We develop new list decoding algorithms for Tanner codes and distance-amplified codes based on bipartite spectral expanders. We show that proofs exhibiting lower bounds on the minimum distance of these codes can be used as certificates discoverable by relaxations in the Sum-of-Squares (SoS) semidefinite programming hierarchy. Combining these certificates with certain entropic proxies to ensure that the solutions to the relaxations cover the entire list, then leads to algorithms for list decoding several families of codes up to the Johnson bound. We prove the following: - We show that the LDPC Tanner codes of Sipser-Spielman [IEEE Trans. Inf. Theory 1996] and Z\'{e}mor [IEEE Trans. Inf. Theory 2001] with alphabet size $q$, block-length $n$ and distance $\delta$, based on an expander graph with degree $d$, can be list-decoded up to distance $\mathcal{J}_q(\delta) - \epsilon$ in time $n^{O_{d,q}(1/\epsilon^4)}$, where $\mathcal{J}_q(\delta)$ denotes the Johnson bound. - We show that the codes obtained via the expander-based distance amplification procedure of Alon, Edmonds and Luby [FOCS 1995] can be list-decoded close to the Johnson bound using the SoS hierarchy, by reducing the list decoding problem to unique decoding of the base code. In particular, starting from \emph{any} base code unique-decodable up to distance $\delta$, one can obtain near-MDS codes with rate $R$ and distance $1-R - \epsilon$, list-decodable up to the Johnson bound in time $n^{O_{\epsilon, \delta}(1)}$. - We show that the locally testable codes of Dinur et al. [STOC 2022] with alphabet size $q$, block-length $n$ and distance $\delta$ based on a square Cayley complex with generator sets of size $d$, can be list-decoded up to distance $\mathcal{J}_q(\delta) - \epsilon$ in time $n^{O_{d,q}(1/\epsilon^{4})}$, where $\mathcal{J}_q(\delta)$ denotes the Johnson bound.

翻译：我们针对基于二分谱扩展图的Tanner码和距离放大码，提出了新的列表译码算法。研究表明，这些码的最小距离下界证明可作为可被平方和（SoS）半定规划层级中的松弛方法发现的证书。通过将这些证书与确保松弛解覆盖完整列表的特定熵代理相结合，即可导出针对多类码族在Johnson界内进行列表译码的算法。我们证明如下结果：- 对于Sipser-Spielman[IEEE Trans. Inf. Theory 1996]和Zémor[IEEE Trans. Inf. Theory 2001]提出的LDPC Tanner码（字母表大小$q$、码长$n$、距离$\delta$，基于度为$d$的扩展图），可在$n^{O_{d,q}(1/\epsilon^4)}$时间内实现至多距离$\mathcal{J}_q(\delta) - \epsilon$的列表译码，其中$\mathcal{J}_q(\delta)$表示Johnson界。- 对于Alon、Edmonds和Luby[FOCS 1995]提出的基于扩展图距离放大过程所得码，通过将列表译码问题归结为基码的唯一译码，可利用SoS层级在接近Johnson界处实现列表译码。特别地，从任意可唯一译码至距离$\delta$的基码出发，可构造出速率$R$、距离$1-R - \epsilon$的近似MDS码，并在$n^{O_{\epsilon, \delta}(1)}$时间内实现至多Johnson界的列表译码。- 对于Dinur等人[STOC 2022]提出的局部可测试码（字母表大小$q$、码长$n$、距离$\delta$，基于生成集大小为$d$的平方Cayley复形），可在$n^{O_{d,q}(1/\epsilon^{4})}$时间内实现至多距离$\mathcal{J}_q(\delta) - \epsilon$的列表译码，其中$\mathcal{J}_q(\delta)$表示Johnson界。