storage system designers

Abstract: A $[n,k]_q$ Generalized Reed--Solomon (GRS) code can be represented, up to monomial equivalence, by a projective system consisting of $n$ points on a degree $(k-1)$ rational normal curve (RNC) $C_{k-1}$ in PG$(k-1,q)$. We use the term \textit{Column-Twisted Reed--Solomon (CTRS)} codes broadly for $[n,k]_q$ codes whose projective system contains at least $(k+2)$ points on a $C_{k-1}$ of PG$(k-1,q)$. By a chord of $C_{k-1}$, we mean a line which is either a tangent, a real secant, or an imaginary secant of $C_{k-1}$. Roth and Lempel constructed non-GRS MDS $[n,k]_q$ codes by considering projective systems consisting of $(n-1)$ points on $C_{k-1}$ together with a point on a tangent line. More recently, Liu et al. obtained non-GRS $[n, k]_q$ MDS codes by considering projective systems consisting of $(n-2)$ points on $C_{k-1}$ together with a pair of points on a real secant. In the case when $q$ is odd, the length of these codes satisfies $k+4 \leq n \leq (q+3)/2$. In this article, we construct non-GRS $[n,k]_q$ MDS codes by considering projective systems consisting of $(n-2)$ points on $C_{k-1}$ together with a pair of Galois conjugate points on an imaginary secant. For odd $q$, these codes have lengths satisfying $k+4 \le n\le (q+5)/2$, thereby improving both the attainable lengths and dimension ranges of previous TRS and CTRS constructions. In particular, our code attains the length $(q+5)/2$ for every $3\le k\le (q-3)/2$. We use a geometric argument to establish the non-GRS property of CTRS codes for the range $3\le k \le n-4$, while the traditional method of comparison with the Schur square of GRS codes proves the non-GRS property only for the narrower range $3\le k \le n/2$. We also determine the structure of the dual of any CTRS code.

Abstract: Subspace-design codes form a central unifying framework in coding theory, connecting probabilistic existence results and explicit algebraically structured constructions. The two core families are folded Reed-Solomon codes and univariate multiplicity codes, both possessing the subspace-design property. These codes reach list-decoding capacity with polynomial-size lists, but their alphabet size grows with code length and hurts practical efficiency. Goyal, Guruswami, and Hsieh achieved a major breakthrough: they constructed the first explicit subspace-design codes over constant-size alphabets via the Alon-Edmonds-Luby expander framework, settling an open problem and showing subspace-design properties can hold without large alphabets. Despite recent progress, constructing subspace-design codes with low-complexity efficient encoding remains a critical open challenge. This work gives an explicit construction of r-dimensional subspace-design codes over a constant alphabet whose size grows exponentially with r. By comparison, Goyal et al. (FOCS 2026) built r-dimensional subspace-design codes with alphabet size doubly exponential in r squared. Our construction is computationally efficient: encoding scales quadratically with block length and linearly with dimension r in field operations. We use modified algebraic geometry codes, replacing rational evaluation points with higher-degree places to obtain a valid additive code. To confirm parameter bounds, we use Moore matrices to analyze determinants with many zero entries. We also generalize univariate multiplicity codes, originally built from Reed-Solomon codes, to algebraic geometry codes, and prove these AG-multiplicity codes achieve optimal parameters.

Abstract: We develop q-ary Guessing Random Additive Noise Decoding (GRAND) for linear codes over a q-ary alphabet. The decoder works on a sorted symbol-likelihood array and uses three local child generation rules to generate symbol deviation patterns. We then prove that these rules induce a monotone rooted spanning tree of the full row-index space, so best-first traversal gives maximum-likelihood (ML) decoding under unlimited search. Reed-Solomon simulations verify ML agreement and show the finite-budget performance-complexity tradeoff.

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.

Abstract: We show that binary codes of length $n$ correcting two deletions exist with redundancy $3\log_2n+O(\log_2\log_2n)$. The previous best upper bound had leading coefficient $4$, unchanged since 1965, while the best known lower bound has coefficient $2$. More generally, codes correcting $t\ge2$ deletions exist with redundancy $(2t-1)\log_2n+O_t(\log_2\log_2n)$, improving the coefficient $2t$. We extract a code from one label class of a random linear hash of substring counts, with about $n$ times fewer labels than a direct construction. Confusable words that still share a label are separated by a two-colouring after discarding the words in components with odd cycles, and these are few because an odd cycle forces the edits along it to overlap.

Abstract: We study binary quasi-cyclic LDPC codes whose parity-check matrices are full arrays of single circulant permutation matrices (CPMs), referred to here as CPM-LDPC codes. Their minimum distance is at most $(J+1)!$, where $J$ is the column weight. For every fixed pair of column and row weights $2\le J<L$, we show that this bound is attained for all sufficiently large integer lift sizes. First, we give one integer exponent matrix independent of the lift size $P$. Second, we show that independent uniform exponent choices attain the bound with probability $1-O_{J,L}(P^{-1})$. Both proofs use cycle conditions required by low-weight codewords and a lower bound on the number of terms in vectors satisfying polynomial check equations. Neither construction requires $P$ to be prime. We also give small-lift arrays attaining the bound 24 for $J=3$, $L=4,\ldots,8$, and arrays with distance at least 28 for $J=4$, $L=5,\ldots,8$, together with computational distance verification.

Abstract: In recent years, simplicial complexes have gained considerable attention as a useful tool for constructing distance-optimal codes over finite fields. In this article, we construct four infinite families of linear codes over the ring $\mathcal{R}=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^2=0$ using simplicial complexes with one or two maximal elements, and completely determine their Lee weight distributions via exponential-sum techniques. By employing a Gray map on $\mathcal{R}$, we obtain infinite families of distance-optimal codes over $\mathbb{F}_{q}$, including a near-Griesmer family, and establish sufficient conditions for their minimality. Furthermore, we investigate the corresponding subfield codes and derive sufficient conditions for their distance-optimality and minimality, yielding infinite families of Griesmer and near-Griesmer codes.