Self-orthogonal codes with dual distance three and quantum codes with distance three over \(\mathbb F _5\)
Self-orthogonal codes with dual distance three and quantum codes with distance three constructed from self-orthogonal codes over \(\mathbb F _5\) are discussed in this paper. Firstly, for given code length \(n\ge 5\), a \([n,k]_{5}\) self-orthogonal code with minimal dimension \(k\) and dual distance three is constructed. Secondly, for each \(n\ge 5\), two nested self-orthogonal codes with dual distance two and three are constructed, and consequently quantum code of length \(n\) and distance three is constructed via Steane construction. All of these quantum codes constructed via Steane construction are optimal or near optimal according to the quantum Hamming bound.
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