随机线性码的平均半径列表可译性
Average-Radius List-Decodability of Random Linear Codes
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中文总结 AI 辅助
该研究证明速率为1−h_q(p)−ε的随机F_q线性码具有平均半径列表可译性，将标准列表译码结果推广到平均半径场景，补充了二元线性码及非线性随机码之外的相关结论。
中文摘要 AI 辅助
我们证明，对于每个素数幂q和每个p∈(0,1−1/q)，速率为1−h_q(p)−ε的随机F_q线性码，以至少1−q^{−Ω(n)}的概率是(p, C_{p,q}/ε)平均半径列表可译的，即对于每个中心y∈F_q^n，距离y最近的C_{p,q}/ε个码字与y的平均分数汉明距离至少为p。这将Guruswami、Håstad和Kopparty（2010）关于（标准）列表译码的类似结果推广到更强的平均半径保证，且列表大小仍为O(1/ε)。此前，平均半径列表译码的此类结果仅对二元线性码（Guruswami、Li、Mosheiff、Resch、Silas和Wootters，2021）以及任意字母表上的一般（非线性）随机码（Elias，1991）已知。
英文摘要
We prove that for every prime power $q$ and every $p \in (0, 1-1/q)$, a random $\mathbb{F}_q$-linear code of rate $1 - h_q(p) - ε$ is $(p, C_{p,q}/ε)$-average-radius list-decodable with probability at least $1 - q^{-Ω(n)}$, i.e., for every center $y \in \mathbb{F}_q^n$, the $C_{p,q}/ε$ codewords closest to $y$ have average fractional Hamming distance at least $p$ from $y$. This extends a similar result for (standard) list-decoding due to Guruswami, Håstad, and Kopparty (2010) to the stronger average-radius guarantee, with the same $O(1/ε)$ list size. For average-radius list-decoding, such a result was previously known only for binary linear codes (Guruswami, Li, Mosheiff, Resch, Silas, and Wootters, 2021) and for general (non-linear) random codes over arbitrary alphabets (Elias, 1991).
