ternary linear codes. (English) Zbl 0756.94008
Summary: Let $n\sb q(k,d)$ denote the smallest value of $n$ for which there exists a linear $[n,k,d]$-code over $\text{GF}(q)$. An $[n,k,d]$-code whose length is equal to $n\sb q(k,d)$ is called optimal. The problem of finding $n\sb q(k,d)$ has received much attention for the case $q=2$. We generalize several results to the case of an arbitrary prime power $q$ as well as introducing new results and a detailed methodology to enable the problem to be tackled over any finite field. In particular, we study the problem with $q=3$ and determine $n\sb 3(k,d)$ for all $d$ when $k\le 4$, and $n\sb 3(5,d)$ for all but 30 values of $d$.

94B05General theory of linear codes
Full Text: DOI
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