On the List-Decodability of Random Self-Orthogonal Codes

In 2011, Guruswami-Håstad-Kopparty \cite{Gru} showed that the list-decodability of random linear codes is as good as that of general random codes. In the present paper, we further strengthen the result by showing that the list-decodability of random {\it Euclidean self-orthogonal} codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. Specifically, we show that, for any fixed finite field $\F_q$, error fraction $δ\in (0,1-1/q)$ satisfying $1-H_q(δ)\le \frac12$ and small $ε>0$, with high probability a random Euclidean self-orthogonal code over $\F_q$ of rate $1-H_q(δ)-ε$ is $(δ, O(1/ε))$-list-decodable. This generalizes the result of linear codes to Euclidean self-orthogonal codes. In addition, we extend the result to list decoding {\it symplectic dual-containing} codes by showing that the list-decodability of random symplectic dual-containing codes achieves the quantum Gilbert-Varshamov bound as well. This implies that list-decodability of quantum stabilizer codes can achieve the quantum Gilbert-Varshamov bound. The counting argument on self-orthogonal codes is an important ingredient to prove our result.

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