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We devise an analytically simple as well as invertible approximate expression, which describes the relation between the minimum distance of a binary code and the corresponding maximum attainable code-rate. For example, for a rate-(1/4),…
In this article we address the computational hardness of optimally decoding a quantum stabilizer code. Much like classical linear codes, errors are detected by measuring certain check operators which yield an error syndrome, and the…
Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study binary linear complementary dual $[n,k]$ codes with the largest minimum weight among all binary…
Linear computation coding is concerned with the compression of multidimensional linear functions, i.e. with reducing the computational effort of multiplying an arbitrary vector to an arbitrary, but known, constant matrix. This paper…
Minimal codewords have applications in decoding linear codes and in cryptography. We study the maximum number of minimal codewords in binary linear codes of a given length and dimension. Improved lower and upper bounds on the maximum number…
In this article we consider linear codes coming from skew-symmetric determinantal varieties, which are defined by the vanishing of minors of a certain fixed size in the space of skew-symmetric matrices. In odd characteristic, the minimum…
We examine an error-correcting coding framework in which each coded symbol is constrained to be a function of a fixed subset of the message symbols. With an eye toward distributed storage applications, we seek to design systematic codes…
Binary codes are widely used to represent the data due to their small storage and efficient computation. However, there exists an ambiguity problem that lots of binary codes share the same Hamming distance to a query. To alleviate the…
In the 2017 paper by Dougherty, Kim, Ozkaya, Sok, and Sol\'e about the linear programming bound for LCD codes the notion $\mathrm{LCD}[n,k]$ was defined for binary LCD $[n,k]$-codes. We find the formula for $\mathrm{LCD}[n,2]$.
In this paper we give a randomized reduction for the Rank Syndrome Decoding problem and Rank Minimum Distance problem for rank codes. Our results are based on an embedding from linear codes equipped with Hamming distance unto linear codes…
New lower bounds on the minimum average Hamming distance of binary codes are derived. The bounds are obtained using linear programming approach.
We consider linear codes over a finite field of odd characteristic, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a code word is derived. Using this formula, we have…
The study of linear codes over a finite field of odd cardinality, derived from determinantal varieties obtained from symmetric matrices of bounded rank, was initiated in a recent paper by the authors. There, one found the minimum distance…
Finding sparse vectors is a fundamental problem that arises in several contexts including codes, subspaces, and lattices. In this work, we prove strong inapproximability results for all these variants using a novel approach that even…
Minimum $k$-Section denotes the NP-hard problem to partition the vertex set of a graph into $k$ sets of sizes as equal as possible while minimizing the cut width, which is the number of edges between these sets. When $k$ is an input…
In this paper, we consider the problem of distinguishing the noisy codewords of a known binary linear block code from a random bit sequence. We propose to use the generalized likelihood ratio test (GLRT) to solve this problem. We also give…
The efficient sparse coding and reconstruction of signal vectors via linear observations has received a tremendous amount of attention over the last decade. In this context, the automated learning of a suitable basis or overcomplete…
Constant dimension codes, with a prescribed minimum distance, have found recently an application in network coding. All the codewords in such a code are subspaces of $\F_q^n$ with a given dimension. A computer search for large constant…
Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study the largest minimum weight $d(n,k)$ among all binary linear complementary dual $[n,k]$ codes. We…
We classify 8-divisible binary linear codes with minimum distance 24 and small length. As an application we consider the codes associated to nodal sextics with 65 ordinary double points.