A longstanding open problem in coding theory is to determine the best (asymptotic) rate $R_2(\delta)$ of binary codes with minimum constant (relative) distance $\delta$. An existential lower bound was given by Gilbert and Varshamov in the 1950s. On the impossibility side, in the 1970s McEliece, Rodemich, Rumsey and Welch (MRRW) proved an upper bound by analyzing Delsarte's linear programs. To date these results remain the best known lower and upper bounds on $R_2(\delta)$ with no improvement even for the important class of linear codes. Asymptotically, these bounds differ by an exponential factor in the blocklength. In this work, we introduce a new hierarchy of linear programs (LPs) that converges to the true size $A^{\text{Lin}}_2(n,d)$ of an optimum linear binary code (in fact, over any finite field) of a given blocklength $n$ and distance $d$. This hierarchy has several notable features: (i) It is a natural generalization of the Delsarte LPs used in the first MRRW bound. (ii) It is a hierarchy of linear programs rather than semi-definite programs potentially making it more amenable to theoretical analysis. (iii) It is complete in the sense that the optimum code size can be retrieved from level $O(n^2)$. (iv) It provides an answer in the form of a hierarchy (in larger dimensional spaces) to the question of how to cut Delsarte's LP polytopes to approximate the true size of linear codes. We obtain our hierarchy by generalizing the Krawtchouk polynomials and MacWilliams inequalities to a suitable "higher-order" version taking into account interactions of $\ell$ words. Our method also generalizes to translation schemes under mild assumptions.

翻译：在编码理论中,一个长期的开放问题在于确定最佳(隐性)标准值$R_2(\delta) $R_2(delta) 二进制代码的最佳(隐性) 标准值为$@delta$。 Gilbert 和 Varshamov 在 1950 年代给出了较低的存在约束值。 在20世纪70年代, McEliece、 Rodemich、 Rumsey 和 Welch (MRW) 被分析为对 Delsart 线性程序进行上层分析的上限。 这些结果仍然是最已知的 $R_2(\delta) 的下层和上层标准值。 即使对于重要的线性代码类别来说, 也没有任何改进。 Asymptotoal 级别的界限因块长因素而不同。 在这项工作中,我们引入了一个新的线性程序等级(LPSOL), 它的直线性值值值值值值值(事实上, 超过任何固定的域域域内, 美元和远方$。