coding theory developers

Abstract: We study completely normal elements of $\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, that is, elements that are normal over every intermediate field, through partial Lagrange resolvents. In these coordinates, complete normality splits into independent components, in a way closely related to Hachenberger's decomposition theory. Our main tool is a structure theorem for the components; together with a finite computation, it shows that every component has weight (the sum of the probabilities that its conditions fail) at most $3/4$, uniformly in $n$. Combined with the classical character-sum method, this gives an explicit sufficient criterion for the existence of a primitive completely normal element. We verify the criterion for all $n>q$ except $70$ pairs $(q,n)$, which we settle with explicit elements found by computer. Since the case $n\le q$ is due to Garefalakis and Kapetanakis, this confirms the conjecture of Morgan and Mullen (1996) that every $\mathbb{F}_{q^n}$ contains a primitive element that is completely normal over $\mathbb{F}_q$. The same estimates give two further results. First, Hachenberger's conjectured lower bound $(q-1)^{n'}q^{(p^a-1)n'}$ for the number of completely normal elements, where $p$ is the characteristic and $n=p^an'$ with $p\nmid n'$, holds for all pairs $(q,n)$; it was known for regular pairs. Second, primitivity and complete normality are asymptotically independent: as $q^n\to\infty$, the number of primitive completely normal elements is $(1+o(1))\,\varphi(q^n-1)/(q^n-1)$ times the number of completely normal elements (for fixed $n$ this was known). Our arguments for $n>q$ and for both further results, including every computer-assisted step, are formalised in Lean 4 with Mathlib; the formalisation takes the case $n\le q$ and an explicit bound of Rosser and Schoenfeld as axioms and relies on Lean's compiler and interpreter for the large computations.

Abstract: Let $1\le h\le n-2$ and $g=\gcd(n,h+1)$. We prove that, for every prime power $q$, a maximum $h$-scattered $\mathbb{F}_q$-subspace of $V(r,q^n)$ exists whenever $(h+1)\mid rn$ and $$r\ge\frac{h+1}{g}\bigl(h(n-h-2)+1\bigr).$$ In particular, for fixed $n$ and minimum distance $δ\in\{2,\ldots,n-1\}$, $\mathbb{F}_{q^n}$-linear MRD codes of length $N$ exist for every $q$ and every sufficiently large $N$ such that $n$ divides $(n-δ+1)N$, with a bound on $N$ depending only on $n$ and $δ$. We take the $\mathbb{F}_{q^s}$-span of the columns of a random matrix $A$ over the algebra $R=\mathbb{F}_{q^n}\otimes_{\mathbb{F}_q}\mathbb{F}_{q^s}$, for a suitable $s$. Such a subspace is maximum $h$-scattered if and only if the kernel of $A$ contains no nonzero vector whose expansion with respect to an $\mathbb{F}_q$-basis of $\mathbb{F}_{q^s}$ has rank weight at most $h+1$. When $\gcd(n,s)=1$, $R$ is a field, and a first-moment argument, counting these vectors up to multiplication by $\mathbb{F}_{q^n}^*\mathbb{F}_{q^s}^*$, shows that with positive probability the kernel contains none of them if $s>h(n-h-2)$. When $d=\gcd(n,s)>1$, $R$ is a product of $d$ fields, and vectors vanishing in some factors lie in the kernel with larger probability. We control them by proving that, for an $(h+1)$-dimensional $\mathbb{F}_q$-subspace $D$ of $\mathbb{F}_{q^n}$, the elements of $D\otimes_{\mathbb{F}_q}\mathbb{F}_{q^s}$ vanishing outside $k$ given factors, $0<k<d$, form an $\mathbb{F}_{q^s}$-space of dimension at most $(h+1)k/d$, with equality only if $D$ is linear over a subfield of $\mathbb{F}_{q^n}$ larger than $\mathbb{F}_q$. Taking $s=gr/(h+1)$ gives the bound on $r$. Finally, if $(n-h-1)\mid(h+1)$, then for every $r$ satisfying the above inequality there is an $h$-scattered $\mathbb{F}_q$-subspace of $V(r,q^n)$ of dimension $\lfloor rn/(h+1)\rfloor$, without a divisibility condition on $rn$.

Abstract: In this paper, we investigate the Linear Complementary Pairs (LCPs) of Maximum Distance Separable (MDS) codes that arise from elliptic curves. We establish geometric conditions for LCPs of elliptic codes and prove that they cannot consist entirely of MDS codes when evaluated over all rational points of the curve. To address this, we construct LCPs of MDS elliptic codes by restricting evaluation divisors to subgroup cosets. We apply this method to obtain explicit infinite families of LCPs of MDS codes over Mordell and supersingular elliptic curves, including large-length constructions over quadratic extensions.

