Related papers: On zero neighbours and trial sets of linear codes
In his Ph.D. disseration, Feldman and his collaborators define the linear programming decoder for binary linear codes, which is a linear programming relaxation of the maximum-likelihood decoding problem. This decoder does not, in general,…
We describe a generalization of the group testing problem termed symmetric group testing. Unlike in classical binary group testing, the roles played by the input symbols zero and one are "symmetric" while the outputs are drawn from a…
We show that errors in data transmitted through linear codes can be thought of as codewords of minimum weight of new linear codes. To determine errors we can then use methods specific to finding such special codewords. One of these methods…
Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for an odd prime power $q$, we construct some class of linear code over finite field $\mathbb{F}_q$ with defining set be the preimage of…
Our main technical result is that, in the coset leader graph of a linear binary code of block length n, the metric balls spanned by constant-weight vectors grow exponentially slower than those in $\{0,1\}^n$. Following the approach of…
We develop an algebraic theory of supports for $R$-linear codes of fixed length, where $R$ is a finite commutative unitary ring. A support naturally induces a notion of generalized weights and allows one to associate a monomial ideal to a…
This paper considers the problem of testing whether there exists a non-negative solution to a possibly under-determined system of linear equations with known coefficients. This hypothesis testing problem arises naturally in a number of…
Let n be an even positive integer and F be the field \GF(2). A word in F^n is called balanced if its Hamming weight is n/2. A subset C \subseteq F^n$ is called a balancing set if for every word y \in F^n there is a word x \in C such that y…
A decoding algorithm for $q$-ary low-density parity-check codes over the $q$-ary symmetric channel is introduced. The exchanged messages are lists of symbols from $\Fq$. A density evolution analysis for maximum list sizes $1$ and $2$ is…
The weight spectra of MDS codes of length $ n $ and dimension $ k $ over the arbitrary alphabets are studied. For all $ q $-ary MDS codes of dimension $ k $ containing the zero codeword, it is shown that all $ k $ weights from $ n $ to $…
A locally testable code (LTC) is an error correcting code with a property tester. The tester tests if a word is codeword by reading constant random bits and rejects the word with probability proportional to the distance from the word to the…
This paper considers the problem of testing whether there exists a solution satisfying certain non-negativity constraints to a linear system of equations. Importantly and in contrast to some prior work, we allow all parameters in the system…
We construct linear codes over the finite field Fq from arbitrary simplicial complexes, establishing a connection between topological properties and fundamental coding parameters. First, we study the behaviour of the weights of codewords…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
We present simple, self-contained proofs of correctness for algorithms for linearity testing and program checking of linear functions on finite subsets of integers represented as n-bit numbers. In addition we explore a generalization of…
Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. Most previous results obtained so far were for cyclic codes with no more than three zeroes. Inspired by the works…
Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for a prime $p$, we determine the explicit complete weight enumerators of a family of linear codes over $\mathbb{F}_p$ with defining set…
We suggest a new approach to obtain bounds on locally correctable and some locally testable binary linear codes, by arguing that these codes (or their subcodes) have coset leader graphs with high discrete Ricci curvature. The bounds we…
In this paper, we have introduced the concepts of support distribution and the support enumerator as refinements of the classical weight distribution and weight enumerator respectively, capturing coordinate level activity in linear block…
We give a characterization for the binary linear constant weight codes by using the symmetric difference of the supports of the codewords. This characterization gives a correspondence between the set of binary linear constant weight codes…