coding engineers

Abstract: Every perfect $r$-error-correcting Lee code in $\mathbb{Z}^n$ induces a finite configuration around each codeword: a family of pairwise disjoint radius-$r$ Lee spheres, all disjoint from the central sphere, that covers every lattice point at Lee distance $r+1$ from the codeword. We call such a configuration a complete covering. Under projection onto a suitable Cartesian product of cycles, a complete covering corresponds to an optimal local structure in the sense of Špacapan. The existence of a complete covering is necessary but, in general, not sufficient for the existence of a perfect Lee code, as our construction for $(n,r)=(3,2)$ shows. We prove that no complete covering exists whenever $n\ge 8$ and $r^2+\lfloor 25r/34\rfloor-3\ge 2n$. For the corresponding Cartesian products of cycles, this strengthens Špacapan's nonexistence theorem by replacing the linear radius condition $r\ge n$ with a threshold of order $\sqrt{2n}$. We also obtain an improved large-radius nonexistence bound for perfect Lee codes, strengthening the bound of Horak and Kim. Furthermore, we show that, for every $\varepsilon>0$, no complete covering exists when $n\ge 3$, $r$ is sufficiently large, and $r^2+(2/\mathrm{e}-\varepsilon)r\ge 2n$. On the positive side, we use near-optimal ternary constant-weight codes to construct almost complete coverings whose pairwise disjoint shadows cover a $1-o(1)$ proportion of the first outer Lee shell as the dimension tends to infinity, for every fixed radius.

Abstract: Let $\mathcal{C}$ be an almost maximum distance separable (AMDS) code over the finite fields with parameters $[n,k,n-k]$ whose minimum-weight codewords support an $(n-k-1)$-$(n,n-k,λ)$ design. We first determine the parameters of the dual code $\mathcal{C}^{\bot}$. We also characterize the design and its complementary design supported by the minimum weight codewords of $\mathcal{C}^{\bot}$. In particular, we present explicit formulas for the weight enumerators of all subset codes of $\mathcal{C}^{\bot}$ with length at least $k+1$. These formulas are completely determined by the cardinality of the subset, the dimension, together with the parameter $λ$, and are independent of the specific choice of subsets. As an application, we answer an open problem posed by Wang, Tang and Ding in 2023 by proving that the $3$-design supported by the minimum-weight codewords in their proposed cyclic dual code is isomorphic to the complementary design of the Witt spherical geometry design.

Abstract: Let \(A=\GR(p^e,r)\) have residue field \(\F_q\), and let \(λ\in A^\times\). We relate coefficient compression of monic divisors of \(X^N-λ\) to support-primitive decomposition of constacyclic codes. At a simple-root length, the support period of every nonconstant monic divisor is preserved at each coefficient precision and equals the order of the multiplicative stabilizer of its root set. For a cyclic Frobenius-orbit polynomial, the multiple equal-difference structure determines its support period, core degree, and least relative degree of an unramified extension over which all its basic-irreducible factors are binomials. Coordinate projections give every constacyclic code over a finite commutative ring a unique maximal interleaving into copies of one support-primitive code. At simple-root lengths over \(A\), the intrinsic period of a code is the greatest common divisor of the periods of its torsion codes. The decomposition yields distance and duality formulas and the arithmetic Singleton bound for free codes. At every length, a nonzero code over \(A\) has the same minimum Hamming distance as its highest torsion code, which gives a refined bound. At repeated-root lengths, identical torsion filtrations can occur with different intrinsic periods, while a class of nonzero codes obtained by substitution from simple-root generators retains the same core and minimum distance. Trace matrices recover the coefficients of basic-irreducible cores using unit pivots, and local Hensel corrections lift compressed factors directly. For \(X^n-1\) with \(p\nmid n\), lifting and output have quasi-linear bit cost in the compressed representation when the arithmetic data and residue factorizations are given.

Abstract: The deep hole problem is a fundamental problem in coding theory, and it has many important applications in code constructions and cryptography. The deep hole problem of Reed-Solomon codes has gained a lot of attention. As a generalization of Reed-Solomon codes under Hamming metric, we study the covering radius and deep holes of Hyperderivative Reed-Solomon (HRS) codes under the Niederreiter-Rosenbloom-Tsfasman (NRT) metric. We determine the NRT covering radius of HRS code and characterize its NRT deep holes via Maximum Distance Separable (MDS) extensions and polynomial remainder conditions. We further relate the enumeration of NRT deep holes to the characteristic polynomial of an associated hyperplane arrangement. We also provide three classes of explicit NRT deep-hole constructions, propose a Cheng-Murray type conjecture for HRS codes under the NRT metric and obtain a conditional low-layer description. Using these characterizations, we derive exact enumeration formulas in several special cases.

