error-correcting code designers

Abstract: Let $X$ be a random vector in $\mathbb{F}_2^m$, and let $X'$ be an independent copy. We prove that, for every $\varepsilon>0$, there is a subspace $G$ with $\dim G\le 2^{320(1+\varepsilon)/\varepsilon}H(X)$ such that $H(q_GX)\le(1+\varepsilon)(H(q_GX+q_GX')-H(q_GX))$. In particular, the quotient entropy is at most $(1+\varepsilon)(H(X+X')-H(X))$. The coefficient is independent of the ambient dimension, and its exponential order in $1/\varepsilon$ is necessary. The structural result is a logarithmic rigidity estimate for finite weighted families of laws with averaged subspace entropy sparsity. A common variational quotient controls both a family and its convolution family without decreasing the original family's normalized mean entropy defect. Posterior families preserve averaged sparsity, and a fresh-index argument combines a local-to-global estimate with the entropic Marton theorem to increase the defect ratio at a fixed multiplicative sparsity cost. We also obtain two-input almost-additivity and concentration of moderate-doubling sets on subspace cosets at the same exponential scale. The constants are not optimized.

Abstract: We construct three binary extremal self-dual codes of length $60$ with weight enumerators for which no binary extremal self-dual codes of length $60$ were previously known to exist. This completes the determination of the weight enumerators that occur for binary extremal self-dual codes of length $60$.

Abstract: It is well known that the sum of two bent functions on $\mathbb{F}^n$ has algebraic degree at most $n/2$ for even $n\ge 4$. Tokareva conjectured that the converse holds: every Boolean function with degree at most $n/2$ can be written as a sum of two bent functions. The conjecture has been verified for even $n\leq 6$ and for several special classes of Boolean functions. We disprove this conjecture by showing that $x_1x_2(x_3x_4+x_5x_6+x_7x_8)$ is not a sum of two bent functions on $\mathbb{F}^8$. We extend this construction to obtain counterexamples in every even dimension $n\ge 8$. Using Shaporenko's reduction, we also obtain explicit counterexamples to Tokareva's derivative conjecture in every even dimension $n\ge 10$.

Abstract: A set $B$ in an abelian group is a $B_h$ set if every $h$-term sum has a unique representation up to permutation; for $h=2$ these are the Sidon sets. We study a weighted removal problem for this collision-free property: if the $h$-fold sum map has small Rényi entropy loss, how much probability mass must be deleted to leave a $B_h$ support? Two Rényi orders naturally arise: a collision order $α$, measuring the entropy loss, and a budget order $β$, controlling how spread out the weighting may be. Existing one-order formulations tie the two together on the diagonal $β=α$. We determine the resulting stability problem on the full $(α,β)$-plane. Stability holds exactly when $β\le1$ and $α\geβ$. Inside this region the optimal deletion rate is polynomial for $β<1$ and logarithmic on the boundary $β=1$, where the leading constant is exact; outside it, stability fails through two distinct mechanisms: a supercritical budget and dilution by light atoms. In each case the limiting defect is computed exactly. The upper bounds follow from a sharp list-coarsening inequality with optimal constant, which also yields an entropy-free removal theorem, a finite combinatorial consequence for moments of the representation function, and extensions to $B_h[g]$ sets. Matching constructions show that the phase boundaries and rates are sharp.

Abstract: Deciding whether a system of multivariate quadratic equations over $\mathbb F_2$ has a solution is a classical NP-complete problem, and remains so for square systems, with as many equations as variables. The hardness of this problem is one of the cornerstones of nowadays post-quantum cryptography. Let $\MQ_0(n)$ and $\MQ_1(n)$ denote the sets of square quadratic systems in $n$ variables having respectively no solutions and exactly one solution. $\cup_{n\geq 2} \MQ_0(n)$ is a coNP-complete language, while $\cup_{n\geq 2} \MQ_1(n)$ lies in DP. It is known that $\lim_{n\to \infty} |\MQ_1(n)|/|\MQ_0(n)|=1$. Here we prove the explicit finite-$n$ bounds \[ |\MQ_0(n)|<|\MQ_1(n)| \le \left(1+\frac{1}{2^n-1}\right)|\MQ_0(n)|, \] More generally, let $Q_d$ be the space of polynomial functions $(\FF_2)^n\to\mathbb F_2$ of degree at most $d$, and let $α_k$ count square systems in $(Q_d)^n$ having exactly $k$ solutions. Then \[ α_0<α_1 \le \left(1+\frac{1}{2^n-1}\right)α_0\,, \qquad 2\le d\le n \,. \] The proof combines matroid and coding-theoretic methods. We interpret $(\FF_2)^n$ as the ground set of the evaluation matroid of $Q_d$, express $α_0$ and $α_1$ through characteristic polynomials, and use a Whitney-type sign-reversing involution to show that the only terms that can push $α_1-α_0$ below $α_1/2^n$ come from the elements of a matroid port. These are identified with minimal-support words of the Reed--Muller code $\RM(n-d-1,n)=\RM(d,n)^\perp$; the required estimate then follows from the MacWilliams identity, the minimum-distance bound $2^{d+1}$, and the even-weight structure of the code.