Evolving Towards Better Codes: LLM-Guided Search for High-Distance Binary Linear Codes

Amal Seddas; Vladyslav Shashkov; Maryna Viazovska; Emmanuel Abbe

cs.ITSep 29, 2026

Evolving Towards Better Codes: LLM-Guided Search for High-Distance Binary Linear Codes

Authors: Amal Seddas, Vladyslav Shashkov, Maryna Viazovska, Emmanuel Abbe

Organizations: EPFL

Abstract

Evolutionary program search driven by large language models (LLMs) has produced record-breaking constructions for open problems in combinatorics and beyond. We apply this approach to the longstanding problem of improving the best-known bounds for binary linear codes. Building on the EvoTune evolutionary framework and the ShinkaEvolve codebase, we introduce LinCodeEvolve, which evolves code-construction programs against an exact minimum-distance evaluator. A strategy loop combines diversity-driven search and expert supervision: when progress plateaus, new strategies are used to redirect the search. LinCodeEvolve discovers seven record-breaking codes, $[172,21,66]$ , $[173,20,68]$ , $[176,21,68]$ , $[181,21,70]$ , $[184,21,72]$ , $[189,22,72]$ and $[200,21,77]$ , six of which have concise quasi-cyclic descriptions. With standard code modification techniques, they improve $22$ entries of the tables. Every code is verified by exhaustive enumeration. These results suggest that LLM-guided search can help find improved codes and complement existing methods in coding theory.

Figures & tables

cell	LB	GV	random best	random median $d$	gen 0	lane	s
$[41,12]$	14	11	$(11,5)$	10	$\mathbf{(12,11)}$	construction_core	214
$[48,16]$	16	11	$(11,3)$	10	$\mathbf{(14,256)}$	qc_core	321
$[50,20]$	13	10	$(10,6)$	9	$\mathbf{(11,30)}$	qc_core	413
$[60,20]$	17	13	$(13,1)$	12	$\mathbf{(16,245)}$	qc_core	307

Table 1: Generation-0 against random full-rank matrices, exact

(d,A_{d})

. Random: best of 50 draws, and the median

d

of the 50. GV: the Gilbert–Varshamov guarantee for

(n,k)

. the portfolio’s pick and its winning lane.

Code $[n,k,d]$	Previous bound	Gain
$[172,21,66]$	64	$+2$
$[173,20,68]$	67	$+1$
$[176,21,68]$	66	$+2$
$[181,21,70]$	68	$+2$
$[184,21,72]$	69	$+3$
$[189,22,72]$	71	$+1$

Table 2: Seven constructions improving CodeTables lower bounds in the 25 September 2026 snapshot. The gain is the difference between the new minimum distance and the previous lower bound.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

$n$	$k$	$d_{\mathrm{old}}$	$d^{\mathrm{ub}}$	$A_{d}$
172	20	66	67	272
173	20	67	68	1156
171	21	64	65	2040
172	21	64	66	2720
173	21	64	66	680
174	21	65	66	680

Appendix

Table 3: Improved entries, with the best previously known minimum distance ( old ), the minimum distance of our code ( new ), and the number

A_{d}

of minimum-weight codewords of the released code.

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Evolutionary Discovery of Bivariate Bicycle Codes with LLM-Guided Search

Juan Cruz-Benito, Andrew W. Cross, David Kremer, Ismael Faro

IBM Research, IBM T. J. Watson Research Center, NY, USA

Quantum Error Correction Quantum Annealing Codebooks Alphaevolve Evolutionary Optimization Methods LLM Inference Optimization+1

Open paper AskPDF arXiv

Quantum LDPC code discovery requires searching large algebraic design spaces while reliably certifying the parameters and equivalence classes of any candidates found. We introduce an LLM-guided evolutionary workflow in which language models mutate Python programs that generate bivariate-bicycle and perturbed bivariate-bicycle code ansätze. Across five campaigns, the system performed approximately 1{,}650 evolutionary iterations, screened about

2 \times 10^5

candidate codes, and required

{\sim}140

hours of computation and

{\sim}

US$400 in LLM inference cost. Candidate codes are evaluated through a staged validation pipeline combining

\mathrm{GF}(2)

rank computation, distance estimation and certification, mixed-integer linear programming, BLISS Tanner-graph deduplication, decomposability analysis, and local-Clifford equivalence checks. At block length

n \leq 360

, the workflow identifies 465 distinct candidate codes: 97 CSS bivariate-bicycle codes and 368 non-CSS perturbed variants. The CSS search recovers known high-performing codes and finds new finite-length representatives, including an indecomposable [[288,16,12]] code and higher-weight codes with up to

k = 50

at distance

d = 8

. The non-CSS search produces perturbed codes matching the gross-code figure of merit at [[144,12,12]], along with additional high-distance candidates reported as certified values or upper bounds according to MILP status. Overall, these results show that LLM-guided program evolution can serve as a practical tool for structured quantum-code discovery when paired with independent evaluation.

Abstract

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