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Optimal ternary linear codes. (English) Zbl 0756.94008
Summary: Let ${n}_{q}\left(k,d\right)$ denote the smallest value of $n$ for which there exists a linear $\left[n,k,d\right]$-code over $\text{GF}\left(q\right)$. An $\left[n,k,d\right]$-code whose length is equal to ${n}_{q}\left(k,d\right)$ is called optimal. The problem of finding ${n}_{q}\left(k,d\right)$ 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 ${n}_{3}\left(k,d\right)$ for all $d$ when $k\le 4$, and ${n}_{3}\left(5,d\right)$ for all but 30 values of $d$.

##### MSC:
 94B05 General theory of linear codes
##### Keywords:
optimal ternary linear codes
##### References:
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