Abstract: Function-correcting codes (FCCs) protect the value of a function $f$ of the message against $t$ errors. In the original formulation of Lenz et al. (2023), the encoding is systematic, and the lower bound of $2t$ on the redundancy relies on this form. Recently, D. Ho (arXiv, 2026) showed, for linear functions and linear encodings, that systematicity can cost redundancy. We begin with the OR function to show that the cost is not confined to the linear setting: a non-systematic encoding attains redundancy $1$ while every systematic encoding needs $2t$. For a linear function $f$ and a fixed linear code $C$, let $d_f$ denote the minimum distance between codewords of messages with different function values. Different generator matrices of $C$ assign different codewords to the messages and can give different values of $d_f$. We study which generator matrix of $C$ gives the largest $d_f$. We give an algorithm that constructs an optimal generator matrix and determines the optimal value as the weight of a codeword in a greedy basis of $C$. We then characterize, in terms of information sets, when a generator matrix in systematic form attains this optimum, and give a necessary condition that is checked on the low-weight codewords of $C$ alone. For two-valued functions, we drop both linearity and systematicity. Using the minimality of initial segments of the simplicial order with respect to Hamming neighbourhoods, we show that a non-systematic $(f,t)$-FCC of length $n$ exists if and only if a condition depending on $f$ only through the size of its smaller preimage holds. For $t=1$, redundancy $1$ is sufficient for every nonconstant two-valued function on $\mathbb{F}_2^k$ with $k\ge 10$. For general $t$, redundancy $1$ suffices for all two-valued functions once $k$ is large enough, and for each $s<2t$ we give an upper bound on the threshold in $k$ beyond which redundancy $s$ suffices.

Abstract: We give a proof of the existence of two-sided product expanding codes which, unlike the earlier result of Kalachev and Panteleev (FOCS, 2025), does not rely on explicit constructions of asymptotically optimal locally testable codes ($c^3$-LTCs). For every fixed number of component codes and dimensions whose rates are bounded away from zero and one, we show that independent random linear codes over a sufficiently large prime field have constant two-sided product expansion with probability tending to one. The tradeoff is that our proof requires the characteristic to grow with the block length, while [KP25] takes extensions of $\mathbb{F}_2$. To replace the use of $c^3$-LTCs, we develop several new techniques that we view as interesting in their own right. Instead of working directly over finite fields, we work over the reals and use random Rademacher matrices for the generator matrices of the component codes. From here, we we show that the extendability of $\varepsilon$-closed sets can be reduced to controlling the operator norm of sparse restrictions of carefully chosen Gram matrices. Applying the trace power method to bound this norm reduces to bounding the number of possible labelings of certain closed walks on bipartite graphs, which we bound using the sparsity of the operators and bounds on the number of equivalence classes of weak Wigner words from Anderson and Zeitouni (Probab. Theory Relat. Fields., 2006), which were originally applied to band matrices. Due to our techniques not relying on explicit $c^3$-LTCs, we believe that they form a promising starting point towards showing the existence of product expanding tensor codes where the component codes have non-trivial automorphism groups and coboundary expanding codes that could be used as the local codes of non-cubical complexes, such as simplicial complexes.

Abstract: A subspace design is a collection of subspaces $H_1,\ldots,H_n$ of $\mathbb{F}_q^k$ with the property that no low-dimensional subspace $W$ intersects the collection "too much". Subspace designs and related objects in linear-algebraic pseudorandomness have found a broad range of applications, ranging from list decoding, to derandomizing algorithms. We construct explicit strong subspace designs over every finite field. In the extremal case where the co-dimension $t$ of each $H_i$ is equal to the dimension of $W$, for every constant field size our construction attains $n=Ω(k)$ and matches the probabilistic intersection bound up to a constant factor. All previous constructions required the field size to grow with $t$ (or $k$). Our subspace designs also imply new construction of rank condensers over arbitrary finite fields. This result is the first to achieve an optimal dependence on $k$ while maintaining both a constant output entropy rate and a constant field size. As an application, we construct lossless rank extractors for linear sources of rank $r$, for all $r < q$, with parameters matching those of Guo, Raj, Shangguan and Zhang (FOCS '26), thereby generalizing their result to prime fields and smaller field sizes. Our construction is based on an algebraic-geometric version of the Parvaresh-Vardy codes (Parvaresh-Vardy FOCS '05, Guruswami ECCC '05), extending the framework underlying the condensers of Guruswami, Umans and Vadhan (JACM '09). We view our construction as a linear-algebraic analysis - tailored to affine sources - of the GUV construction, generalized to functions over algebraic curves. More specifically, inspired by Ta-Shma and Umans (CCC 12') we develop a two-level evaluation scheme, where we first evaluate a function on a curve at extension-field points, and then evaluate a corresponding affine-linear polynomial to obtain outputs over the base field.