Abstract: The $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive codes are subgroups of $\mathbb{Z}_p^{α_1}\times\mathbb{Z}_{p^2}^{α_2}$, and they can be seen as linear codes over $\mathbb{Z}_p$ when $α_2=0$, as $\mathbb{Z}_{p^2}$-additive codes when $α_1=0$, and as $\mathbb{Z}_2\mathbb{Z}_4$-additive codes when $p=2$. A $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear generalized Hadamard (GH) code is a GH code over $\mathbb{Z}_p$ which is the Gray map image of a $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive code. For every prime $p$, there are recursive constructions of $\mathbb{Z}_p\mathbb{Z}_{p^2}$-linear GH codes $H_p^{t_1,t_2}$ with $α_1\neq0$ and $α_2\neq0$, and of $\mathbb{Z}_{p^s}$-linear GH codes $\bar H_p^{t_1,\dots,t_s}$. We show that the kernel blocks of a nonlinear code $H_p^{t_1,t_2}$ give equivalent punctured codes, and that these punctured codes are linear. Any of them is called the derived code of $H_p^{t_1,t_2}$. We also determine the ranks of the codes $H_p^{t_1,t_2}$ and $\bar H_p^{t_1,t_2}$. For odd $p$, the rank of $H_p^{t_1,t_2}$ depends on the Bernoulli numbers modulo $p$. By comparing the derived codes and the ranks, we show that, for odd $p$, no code $H_p^{t_1,t_2}$ is equivalent to any of these $\mathbb{Z}_{p^s}$-linear GH codes. For $p=2$, a nonlinear code $H_2^{t_1,t_2}$ is equivalent to a $\mathbb{Z}_{2^s}$-linear Hadamard code if and only if $t_2\geq2$. Finally, we count the nonequivalent codes in the union of both families.

Abstract: We analytically study the evolution of the energy function of Dense Associative Memories (DenseAMs) as they train on randomly sampled codewords of random expander low-density parity-check codes (a.k.a. solutions of random XORSAT problems). We identify two distinct regimes and a transition between them, reminiscent of grokking in neural networks. A regime of small training set size is where the model memorizes a few training examples but cannot discover the underlying rules of the entire codeword dictionary generative process. The opposite regime, corresponding to a larger training set size, is where the model has uncovered the sparse constraints that generated the entire data set after seeing only polynomially many patterns. We also design a novel class of DenseAMs with a thresholding learning rule, and show that any amount of thresholding enables the model to perform simplified sparse Hamiltonian learning, which exponentially reduces the number of training examples needed to learn the true underlying data set. In this regime, the model learns how to discover $\exp[\mathrm{O}(n)]$ solutions from only $\mathrm{O}(\log n)$ training samples, which is asymptotically optimal for any learning algorithm. When training on codewords of certain random codes, the model can even learn the underlying code when a small O(1) fraction of bits are latent variables. These latent variables are not shown to the model during learning, but will be discovered by sparsifying observed higher-order correlations. For certain random codes, we show that a simple and conservative algorithm learns a minimal latent representation for the data set. This provides a transparent, physics-motivated, and analytically tractable toy model for how deep latent representations emerge in an associative memory training.

Abstract: Ferrers diagram rank-metric codes are essential components of the multilevel construction of constant dimension codes for random network coding. This paper aims to construct optimal Ferrers diagram rank-metric codes, focusing on the diagram \([3,3,3,5]\) with rank distance \(3\). We construct an optimal \([[3,3,3,5],6,3]_q\) Ferrers diagram rank-metric code for every prime power \(q\), thereby answering an open problem posed by Etzion et al. in 2016~\cite{Etzion2016}. The construction first builds a \([3\times4,4,3]_q\) MRD code and then proves that the Delsarte dual of its puncturing on the last column contains a \([3\times3,3,3]_q\) MRD code, which comes from semifield codes. We further construct optimal Ferrers diagram rank-metric codes for two infinite families of Ferrers diagrams: the first relies on the \([[3,3,3,5],6,3]_q\) construction and holds for every prime power \(q\), while the second is binary and obtained directly by the puncturing-inclusion construction.

