A central and longstanding open problem in coding theory is the rate-versus-distance trade-off for binary error-correcting codes. In a seminal work, Delsarte introduced a family of linear programs establishing relaxations on the size of optimum codes. To date, the state-of-the-art upper bounds for binary codes come from dual feasible solutions to these LPs. Still, these bounds are exponentially far from the best-known existential constructions. Recently, hierarchies of linear programs extending and strengthening Delsarte's original LPs were introduced for linear codes, which we refer to as higher-order Delsarte LPs. These new hierarchies were shown to provably converge to the actual value of optimum codes, namely, they are complete hierarchies. Therefore, understanding them and their dual formulations becomes a valuable line of investigation. Nonetheless, their higher-order structure poses challenges. In fact, analysis of all known convex programming hierarchies strengthening Delsarte's original LPs has turned out to be exceedingly difficult and essentially nothing is known, stalling progress in the area since the 1970s. Our main result is an analysis of the higher-order Delsarte LPs via their dual formulation. Although quantitatively, our current analysis only matches the best-known upper bounds, it shows, for the first time, how to tame the complexity of analyzing a hierarchy strengthening Delsarte's original LPs. In doing so, we reach a better understanding of the structure of the hierarchy, which may serve as the foundation for further quantitative improvements. We provide two additional structural results for this hierarchy. First, we show how to \emph{explicitly} lift any feasible dual solution from level $k$ to a (suitable) larger level $\ell$ while retaining the objective value. Second, we give a novel proof of completeness using the dual formulation.

Abstract. A Sidorenko bigraph is one whose density in a bigraphon [Formula: see text] is minimized precisely when [Formula: see text] is constant. Several techniques in the literature to prove the Sidorenko property consist of decomposing (typically in a tree decomposition) the bigraph into smaller building blocks with stronger properties. One prominent such technique is that of [Formula: see text]-decompositions of Conlon and Lee, which uses weakly Hölder (or weakly norming) bigraphs as building blocks. In turn, to obtain weakly Hölder bigraphs, it is typical to use the chain of implications reflection bigraph [Formula: see text] cut-percolating bigraph [Formula: see text] weakly Hölder bigraph. In an earlier result by the author with Razborov, we provided a generalization of [Formula: see text]-decompositions, called reflective tree decompositions, that uses much weaker building blocks, called induced-Sidorenko bigraphs, to also obtain Sidorenko bigraphs. In this paper, we show that “left-sided” versions of the concepts of reflection bigraph and cut-percolating bigraph yield a similar chain of implications: left-reflection bigraph [Formula: see text] left-cut-percolating bigraph [Formula: see text] induced-Sidorenko bigraph. We also show that under mild hypotheses the “left-sided” analogue of the weakly Hölder property (which is also obtained via a similar chain of implications) can be used to improve bounds on another result of Conlon and Lee that roughly says that bigraphs with enough vertices on the right side of each realized degree have the Sidorenko property.

Determining the maximum size $A_2(n,d)$ of a binary code of blocklength $n$ and distance $d$ remains an elusive open question even when restricted to the important class of linear codes. Recently, two linear programming hierarchies extending Delsarte's LP were independently proposed to upper bound $A_2^{\text{Lin}}(n,d)$ (the analogue of $A_2(n,d)$ for linear codes). One of these hierarchies, by the authors, was shown to be approximately complete in the sense that the hierarchy converges to $A_2^{\text{Lin}}(n,d)$ as the level grows beyond $n^2$. Despite some structural similarities, not even approximate completeness was known for the other hierarchy by Loyfer and Linial. In this work, we prove that both hierarchies recover the exact value of $A_2^{\text{Lin}}(n,d)$ at level $n$. We also prove that at this level the polytope of Loyfer and Linial is integral.Even though these hierarchies seem less powerful than general hierarchies such as Sum-of-Squares, we show that they have enough structure to yield exact completeness via pseudoprobabilities.

A longstanding open problem in coding theory is to determine the best (asymptotic) rate $R_2(δ)$ of binary codes with minimum constant (relative) distance $δ$. 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 $R_2(δ)$ 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.