Isodual and self-dual codes from graphs (original) (raw)
Related papers
Constructing self-dual codes using graphs
Journal of Combinatorial Theory, Series B, 1991
We show that the rows of the face-vertex indicence matrix of a connected cubic planar bipartate graph on n vertices generate a self-dual binary code of length n.
Graph theoretic aspects of minimum distance and equivalence of binary linear codes
Australas. J Comb., 2021
A binary linear [2n, n]-code with generator matrix [In|A] can be associated with a digraph on n vertices with adjacency matrix A and vice versa. We use this connection to present a graph theoretic formula for the minimum distance of codes with information rate 1/2. We also formulate the equivalence of such codes via new transformations on corresponding digraphs.
On the dual binary codes of the triangular graphs
European Journal of Combinatorics, 2007
The stabilizers of the minimum-weight codewords of dual binary codes obtained from the strongly regular graphs T (n) defined by the primitive rank-3 action of the alternating groups A n where n ≥ 5, on Ω {2} , the set of duads of Ω = {1, 2, . . . , n}, are examined.
1989
I would like to express my appreciation for the useful discussions I had with Louis Goddyn and Gordon F. Royle, and for the friendship and support of everybody at the University of Waterloo Combinatorics and Optimization Department during the last year of my Ph.D. studies. I also thank Steve White and Rob Ballantyne for their help in the printing of this thesis.
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.
New extremal binary self-dual codes of lengths 64 and 66 from bicubic planar graphs
ArXiv, 2016
In this work, connected cubic planar bipartite graphs and related binary self-dual codes are studied. Binary self-dual codes of length 16 are obtained by face-vertex incidence matrices of these graphs. By considering their lifts to the ring R_2 new extremal binary self-dual codes of lengths 64 are constructed as Gray images. More precisely, we construct 15 new codes of length 64. Moreover, 10 new codes of length 66 were obtained by applying a building-up construction to the binary codes. Codes with these weight enumerators are constructed for the first time in the literature. The results are tabulated.
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.
Symmetric configurations for bipartite-graph codes
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.
Some Combinatorial Aspects of Constructing Bipartite-Graph Codes
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.
Matrices for graphs designs and codes
The adjacency matrix of a graph can be interpreted as the incidence matrix of a design, or as the generator matrix of a binary code. Here these relations play a central role. We consider graphs for which the corresponding design is a (symmetric) block design or (group) divisible design. Such graphs are strongly regular (in case of a block design) or very similar to a strongly regular graph (in case of a divisible design). Many construction and properties for these kind of graphs are obtained. We also consider binary code of a strongly regular graph, work out some theory and give several examples.