Hengzhuo Li | Searcharxiv

arXiv2026

The Optimal Asymptotic Rate of Generalized Covering Codes

Let $G_q$ be an alphabet of size $q\geq2$. We determine the optimal asymptotic rate of generalized covering codes $C\subseteq G_q^n$, whose covering centers in $G_q^{t\times n}$ are constrained to the product form $C^t$. For every fixed integer $t\geq1$ and every $ρ\in[0,1]$, we prove that \[ κ_t(ρ,q)= \begin{cases} 1-H_{q^t}(ρ),&0\leqρ<1-q^{-t},\\ 0,&1-q^{-t}\leqρ\leq1, \end{cases} \] where $κ_t(ρ,q)$ denotes the minimum asymptotic rate $n^{-1}\log_q|C|$ among codes whose $t$-th covering radius is at most $ρn$, and $H_{q^t}$ is the $q^t$-ary entropy function. When $q$ is a prime power, we prove that the same formula holds under the additional requirement that $C\leq\mathbb F_q^n$. Thus, both the product-form constraint and linearity are asymptotically cost-free: the resulting rate is the ordinary sphere-covering rate over an alphabet of size $q^t$. This extends the recent $t=2$ result of Elimelech and Schwartz for codes without a linearity constraint and the classical $t=1$ result of Cohen and Frankl for linear codes, thereby resolving both open problems posed by Elimelech and Schwartz. Our proofs are probabilistic and combine tools from information theory and probabilistic combinatorics, including the method of types, Janson's inequality, the second-moment method, and a structured alteration argument. Direct applications of Janson's inequality and the second-moment method are obstructed by highly dependent pairs of candidate error matrices. We overcome this obstruction by restricting the errors to a balanced exact-type class of optimal exponential size. Standard type-class estimates, together with Shearer's inequality, then give the required bounds on the number of error-matrix pairs whose selected rows have a prescribed difference.

cs.IT↗

arXiv2026

Sharp Bounds and New Constructions for Single-Error Detection and Correction in Analog Codes

We study single-error detection and correction for analog codes over $\mathbb{R}$. The key performance measures are the parameters $Γ_1(\mathcal{C})$ and $Γ_2(\mathcal{C})$, which quantify, respectively, the minimum separation required between large outlying errors that must be detected or located and the magnitude of tolerable perturbations. First, we prove that every real linear $[n,k]$ code $\mathcal{C}$ satisfies \[ Γ_1(\mathcal{C})\ge 2\left\lceil\frac{n}{n-k}\right\rceil. \] Moreover, when $k=n-2$, we prove that every real linear $[n,n-2]$ code $\mathcal{C}$ satisfies \[ Γ_2(\mathcal{C})\ge \frac{1}{\sin^2(π/2n)}. \] Together, these two lower bounds settle all four open problems of Roth concerning the optimality of single-error-detecting and single-error-correcting analog codes. The proof of the first bound is based on a double-induction argument, while the proof of the second combines a zonotope-based geometric characterization of $Γ_2(\mathcal{C})$ with a cyclic sine-product inequality. In addition, we construct analog codes with higher fixed redundancy and show that, for every fixed $r\ge 2$, there exists a class of linear $[n,\ge n-r]$ codes over $\mathbb{R}$ such that \[ Γ_2(\mathcal{C})\le O\left(n^{1+\frac{1}{r-1}}\right). \] This gives a new upper bound in the fixed-redundancy regime, which was not covered by previously known constructions.

cs.IT↗