Abstract: Let $\mathcal C=(f)$ be a $λ$-constacyclic code over $\mathbb F_q$, where $f(X)$ is a monic factor of $X^N-λ$ with nonzero constant term. We introduce the support period $\operatorname{sp}(f)$ of $f(X)$, and define its support-primitive core $f_{\mathrm{sp}}(X)$ as the unique support-primitive polynomial satisfying $f(X)=f_{\mathrm{sp}}(X^s)$, where $s=\operatorname{sp}(f)$. We show that this polynomial relation induces a Hamming-weight-preserving linear isomorphism $\mathcal C\cong\mathcal C_{\mathrm{sp}}^{\, s}$, where $\mathcal C_{\mathrm{sp}}$ is the support-primitive core of $\mathcal C$, and prove that this decomposition is intrinsic to the code. We give two equivalent descriptions of the support period: a coefficient-based one and a roots-based one. In the repeated-root case, the latter is determined by the $p$-adic structure and the stabilizer of the defining function, while in the simple-root case it is determined by the coarsest multiple equal-difference representation of the defining set. We then derive coding-theoretic consequences for the Hamming distance, weight enumerator, covering radius, and Euclidean duality. In particular, the arithmetic Singleton bound of a simple-root constacyclic code is identified with the classical Singleton bound of its support-primitive core. Finally, we apply the decomposition to cyclic codes with reducible generator polynomials and obtain bounds for their arithmetic Singleton values.

Abstract: We study the complexity of the Code Equivalence problem and show that it is polynomially equivalent to several computational automorphism problems for codes. These problems ask for the cardinality (ACOUNT), an orbit partition (APART), and a generating set (AGEN) for the permutation automorphism group of a code. We present deterministic, polynomial-time reductions between Permutation Code Equivalence (PCE) and each of these problems, including a one-shot reduction from search-PCE to AGEN that makes a single oracle call. We present similar reductions between Linear Code Equivalence (LCE) and analogous problems for the monomial automorphism group of a code. All of our reductions work for any linear codes.

Abstract: The minimal Kolmogorov complexity of a string within Hamming distance r of a given string x is the algorithmic rate-distortion function of x, and Vereshchagin and Vitanyi characterized completely which shapes it can have. This paper is about the opposite extreme. For a binary string x of length n let g_x(r) denote the maximal Kolmogorov complexity of a string within Hamming distance r of x; we study which values, and more generally which functions of r, this quantity can attain. First we characterize, up to an additive error O(log n), the possible values of the triple (C(x),r,g_x(r)): writing r_k for the radius of a Hamming ball of cardinality about 2^k, a triple (k,r,l) is realizable if and only if log V(r_k + r) < l < min{n, k+log V(r)}, where V(a) is the cardinality of a ball of radius a. In particular, for r_k+r > n/2 both bounds collapse to n and only l = n is realizable. The two ends of this interval correspond to the two extreme ways of placing a set of complexity k in the cube: a single Hamming ball, where the lower bound comes from Harper's isoperimetric inequality, and an error-correcting code, which for the intermediate parameters we relax to a family of centers with bounded covering multiplicity, in the spirit of list decoding. Then we turn to the function r -> g_x(r) as a whole: we establish four properties that it always has, and show that the minimal and the maximal functions consistent with these properties are both attained, for every complexity level k. Which intermediate profiles are attainable remains open.

Abstract: In this article, we consider the affine Grassmann code $C^{\mathbb A}(3,6)$, obtained from the affine open cell ${\mathbb A}^9$ of the Grassmannian $G_{3,6}$. We exploit the representation of codewords as linear combinations of minors of all sizes of a generic $3\times3$ matrix and classify them according to the largest size of a minor occurring with a nonzero coefficient. Using this classification, we determine all possible Hamming weights of codewords of $C^{\mathbb A}(3,6)$ and, for each weight, compute the number of codewords attaining that weight. Consequently, we obtain the complete weight spectrum of the affine Grassmann code $C^{\mathbb A}(3,6)$.