Abstract: We study the behavior of the $k$-th-order nonhomomorphicity of Boolean functions under restriction to affine subspaces. For odd $k\ge3$ and $0\le t<n$, we introduce the $t$-restriction $k$-th-order nonhomomorphicity $P_{k,t}(f)$, defined as the minimum of $P_k(f_{\mid A})$ over affine spaces of codimension at most $t$. For a direct sum of $m$ monomials of degrees at least two, we determine $P_{k,t}$ exactly. When $t<m$, a minimum is obtained by setting one variable equal to zero in each of the $t$ monomials of lowest degree. When $m\le t<n$, the minimum is zero. The proof combines multiplicativity of even Walsh moments with a contraction argument for affine restrictions. We establish a general formula for the Walsh transform of a Boolean function restricted to an affine subspace of arbitrary codimension, yielding an exact spectral characterization of the $t$-restriction normalized nonlinearity $\rnnl_t(f)$. As odd $k$ tends to infinity, $(1-2P_{k,t}(f))^{1/(k+1)}$ converges to $1-2\rnnl_t(f)$. We further prove that every non-affine function has a hyperplane restriction with strictly smaller odd-order nonhomomorphicity. Finally, we determine $\rnnl_1(T_{d,n})$ exactly for all threshold functions with $n\ge3$. This proves the two threshold-function conjectures left open in~\cite{SergeAnaPante2025}, including the previously unresolved odd-majority case, where a minimizing hyperplane is $x_1+x_2=0$.

Abstract: A verifier that accepts or rejects whole answers reveals little: under a flat prior over $k$-bit answers, zero error needs $2^k-1$ verifications. The usual remedy is to solve related source tasks, either all first, as a curriculum does, or interleaved with verification. We price this remedy in information and in calls. With an exact verifier, the least causal information that any interleaving of source calls and $n$ verifications needs to succeed with probability $s$ is a list rate-distortion function, attained by one observation before any verification. It lower-bounds the expected number of binary source calls, which designed sources meet within $1+\log_25$ calls for unique answers and within a logarithmic term in general, where no additive constant suffices. With an exact verifier and fixed sources, moving every call before the first verification preserves all hard caps on calls, although interleaving can save unboundedly many expected calls; under a noisy verifier, source-first protocols can lose unbounded factors in information and in error. For linear banks over $\mathbb{F}_2$, optimal accuracy has a closed form, and after a polynomial-time reduction the budget profile is computable in time $2^{O(h^2)}\operatorname{poly}(J,k+h)$ for $J$ sources and nuisance dimension $h$. In these banks, for zero error under a hard cap, the calls beyond the rounded-up information price are exactly those spent on nuisance. Every numbered result apart from two clauses about the planner is machine-checked in Lean 4, assuming two published results. Used as a ruler, the frontier shows a small transformer using all delivered bits at latent dimension $5$ and none at $11$ within fixed training budgets; in a test with predictions recorded before training, low XOR degree of the target bits did not suffice for their use.

Abstract: Let $k>t\ge 2$ be fixed integers, and let $G$ be a permutation group on a set of $v$ points. A $G$-invariant packing is free if every block has $|G|$ distinct images under $G$. We give two conditions on $G$ under which, as $v\to\infty$, there is a free $G$-invariant $t$-$(v,k,1)$ packing with $(1-o(1))\binom{v}{t}/\binom{k}{t}$ blocks. This is a $G$-invariant form of the Erdős--Hanani theorem on asymptotically optimal packings. For semiregular groups $G$, packings with these properties exist if and only if all but $o(v^t)$ of the $t$-subsets of points have trivial setwise stabiliser. This property is satisfied by every semiregular group for $t\ge 3$, and by every cyclic semiregular group for $t=2$. As a consequence, the Johnson bound is asymptotically attained at all lengths by optical orthogonal codes of constant weight $w$ and constant correlation $λ$ with $w\geλ+2$, by their multidimensional and signature pattern versions, and by cyclic and quasi-cyclic constant-weight codes. The same method gives a form of the theorem for group divisible packings that are invariant under a cyclic group. This determines the asymptotic maximum size of two-dimensional optical orthogonal codes with at most one pulse per wavelength.

