Higher-order Delsarte Dual LPs: Lifting, Constructions and Completeness

TL;DR

This work analyzes higher-order Delsarte LP hierarchies for binary codes through their dual formulations. It introduces explicit dual-constructive techniques: (i) a lifting method to transport dual solutions from level $k$ to level $\ell$ (when $k\mid\ell$) while preserving the objective, and (ii) a spectral-based construction yielding constant-level duals for $\varepsilon$-balanced linear codes whose objective matches the state-of-the-art MRRW upper bounds up to lower-order terms. Additionally, it provides a dual-based completeness proof, strengthening the theoretical foundation that these hierarchies converge to the true optimum. The results offer structural understanding and a technical framework that may enable quantitative improvements and extensions to other combinatorial hierarchies. Overall, the paper advances dual certificates for complete convex-programming hierarchies in coding theory and paves the way for tighter rate-vs-distance bounds.

Abstract

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.

Figures (6)

• Figure 1: Different projections of the space of all possible Hamming weight combinations when $\ell=2$ (the picture rescales $n$ out). Three of the six edges of the tetrahedron are contained on the coordinate planes. The top left projection is isometric.
• Figure 2: Different projections of $\mathop{\mathrm{Valid}}\nolimits_{n,\ell}^\varepsilon$ (in Hamming weight coordinates) with $\varepsilon=0.2$ (the picture rescales $n$ out) when $\ell=2$. The region $\mathop{\mathrm{Valid}}\nolimits_{n,\varepsilon}$ consists of the origin, the three line segments on the coordinate planes and the cube (with interior) in the middle. The cube faces are paralel to the coordinate planes. The top left projection is isometric.
• Figure 3: Different projections of $\mathop{\mathrm{Valid}}\nolimits_{n,\ell}^\varepsilon$ (in Hamming weight coordinates) with $\varepsilon=0.2$ (the picture rescales $n$ out) and the cylinders when $\ell=2$. The region $\mathop{\mathrm{Valid}}\nolimits_{n,\varepsilon}$ consists of the origin, the three line segments on the coordinate planes and the cube (with interior) in the middle. The vertices of the cube are precisely the points in which all three cylinder surfaces intersect. The cube faces and cylinder bases are parallel to the coordinate plane. The top left projection is isometric.
• Figure 4: Different projections of the space of all possible Venn diagram configurations when $\ell=2$ (the picture rescales $n$ out). Three of the tetrahedron faces are on coordinate planes. The top left projection is isometric.
• Figure 5: Different projections of $\mathop{\mathrm{Valid}}\nolimits_{n,\ell}^\varepsilon$ in Venn diagram configuration space with $\varepsilon=0.2$ (the picture rescales $n$ out) when $\ell=2$. Here $X_i\stackrel{\text{def}}{=}\mathop{\mathrm{supp}}\nolimits(x_i)$. The region $\mathop{\mathrm{Valid}}\nolimits_{n,\varepsilon}$ consists of the origin, the three line segments on the coordinate axes and the cube (with interior) in the middle. None of the cube faces are parallel to the coordinate planes. The top left projection is isometric. On the top right and bottom left projections, two of the cube faces are parallel to the projection plane.

Theorems & Definitions (42)

• Theorem 1.1: Lifting Dual Solutions (Informal version of \ref{['thm:lift']})
• Remark 1.2
• Theorem 1.3: Higher-order Dual Solution (Informal version of \ref{['cor:spectral_construction']} of \ref{['theo:spectral_construction']})
• Theorem 1.4: Completeness from the Dual (Informal version of \ref{['thm:completeness_from_dual']})
• Remark 1.5
• Remark 4.1
• Lemma 4.2
• proof
• Remark 4.3
• Lemma 4.4