error correcting code developers

Abstract: We prove that the covering radius of the binary first-order Reed-Muller code RM(1,9) is exactly 242. Equivalently, the maximum nonlinearity of a Boolean function in nine variables is 242. Our proof combines a one-point rigidity theorem with a spectral and computational exclusion of the two larger candidate nonlinearities. We first show that every nine-variable Boolean function of nonlinearity 243 admits a unique one-point modification that raises its nonlinearity to 244: the point to be changed is the vector sum of its support. This part uses exact Walsh-update identities, integer interpolation, rigidity of a quadratic residue, and a sharp Walsh l1 estimate for cubic Boolean functions. To exclude nonlinearity 244, we use Parseval's identity and congruences of integral Walsh coefficients to force an exceptionally restricted spectrum. An associated cubic residue phase must attain the extremal Walsh l1 bound 10240. We eliminate one equality spectrum by hyperplane slicing; the other forces the original Boolean function from the signs of the cubic phase and yields a double-complement-pairing identity for its cubic top part. The classification of alternating trilinear forms over F2, an exact filter of their representatives, and two exhaustive enumerations of 218 quadratic supplements then rule out every remaining candidate. Together with the known construction attaining nonlinearity 242, these arguments establish the asserted covering radius.

Abstract: Finding the roots of a polynomial over a finite field $\mathbb F_q$ in deterministic polynomial time is a long-standing open problem. Classical deterministic methods work in the multiplicative group $\mathbb F_q^\times$ and are efficient only when $q-1$ is smooth, which is rarely the case. We propose a method that replaces $\mathbb F_q^\times$ by the groups of rational points of an elliptic curve $E$ and of its quadratic twist $\widetilde E$. The computation splits into an offline phase, which depends only on $\mathbb F_q$ and outputs a short \emph{elliptic preprocessing datum} $\mathcal D$, and an online phase, which processes the input polynomial. Writing $Ξ(\mathcal D)$ for the largest prime factor of $\lvert E(\mathbb F_q)\rvert\cdot\lvert\widetilde E(\mathbb F_q)\rvert$, the online phase is deterministic and finds the roots of any $f\in\mathbb F_q[X]$ of degree $n$ within $\widetilde O(Ξ(\mathcal D)^2\,n\log^2 q+n^ω\log q+n\log^3 q)$ operations in $\mathbb F_q$. A precursor of our method is an algorithm by Rónyai, which recovers the roots through Galois computations over number fields, relies on GRH, and comes with a polynomial bound of unspecified degree; in contrast, our online phase works entirely over $\mathbb F_q$ and is unconditional. Moreover, $\mathcal D$ can be certified deterministically and unconditionally in time polynomial in $\log q$ and $Ξ(\mathcal D)$. We conjecture that every $\mathbb F_q$ of large enough characteristic admits a datum with $Ξ(\mathcal D)\leq(\log q)^{O(1)}$, and support this with a heuristic and with experiments. Constructing such a datum in polynomial time remains open.

Abstract: DNA storage systems retrieve information by randomly sampling synthesized DNA strands, making the number of reads required for successful recovery a fundamental performance measure. This motivates the coverage depth problem: for given code parameters, determine a linear code that minimizes the expected number of randomly sampled columns required to span the entire information space. While MDS codes are known to be optimal whenever they exist, identifying optimal codes in parameter regimes where MDS codes do not exist remains largely open. In this work, we derive a general formula for the expectation in the coverage depth setting, and apply this result to establish the optimality of the $q$-ary simplex code for the parameters that allow its existence. Moreover, we show the optimality of the $q$-ary Hamming code by relating coverage depth to independent sets in the dual code and exploiting log-concavity properties of matroid basis-generating polynomials. We further analyze random linear codes, derive an exact expression for their expected coverage depth, and show that, in the constant-rate regime, their additive gap from the MDS benchmark remains bounded independently of the code length.