Abstract: We study the problem of constructing highly nonlinear vectorial maps $F: \mathbb F_2^n \to \mathbb F_2^m$. Concretely, we want an $F$ and an $A = A(m,n)> 0$ as small as possible, so that for every affine map $L: \mathbb F_2^n \to \mathbb F_2^m$ (of the form $L(x) = M x + b $) we have: $$\mathrm{agree}(F, L) := |\{ x \in \mathbb F_2^n \mid F(x) = L(x) \}| \leq A.$$ Such questions have been studied by Nyberg (1991,1993), Carlet and Ding (2004,2007), Liu, Mesnager and Chen (2017), Nagy (2025), and Biryukov, Turecek, and Udovenko (2026). There is a classical method of constructing such functions from bent-functions and Fourier analytic ideas; the best bound achievable by this method is: $$ A(m,n) = Θ(2^{n-m} + 2^{n/2}),$$ and in particular, is never smaller than $2^{n/2}$. In this work, we show how to construct highly nonlinear functions beyond this Fourier bound. Concretely, we show how to construct for every $γ>0$, a function $F: \mathbb F_2^n \to \mathbb F_2^m$ with $m = O_γ(n)$, achieving $$ A(m,n) \leq (1 + γ)^n.$$ Surprisingly, we even achieve the same quantitative behavior for the much harder question of having low agreement with $m$-tuples of degree $d$ polynomials $Q: \mathbb F_2^n \to \mathbb F_2^m$, with $m = O_{γ, d}(n)$. Here the previously best bounds were of the form $A(m,n) = O( 2^{-\frac{n}{2^{d+1}}} \cdot 2^n )$ of Ben-Sasson and Kopparty (2010), based on Gowers-norm-type arguments. All our results generalize to all finite fields $\mathbb F_q$ in place of $\mathbb F_2$. Our methods are based on a new connection to classical results on counting solutions to systems of polynomial equations via algebraic methods. This connection brings us to basic questions in combinatorics, about graphs and hypergraphs with simultaneously a small number of edges and independent sets.

Abstract: Hyun, Kim, Wu and Yue constructed optimal and minimal binary linear codes from order ideals of hierarchical posets with two levels. Two different generalizations of the underlying antichain (simplicial complex) setting to odd characteristic are known: down-sets of $\mathbb{F}_p^n$ under the componentwise order, and support-closed subsets of $\mathbb{F}_q^m$. No generalization of the poset setting itself has appeared. We introduce generalized order ideals of a poset of order $p-1$, obtained by attaching multiplicities in ${0,\dots,p-1}$ to the elements of a poset, and study the two natural notions of order ideal that arise for hierarchical posets with two levels. Whenever the ideal meets the upper level, the resulting defining sets are neither down-sets nor support-closed. We determine the weight distributions of the associated complement codes, exhibit a family of Griesmer codes in which the upper element carries an arbitrary multiplicity, and, via the characteristic function of a generalized order ideal, obtain an infinite family of minimal $p$-ary codes of length $p^n-1$ and dimension $n+1$ violating the Ashikhmin--Barg condition.

Abstract: We use $m$-dimensional partial spreads of $\fp^{2n}$, where $p$ is an odd prime, $n$ is a positive integer and $m$ divides $n$, to construct two classes of bent functions from $\fp^{2n}$ to $\fp$. Our construction generalizes the classes of $p$-ary $\cPS^{-}$ and $\cPS^{+}$ bent functions proposed by P. Lison\v ek and H. Y. Lu (Des. Codes Cryptogr. 73 (2014), 209--216).

Abstract: The (binary) differential-linear connectivity table (DLCT) measures the dependence between an input difference and a linear mask applied to the corresponding output difference. For vectorial Boolean functions, each DLCT entry is one half of an additive autocorrelation value. We extend this relation to functions over finite fields of arbitrary prime characteristic by introducing a level-resolved p-ary differential-linear profile. Its entries are the centered numbers of inputs for which a derivative component has each prescribed trace value in $\mathbb{F}_p$. The discrete Fourier transform of this profile is the family of additive autocorrelations obtained by multiplying the output mask by the nonzero elements of $\mathbb{F}_p$; when p=2, the usual binary identity is recovered. For a fixed input difference, we show that the profiles over all nonzero output masks determine the corresponding DDT row exactly, and we give an explicit inversion formula. We establish a second-moment identity: the total profile energy in one derivative direction is a constant multiple of the squared Euclidean distance between that DDT row and the balanced row. Thus this energy is determined by the full row differential spectrum, not by differential uniformity alone. It follows that all profiles in a direction vanish exactly when the derivative is balanced; for square maps in odd characteristic, this gives a characterization of planarity. As concrete odd-characteristic examples, we determine the complete profile of the monomial $x^{p^k+1}$ and derive an exact Kloosterman-sum formula for the inverse monomial. Finally, we determine the behavior of the profiles under equivalence. EA-equivalence reindexes the input and output masks and translates the trace level, whereas a general CCZ equivalence may mix several derivative directions. Nevertheless, for square maps the global nontrivial profile energy is CCZ-invariant.

