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.