Codes from Adjacency Matrices of Uniform Subset Graphs

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.

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.

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.

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 $\overline {O(k)} $, is investigated.

Mathematics and Statistics, 2022

The unity product graph of a ring R is a graph which is obtained by setting the set of unit elements of R as the vertex set. The two distinct vertices r i and r j are joined by an edge if and only if r i •r j = e. The subgraphs of a unity product graph which are obtained by the vertex and edge deletions are said to be its induced and spanning subgraphs, respectively. A subset C of the vertex set of induced (spanning) subgraph of a unity product graph is called perfect code if the closed neighbourhood of c, S 1 (c) forms a partition of the vertex set as c runs through C. In this paper, we determine the perfect codes in the induced and spanning subgraphs of the unity product graphs associated with some commutative rings R with identity. As a result, we characterize the rings R in such a way that the spanning subgraphs admit a perfect code of order cardinality of the vertex set. In addition, we establish some sharp lower and upper bounds for the order of C to be a perfect code admitted by the induced and spanning subgraphs of the unity product 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).

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.

Discrete Applied Mathematics, 1990

The study of P-polynomial association schemes, or distance-regular graphs, and their possible classification is one of the main topics of algebraic combinatorics. One way to approach the issue is through the parameters Pkij which characterize the scheme. The purpose of this paper is to deal with a concrete case. This case is also important in the study of the links between P-polynomial schemes and error-correcting codes. We present one way of constructing completely regular binary block-codes taking one kind of distance-regular graph as a starting point. The parameters of the code (length, error-correcting capability, minimum distance, covering radius,. ..) are calculable starting from the parameters pk,, of the graph. We will use this construction to study an extremal case of distance-regular graphs leading to completely regular, uniformly packed codes, and we will use some well-known results concerning this type of code in order to find a solution to the problem of classifying the given graphs.

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).

Graphs and Combinatorics, 2013

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 contain at most one unit in common. These matrices are equivalent to n-regular bipartite graphs without 4-cycles, and therefore can be used for the construction of efficient bipartite-graph codes such that both the classes of its vertices are associated with local constraints. We significantly extend the region of parameters m, n for which there exist an n-regular bipartite graph with 2m vertices and without 4-cycles. In that way we essentially increase the region of lengths and rates of the corresponding bipartite-graph codes. 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.