A graphical representation of binary linear codes (original) (raw)
An introduction to coding sequences of graphs
2016
In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over Z_2 which has the consecutive 1's property (i.e., 1's are consecutive) for columns and obtained a characterization of graphic matroids in terms of this. In fact, we introduce a new representation of simple undirected graphs in terms of some vectors of finite dimensional vector spaces over Z_2 which satisfy consecutive 1's property. The set of such vectors is called a coding sequence of...
The Lattice Structure of Linear Subspace Codes
ArXiv, 2019
The projective space mathbbPq(n)\mathbb{P}_q(n)mathbbPq(n), i.e. the set of all subspaces of the vector space mathbbFqn\mathbb{F}_q^nmathbbFqn, is a metric space endowed with the subspace distance metric. Braun, Etzion and Vardy argued that codes in a projective space are analogous to binary block codes in mathbbF2n\mathbb{F}_2^nmathbbF2n using a framework of lattices. They defined linear codes in mathbbPq(n)\mathbb{P}_q(n)mathbbPq(n) by mimicking key features of linear codes in the Hamming space mathbbF2n\mathbb{F}_2^nmathbbF2n. In this paper, we prove that a linear code in a projective space forms a sublattice of the corresponding projective lattice if and only if the code is closed under intersection. The sublattice thus formed is geometric distributive. We also present an application of this lattice-theoretic characterization.
On the Grassmann graph of linear codes
Finite Fields and Their Applications, 2021
Let Γ(n, k) be the Grassmann graph formed by the k-dimensional subspaces of a vector space of dimension n over a field F and, for t ∈ N \ {0}, let ∆ t (n, k) be the subgraph of Γ(n, k) formed by the set of linear [n, k]-codes having minimum dual distance at least t + 1. We show that if |F| ≥ n t then ∆ t (n, k) is connected and it is isometrically embedded in Γ(n, k).
Discrete Applied Mathematics, 2009
A binary linear code in F n 2 with dimension k and minimum distance d is called an [n, k, d] code. A t-(n, m, λ) design D is a set X of n points together with a collection of m-subsets of X (called a block) such that every t-subset of X is contained in exactly λ blocks. A constant length code which corrects different numbers of errors in different code words is called a non-uniform error correcting code. Parity sectioned reduction of a linear code gives a non-uniform error correcting code. In this paper a new code, [2 n − 1, n, 2 n−1 ], is developed. The error correcting capability of this code is 2 n−2 − 1 = e. It is shown that this code holds a 2-(2 n − 1, 2 n−1 , 2 n−2) design. Also the parity sectioned reduction code after deleting the same g (≤ e) positions of each code word of this code holds a
Linear Codes from Incidence Matrices of Unit Graphs
ArXiv, 2020
In this paper, we examine the binary linear codes with respect to Hamming metric from incidence matrix of a unit graph G(mathbbZn)G(\mathbb{Z}_{n})G(mathbbZn) with vertex set is mathbbZn\mathbb{Z}_{n}mathbbZn and two distinct vertices xxx and yyy being adjacent if and only if x+yx+yx+y is unit. The main parameters of the codes are given.
Weight hierarchies of binary linear codes of dimension 4
Discrete Mathematics, 2001
The weight hierarchy of a binary linear [n; k] code C is the sequence (d1; d2; : : : ; d k ) where dr is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes. The possible weight hierarchies in each class are given. For one class the details of the proofs are included.
On the covering dimension of a linear code
arXiv (Cornell University), 2015
The critical exponent of a matroid is one of the important parameters in matroid theory and is related to the Rota and Crapo's Critical Problem. This paper introduces the covering dimension of a linear code over a finite field, which is analogous to the critical exponent of a representable matroid. An upper bound on the covering dimension is conjectured and nearly proven, improving a classical bound for the critical exponent. Finally, a construction is given of linear codes that attain equality in the covering dimension bound.
On zero neighbours and trial sets of linear codes
In this work we study the set of leader codewords of a non-binary linear code. This set has some nice properties related to the monotonicity of the weight compatible order on the generalized support of a vector in F n q . This allows us to describe a test set, a trial set and the set zero neighbours in terms of the leader codewords.
Binary codes and partial permutation decoding sets from the odd graphs
Central European Journal of Mathematics, 2014
For k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ω{k}, the set of all k-subsets of Ω = {1, 2, …, 2k +1}, and any two of its vertices u and v constitute an edge [u, v] if and only if u ∩ v = /0. In this paper the binary code generated by the adjacency matrix of O(k) is studied. The automorphism group of the code is determined, and by identifying a suitable information set, a 2-PD-set of the order of k 4 is determined. Lastly, the relationship between the dual code from O(k) and the code from its graph-theoretical complement overlineO(k)\overline {O(k)} overlineO(k), is investigated.
