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A lower bound on the number of uncorrectable errors of weight half the minimum distance is derived for binary linear codes satisfying some condition. The condition is satisfied by some primitive BCH codes, extended primitive BCH codes,…
We introduce a linear programming method to obtain bounds on the cardinality of codes in Grassmannian spaces for the chordal distance. We obtain explicit bounds, and an asymptotic bound that improves on the Hamming bound. Our approach…
The code equivalence problem is central in coding theory and cryptography. While classical invariants are effective for Hamming and rank metrics, the sum-rank metric, which unifies both, introduces new challenges. This paper introduces new…
Let $\mathrm{Sym}_q(m)$ be the space of symmetric matrices in $\mathbb{F}_q^{m\times m}$. A subspace of $\mathrm{Sym}_q(m)$ equipped with the rank distance is called a symmetric rank-metric code. In this paper we study the covering…
The quantum Hamming bound was originally put forward as an upper bound on the parameters of nondegenerate quantum codes, but over the past few decades much work has been done to show that many degenerate quantum codes must also obey this…
We present a PAC-Bayes-style generalization bound which enables the replacement of the KL-divergence with a variety of Integral Probability Metrics (IPM). We provide instances of this bound with the IPM being the total variation metric and…
We consider the coding problem in the Stiefel manifold with chordal distance. After considering various low-dimensional instances of this problem, we use Rankin's bounds on spherical codes to prove upper bounds on the minimum distance of a…
We investigate perfect codes in $\mathbb{Z}^n$ under the $\ell_p$ metric. Upper bounds for the packing radius $r$ of a linear perfect code, in terms of the metric parameter $p$ and the dimension $n$ are derived. For $p = 2$ and $n = 2, 3$,…
In this paper, we first generalize the class of linear codes by Ding and Ding (IEEE TIT, 61(11), pp. 5835-5842, 2015). Then we mainly study the augmented codes of this generalized class of linear codes. For one thing, we use Gaussian sums…
The Heisenberg limit provides a quadratic improvement over the standard quantum limit, and is the maximum quantum advantage that quantum sensors could provide over classical methods. This limit remains elusive, however, because of the…
In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and…
This paper investigates the relationship between the rank weight distribution of a linear code and that of its dual code. The main result of this paper is that, similar to the MacWilliams identity for the Hamming metric, the rank weight…
Error control is significant to network coding, since when unchecked, errors greatly deteriorate the throughput gains of network coding and seriously undermine both reliability and security of data. Two families of codes, subspace and rank…
This paper investigates general properties of codes with the rank metric. We first investigate asymptotic packing properties of rank metric codes. Then, we study sphere covering properties of rank metric codes, derive bounds on their…
Linear error-correcting codes form the mathematical backbone of modern digital communication and storage systems, but identifying champion linear codes (linear codes achieving or exceeding the best known minimum Hamming distance) remains…
Spherical codes, with a rich history spanning nearly five centuries, remain an area of active mathematical exploration and are far from being fully understood. These codes, which arise naturally in problems of geometry, combinatorics, and…
The binary Hamming codes with parameters $[2^m-1, 2^m-1-m, 3]$ are perfect. Their extended codes have parameters $[2^m, 2^m-1-m, 4]$ and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes…
We introduce a $W^*$-metric space, which is a particular approach to non-commutative metric spaces where a \textit{quantum metric} is defined on a von Neumann algebra. We generalize the notion of a quantum code and quantum error correction…
We study lower bounds on the optimal error probability in classical coding over classical-quantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for…
This paper presents an achievability bound that evaluates the exact probability of error of an ensemble of random codes that are decoded by a minimum distance decoder. Compared to the state-of-the-art which demands exponential computation…