Paper 2026/2316

Fast Linear Codes over Large Fields: Addition-Only and Small-Coefficient Encoders

Abstract

We construct fast linear codes over large fields using addition-only encoders and variants with small integer coefficients. A local precode is followed by rounds of a random permutation and a scan, a linear recurrence with a short feedback window. At dimension $2^{20}$, rate $1/2$, and $p=2^{128}-159$, two plain accumulators (unweighted prefix sums) give relative distance above $11.7\%$, except with probability below $2^{-49.64}$ over the sampling of the encoder. A binary subset-sum precode instead gives distance above $12.56\%$, except with probability below $2^{-22.05}$. These addition-only encoders use fewer than $6.8125n$ and $3.3125n$ field additions, respectively, for output length $n$. Both exceed the binary rate-$1/2$ Gilbert--Varshamov benchmark. The proof counts the supports and ranks of the zero equations a codeword must satisfy, retaining the dependence between cancellations. Coefficient size and scan width extend the tradeoff. On one core, with a common implementation and setup failure below $2^{-40}$, encoding time ranges from $12.171$ ms at proved distance above $11.7\%$ to $32.342$ ms above $42.987\%$. A coefficient-independent upper bound limits every encoder with two width-one scans after the Newton precode to $39.6855\%$. The distance and setup-failure bounds hold over $\mathbb F_p$ and every finite extension, covering all nonzero messages after setup. Timings are for $\mathbb F_p$. Extension preserves distance exactly because the encoder coefficients lie in the prime subfield. Changing characteristic requires a separate argument; we give field-transfer results in both directions, including counterexamples.