Codes from incidence matrices of graphs

Advances in Mathematics of Communications, 2011

We examine the p-ary codes from incidence matrices of Paley graphs P (q) where q ≡ 1 (mod 4) is a prime power, and show that the codes are [ q(q−1) 4 , q − 1, q−1 2 ] 2 or [ q(q−1) 4 , q, q−1 2 ] p for p odd. By finding PD-sets we show that for q > 9 the p-ary codes, for any p, can be used for permutation decoding for full error-correction. The binary code from the line graph of P (q) is shown to be the same as the binary code from an incidence matrix for P (q).

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.

Designs, Codes and Cryptography, 2014

The hulls of codes from the row span over F p , for any prime p, of incidence matrices of connected k-regular graphs are examined, and the dimension of the hull is given in terms of the dimension of the row span of A + k I over F p , where A is an adjacency matrix for the graph. If p = 2, for most classes of connected regular graphs with some further form of symmetry, it was shown by Dankelmann et al. (Des. Codes Cryptogr. 2012) that the hull is either {0} or has minimum weight at least 2k − 2. Here we show that if the graph is strongly regular with parameter set (n, k, λ, μ), then, unless k is even and μ is odd, the binary hull is non-trivial, of minimum weight generally greater than 2k − 2, and we construct words of low weight in the hull; if k is even and μ is odd, we show that the binary hull is zero. Further, if a graph is the line graph of a k-regular graph, k ≥ 3, that has an -cycle for some ≥ 3, the binary hull is shown to be non-trivial with minimum weight at most 2 (k − 2). Properties of the p-ary hulls are also established.

ArXiv, 2020

In this paper, we examine the binary linear codes with respect to Hamming metric from incidence matrix of a unit graph $G(\mathbb{Z}_{n})$ with vertex set is $\mathbb{Z}_{n}$ and two distinct vertices $x$ and $y$ being adjacent if and only if $x+y$ is unit. The main parameters of the codes are given.

Afrika Matematika, 2018

We exhibit PD-sets for the binary and non-binary codes generated by incidence matrices of triangular graphs T n where n ≥ 5.

Graphs and Combinatorics, 2017

Studies of the p-ary codes from the adjacency matrices of uniform subset graphs Γ (n, k, r) and their reflexive associates have shown that a particular family of codes defined on the subsets are intimately related to the codes from these graphs. We describe these codes here and examine their relation to some particular classes of uniform subset graphs. In particular we include a complete analysis of the p-ary codes from Γ (n, 3, r) for p ≥ 5, thus extending earlier results for p = 2, 3.

Discrete Mathematics, 2014

Linear codes arising from the row span over any prime field Fp of the incidence matrices of the odd graphs O k for k ≥ 2 are examined and all the main parameters obtained. A study of the hulls of these codes for p = 2 yielded that for O2 (the Petersen graph), the dual of the binary hull from an incidence matrix is the binary code from points and lines of the projective geometry P G3(F2), which leads to a correspondence between the edges and vertices of O2 with the points and a collection of ten lines of P G3(F2), consistent with the codes. The study also gives the dimension, the minimum weight, and the nature of the minimum words, of the binary codes from adjacency matrices of the line graphs L(O k).

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 $\mathbb{Z}_n$ and two distinct vertices being adjacent iff their product is zero over $\mathbb{Z}_n.$ The main parameters of the codes are obtained.

Discrete Mathematics, 2007

The binary codes of the line graphs L m (n) of the complete multipartite graphs K n 1 ,...,nm (n i = n for 1 ≤ i ≤ m) n ≥ 2, m ≥ 3 are examined, and PD-sets and s-PD-sets are found.

We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix have at most one unit in the same position. In terms of Design Theory, such a matrix is an incidence matrix of a symmetric configuration. Also, it gives rise to an n-regular bipartite graphs without 4-cycles, which can be used for constructing bipartite-graph codes so that both the classes of their vertices are associated with local constraints (constituent codes). We essentially extend the region of parameters of such matrices by using some results from Galois Geometries. Many new matrices are either circulant or consist of circulant submatrices: this provides code parity-check matrices consisting of circulant submatrices, and hence quasi-cyclic bipartite-graph codes with simple implementation.