error correcting code designers

Abstract: Boolean functions are widely used in cryptography and coding theory. A key challenge lies in finding the weight spectrum for all $n$-variable Boolean functions of degree at most $r$, also known as the Reed--Muller code $\RM(r,n)$. In 2023, Carlet and Solé determined the weight spectra of $\RM(n-c,n)$ for $c=3,4$, and conjectured the weight spectrum of $\RM(n-c, n$) for $n\ge2c$. The weight spectra of $\RM(n-5, n)$ and $\RM(n-6,n)$ were determined by Carlet in 2024 and by Lou and Wang in 2025, respectively, confirming the Carlet-Solé conjecture for the cases $c=5,6$. Very recently, Leuenberger et al. and Wei et al. investigated the weight spectrum of $\RM(7,14)$ independently, and there are eight potential weights which have not been determined. Up until now, the weight spectra of the Reed--Muller codes $RM(r,n)$ are still unknown for $r=3,...,n-7$. In this paper, we determine the exact weight spectrum of $\RM(m-7, m)$ for every $m\ge 14$. By combining affine geometry and Fourier analysis, we exclude the conjectured eight weights for $\RM(7,14)$. We then extend the exclusions to all higher dimensions using a geometric transfer. Consequently, we provide a counterexample to the Carlet-Solé conjecture.

Abstract: Let $q=p^f$, where $p$ is prime. We prove that every $q$-ary linear code meeting the Griesmer bound, with positive dimension and minimum distance $d\ge1$, is $p^{\max\{0,ν_p(d)-f+1\}}$-divisible, as asserted in Ward's conjecture. Arbitrary projective column multiplicities are allowed. The geometric ingredient is a bound independent of dimension: a nonzero scalar-invariant function $F:\mathbb{F}_q^k\to\mathbb{F}_p$, with $F(0)=0$ and prime-field algebraic degree at most $(f+s)(p-1)$, vanishes identically on at most $p^s(q-1)$ linear hyperplanes when $0\le s<f$. A truncated digit-sum inequality forces selected coefficient polynomials to be homogeneous of ordinary degree less than $q$. Weighted moments on a projective line then bound their linear factors, while a second-moment argument handles the complementary support configuration. An elementary finite-difference calculation supplies the degree bound for divided weight residues. Endpoint shortenings prove the basic divisibility layers; endpoint lifting, a surjective linear functional, and residual-code induction extend the argument to every valuation. We include the proofs of the classical ingredients, identify their sources, and distinguish the present argument from an earlier reported unpublished proof.

Abstract: Counting satisfying assignments of Boolean formulas is a basic problem in theoretical computer science, with $\#3\text{-}\SAT$ as the standard $\SharpP$-complete problem. More generally, counting the solutions of a system of polynomial equations over $\F_2$ is $\SharpP$-complete. Here we focus on the more structured problem of counting the solutions of a single polynomial equation. For polynomial equations over finite fields, Ehrenfeucht and Karpinski \cite{computationalcomplexityofxorandcountingproblems1990} showed a sharp difference between degrees two and three: quadratic root counting is solvable in polynomial time, while the degree-three problem is $\SharpP$-complete. Their quadratic algorithm is sequential. For fixed finite fields, Ishai et al.~\cite{ishai2012randomizing} later gave deterministic parallel algorithms in odd characteristic and randomized parallel algorithms in characteristic two. We give a deterministic $\NC$ algorithm for exactly counting the solutions of a quadratic polynomial equation over every fixed finite field of characteristic two. Our algorithm separates the radical and uses the absolute trace to realize the bit distinguishing the two nondegenerate finite-field types as the Arf invariant \cite{arf1941untersuchungen} of a quadratic form over $\F_2$. It then recovers that invariant from an integral matrix using Browder's determinant criterion \cite{browder2006complete}. This replaces the randomized canonical-form step in the algorithm of Ishai et al.

Abstract: The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1}\times\mathbb{Z}_4^{α_2}\times\mathbb{Z}_8^{α_3}$, and a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is the Gray map image of such a code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes $\mathcal H^{t_1,t_2,t_3}$, with all $α_i\neq0$, $t_1\geq1$, $t_2\geq0$, and $t_3\geq1$, is known, as are the linearity, kernel dimension, and rank of the corresponding codes $H^{t_1,t_2,t_3}$ of length $2^t$, where $t+1=3t_1+2t_2+t_3$. Yet these invariants do not completely classify the family. Two infinite families of pairs of distinct types share the length, rank and kernel dimension, and for classified lengths $2^t$, $3\leq t\leq11$, such pairs were separated only by computer equivalence tests. In this paper, we introduce an equivalence invariant that resolves these cases. The kernel partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both; the \emph{kernel-block rank profile} is the multiset of the dimensions of the linear span punctured on these blocks. Unlike rank and kernel dimension, which are global, this invariant records how much of the span survives on each block. We compute it for the whole family: there are $2^{t_1+t_2+t_3-1}$ blocks, all of size $2^{2t_1+t_2}$, and the profile takes at most two values $t_2+\binom{t_1+2}{2}$ and $t_2+2+\binom{t_1+1}{2}$, whose difference is $t_1-1$; it is constant precisely when $t_1=1$. Hence, the profile recovers $t_1$, and the length and kernel dimension recover $t_2$ and $t_3$. Two codes of the family with the same length are therefore equivalent if and only if their types coincide, and the number of pairwise nonequivalent such codes of length $2^t$ is $\lfloor(t^2+6)/12\rfloor$ for every $t\geq3$.

Abstract: Let $R=\mathbb{Z}/(2^n-1)\mathbb{Z}$, where $n\geq 3$, and let $S_w\subseteq R$ be the residues whose canonical $n$-digit binary expansion has Hamming weight $w$. We obtain, in particular, an asymptotic formula for the additive energy of $S_w$ \[ E(S_w)=\frac{\left|S_w\right|^4}{|R|}+ \mathcal{O}\left(|R|^3 n^{-3} \right), \] which holds uniformly in $w$. The error term is optimal in order, with a matching lower bound for $w=\lfloor n/2+\sqrt{n} \rfloor$. It follows that triple sums of arbitrary unit dilates have asymptotically uniform representation counts when $\prod_{j=1}^{3} \left|S_{w_j} \right| /\left(|R| n^{-3/5}\right)^3\to\infty$, and that double sums have asymptotically full support when $\left|S_{w_1}\right| \left|S_{w_2} \right|/\left(|R| n^{-3/4}\right)^2\to\infty$. In the proof, modular collisions are represented using a cyclic binary carry automaton; this appears to be a novel approach in this area of problems.