Abstract: This paper introduces a new polynomial model for the exact variant of the Syndrome Decoding Problem (SDP) in the binary case. The model is based on elementary symmetric polynomials. We estimate the computational complexity of solving the corresponding polynomial system by establishing bounds on the degree of regularity and on the solving degree of the ideal associated to the model. The complexity estimate is lower than for previous polynomial models. We also provide a variant of the model whose complexity depends directly on the specific instance of the SDP and is lower than for the first model. Finally, we discuss how to apply our approach to solve other variants of the SDP.

Abstract: In this paper, we settle a conjecture of Polujan and Pott by constructing an explicit, infinite family of Maiorana--McFarland bent functions $f_t$ in $2(2^t-1)$ variables with algebraic degree $°(f_t) = t + 1$ for any integer $t \ge 2$. Our construction builds upon a minimal commutative algebra $I_t$, which naturally induces a triangular set-partition polynomial permutation $P_t$. By identifying an elementary abelian subgroup within the direct sum $ I_t \oplus I_t^*$, we establish an explicit nonlinear coordinate transformation that pulls $f_t$ back to a canonical quadratic form. This linearizes the translation development $\operatorname{Dev}(D_{f_t})$ under an exotic group structure and proves that it is isomorphic to the classical symplectic design $S^\pm(2(2^t-1))$, thereby fully resolving the conjecture.

Abstract: Proximity gaps are central to the soundness of interactive oracle proofs of proximity (IOPPs) and polynomial commitment schemes (PCSs). An $[n,k,d]$ linear code $C\subseteq\mathbb F^n$ has a $δ$-proximity gap with error $ε$ if, for every $u_0,u_1\in\mathbb F^n$, either all points on $\ell_{u_0,u_1}=\{u_0+αu_1:α\in\mathbb F\}$ are $δ$-close to $C$, or at most an $ε$ fraction are. Although proximity gaps for Hamming-metric codes are well understood, their rank-metric counterparts remain largely unexplored despite their applications in coding theory and cryptography. In this work, we study proximity gaps for linear rank-metric codes and their cryptographic applications. First, we show that every $[n,k,d]$ linear rank-metric code $C$ over $\mathbb F_{q^m}$ admits a proximity gap for every $δ\le(d-1)/(3n)$, with error at most $q^{e+1}/q^m$, where $e=\lfloorδn\rfloor$. For Gabidulin codes, we improve the gap to $(d-1)/(2n)$ with error $10q^{n-1}/q^m$. These two proximity gaps match those for general linear Hamming-metric codes and Reed--Solomon (RS) codes, respectively. We prove the $(d-1)/(2n)$ bound is tight by constructing an infinite family of constant-rate Gabidulin codes and affine lines $\ell_{u_0,u_1}$ on which a $1-o(1)$ fraction of points are $d/(2n)$-close to the code, while $u_1$ is at least $3d/(4n)$-far from it. At the $d/(3n)$ gap, we also give a counterexample establishing a lower bound on $ε$. As applications, we construct an IOPP for interleaved Gabidulin codes by adapting the Ligero IOPP for interleaved RS codes. We then adapt the Ligero-based PCS for ordinary polynomials to obtain a $q$-linearized polynomial commitment scheme. To our knowledge, this is the first PCS framework based on rank-metric error-correcting codes.

Abstract: Let $R_2 = \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ ($u^2 = 0$), where $p$ is an odd prime and $m, s \in \mathbb{N}$. For $p \equiv 3, 5 \pmod 7$ with $\gcd(m, 6) = 1$, the cyclotomic polynomial $Φ_7(x)$ is irreducible over $\mathbb{F}_{p^m}$. This yields a direct sum decomposition $C = C_1 \oplus C_2$ for any cyclic code $C$ of length $7p^s$ over $R_2$, where $C_1$ has length $p^s$ and $C_2$ is a $7$-cyclotomic code of length $6p^s$. We classify $7$-cyclotomic codes into four disjoint generator-based types and calculate exact cardinalities using residue and torsion subcodes. Furthermore, explicit generators for the Euclidean dual codes $C^\perp$ are determined. As operational applications of these classified codes, we construct new families of quantum stabilizer codes via the CSS framework and establish parameter criteria for linear codes with complementary duals (LCD codes).