Codes from incidence matrices of graphs
Designs, Codes and Cryptography, 2013
We examine the p-ary codes, for any prime p, from the row span over Fp of |V | × |E| incidence matrices of connected graphs Γ = (V, E), showing that certain properties of the codes can be directly derived from the parameters and properties of the graphs. Using the edge-connectivity of Γ (defined as the minimum number of edges whose removal renders Γ disconnected) we show that, subject to various conditions, the codes from such matrices for a wide range of classes of connected graphs have the property of having dimension |V | or |V | -1, minimum weight the minimum degree δ(Γ), and the minimum words the scalar multiples of the rows of the incidence matrix of this weight. We also show that, in the k-regular case, there is a gap in the weight enumerator between k and 2k -2 of the binary code, and also for the p-ary code, for any prime p, if Γ is bipartite. We examine also the implications for the binary codes from adjacency matrices of line graphs. Finally we show that the codes of many of these classes of graphs can be used for permutation decoding for full error correction with any information set.
Dcc, 2010
In this work, we investigate linear codes over the ring F 2 + uF 2 + vF 2 + uvF 2 . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43-65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32-45, 1999). We then characterize the F 2 + uF 2 + vF 2 + uvF 2 -linearity of binary codes under the Gray map and give a main class of binary codes as an example of F 2 + uF 2 + vF 2 + uvF 2 -linear codes. The duals and the complete weight enumerators for F 2 + uF 2 + vF 2 + uvF 2 -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over F 2 + uF 2 + vF 2 + uvF 2 are obtained.
Linear codes using simplicial complexes
arXiv (Cornell University), 2022
Certain simplicial complexes are used to construct a subset D of F m 2 n and D, in turn, defines the linear code C D over F 2 n that consists of (v • d) d∈D for v ∈ F m 2 n. Here we deal with the case n = 3, that is, when C D is an octanary code. We establish a relation between C D and its binary subfield code C (2) D with the help of a generator matrix. For a given length and dimension, a code is called distance optimal if it has the highest possible distance. With respect to the Griesmer bound, five infinite families of distance optimal codes are obtained, and sufficient conditions for certain linear codes to be minimal are established.
Codes with few weights arising from linear sets
Advances in Mathematics of Communications, 2019
In this article we present a class of codes with few weights arising from special type of linear sets. We explicitly show the weights of such codes, their weight enumerator and possible choices for their generator matrices. In particular, our construction yields also to linear codes with three weights and, in some cases, to almost MDS codes. The interest for these codes relies on their applications to authentication codes and secret schemes, and their connections with further objects such as association schemes and graphs.
Linear codes from simplicial complexes
Designs, Codes and Cryptography, 2017
In this article we introduce a method of constructing binary linear codes and computing their weights by means of Boolean functions arising from mathematical objects called simplicial complexes. Inspired by Adamaszek (Am Math Mon 122:367-370, 2015) we introduce n-variable generating functions associated with simplicial complexes and derive explicit formulae. Applying the construction (Carlet in Finite Field Appl 13:121-135, 2007; Wadayama in Des Codes Cryptogr 23:23-33, 2001) of binary linear codes to Boolean functions arising from simplicial complexes, we obtain a class of optimal linear codes and a class of minimal linear codes.
Enumeration of linear codes with different hulls
arXiv (Cornell University), 2024
The hull of a linear code C is the intersection of C with its dual code. We present and analyze the number of linear q-ary codes of the same length and dimension but with different dimensions for their hulls. We prove that for given dimension k and length n ≥ 2k the number of all [n, k] q linear codes with hull dimension l decreases as l increases. We also present classification results for binary and ternary linear codes with trivial hulls (LCD and self-orthogonal) for some values of the length n and dimension k, comparing the obtained numbers with the number of all linear codes for the given n and k.
Codes from the incidence matrices of graphs on 3-sets
Discrete Mathematics, 2011
We examine the p-ary linear codes from incidence matrices of the three uniform subset graphs with vertex set the set of subsets of size 3 of a set of size n, with adjacency defined by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively. All the main parameters of the codes and the nature of the minimum words are obtained, and it is shown that the codes can be used for full error-correction by permutation decoding. We examine also the binary codes of the line graphs of these graphs.
A class of algebraic linear codes is introduced in which the parity-check matrix of the code is constructed by using a subset of the Abelian group of Walsh functions. These codes meet the Helgert and Stinaff upper bounds on minimum Hamming distance, and all the codes of this class are easily decodable by a one-step majority-logic algorithm.
The graphs of projective codes
Finite Fields and Their Applications, 2018
Consider the Grassmann graph formed by k-dimensional subspaces of an n-dimensional vector space over the field of q elements (1 < k < n − 1) and denote by Π(n, k)q the restriction of this graph to the set of projective [n, k]q codes. In the case when q ≥ n 2 , we show that the graph Π(n, k)q is connected, its diameter is equal to the diameter of the Grassmann graph and the distance between any two vertices coincides with the distance between these vertices in the Grassmann graph. Also, we give some observations concerning the graphs of simplex codes. For example, binary simplex codes of dimension 3 are precisely maximal singular subspaces of a non-degenerate quadratic form.
Codes from the Incidence Matrices of a zero-divisor Graphs
2020
In this paper, we examine the linear codes with respect to the Hamming metric from incidence matrices of the zero-divisor graphs with vertex set is the set of all non-zero zero-divisors of the ring mathbbZn\mathbb{Z}_nmathbbZn and two distinct vertices being adjacent iff their product is zero over mathbbZn.\mathbb{Z}_n.mathbbZn. The main parameters of the codes are obtained.