Paper 2026/2315

Expander-Based Codes at the Gilbert–Varshamov Bound

Abstract

We prove that Expand--Accumulate (EA) codes approach the Gilbert--Varshamov (GV) distance as the code length grows. The result holds over the binary field and every fixed finite field. At fixed rate, encoding uses $O(n\log n)$ expected field operations, with a tunable inverse-polynomial probability of sampling a bad code. Fixed-memory wrapped binary Expand--Convolute (EC) codes satisfy the same guarantee. Allowing $O(n\log^2 n)$ expected field operations makes sampling failure negligible in the code length. These codes combine a sparse random linear map with an invertible recursive map. Our analysis uses exact input--output weight enumerators in place of the earlier concentration bounds. It also gives tighter constants in the logarithmic expected degree. For EA-based pseudorandom correlation functions, these bounds improve the degree--distance tradeoff that controls local evaluation cost. Separately, we obtain finite-length bounds at small degrees by replacing independent incidence with regional regularity. At rate one half, a two-sided regular EC ensemble with left/right degrees $10/5$ and memory $15$ reaches $99.99\%$ of the binary GV distance. The probability of sampling a generator below this distance is less than $2^{-32.49}$. We extend the construction to finite fields using independent nonzero edge labels and full-field convolution coefficients. These coefficients mix field components; scalar extension of a binary matrix does not. Over $\F_{2^{128}}$, degrees $24/12$ with memory $5$ reach $98.41\%$ of the rate-one-half GV distance with sampling failure below $2^{-45.03}$; degrees $26/13$ with memory $4$ reach the same distance with failure below $2^{-135.20}$. Every finite claim is certified with outward-rounded interval arithmetic. The results concern minimum distance of sampled ensembles; they do not provide a decoder or a complete protocol-security proof.