We study relationships between worst-case and random-noise properties of error correcting codes. More concretely, we consider connections between minimum distance, list decoding radius, and block error probability on noisy channels. A recent result of Pernice, Sprumont, and Wootters established the tight connection between list decoding radius and symmetric channel performance for linear codes. We extend this result to general codes. The proof proceeds by directly bounding the weight distribution rather than by sharp threshold techniques. We then turn to the relation between minimum distance and symmetric channel performance. The results we just described imply that a $q$-ary code of relative distance $δ$ has vanishing error probability on the symmetric channel up to the Johnson radius $J_q(δ)$. We improve upon this bound in the case of linear codes, for a range $δ$, for $q\ge 4$. In our proof we consider the \emph{erasure} properties of codes, and bound their weight distribution through inequalities introduced by Samorodnitsky. The latter part of the proof gives a more general technique that bounds the symmetric channel performance of a linear code with constant relative distance and good erasure channel performance.

翻译：我们研究了纠错码最坏情况与随机噪声性质之间的关系。具体而言，考虑最小距离、列表译码半径以及噪声信道上的误块概率之间的关联。Pernice、Sprumont 和 Wootters 近期结果建立了线性码列表译码半径与对称信道性能的紧致联系。我们将该结果推广至一般码。证明通过直接界维权值分布而非使用尖锐阈值技术实现。随后转向研究最小距离与对称信道性能的关系。上述结果表明，相对距离为 $δ$ 的 $q$ 进制码在对称信道上的错误概率直至 Johnson 半径 $J_q(δ)$ 处趋近于零。针对 $q\ge 4$ 时特定区间 $δ$ 的线性码，我们改进了该界。证明中我们考虑了码的\emph{擦除}性质，并通过 Samorodnitsky 提出的不等式界定了其权值分布。证明的后半部分给出一种更通用的技术，用于界定额定相对距离且具有良好擦除信道性能的线性码的对称信道性能。