The set of all even subgraphs of a connected graph G on p vertices with q edges forms a binary linear code C=C/sub E/(G) with parameters [q,q-p+1,g], where g is the girth of G. Such codes were studied systematically by Bredeson and Hakimi (1967) and Hakimi and Bredeson (1968) who were concerned with the problems of augmenting C to a larger [q,k,g]-code and of efficiently decoding such augmented graphical codes. We give a new approach to these problems by requiring the augmented codes to be graphical. On one hand, we present two construction methods which turn out to contain the methods proposed by Hakimi and Bredeson as special cases. As we show, this not only gives a better understanding of their construction, it also results in augmenting codes of larger dimension. We look at the case of 1-error-correcting graphical codes in some detail. In particular, we show how to obtain the extended Hamming codes as "purely" graphical codes by our approach. On the other hand, we follow a suggestion of Ntafos and Hakimi (1981) and use techniques from combinatorial optimization to give decoding procedures for graphical codes which turn out to be considerably more efficient than the approach via majority logic decoding proposed by Bredeson and Hakimi. We also consider the decoding problem for the even graphical code based on the complete graph K/sub 2n/ in more detail: we discuss an efficient hardware implementation of an encoding/decoding scheme for these codes and show that things may be arranged in such a way that one can also correct all adjacent double errors. Finally, we discuss nonlinear graphical codes.

Graphical codes revisited

This correspondence formulates GF(q) matrix descriptions for a class of weighted, directed graphs. As a result of this formulation, the concept of graph theoretic error-correcting codes is generalized to the q -ary case. It is shown that graph theoretic q -ary codes are completely orthogonalizable and, hence, one-step majority decodable. It is also seen that known techniques for the augmentation of circuit codes can he extended to the q -ary case. The resulting codes remain easily decodable.

Graph theoretic q -ary codes (Corresp.)

In this paper, we conclude five kinds of methods for construction of the regular low-density parity matrix H and three kinds of methods for the construction of irregular low-density parity-check matrix H. Through the analysis of the code rate and parameters of these eight kinds of structures, we find that the construction of low-density parity-check matrix tends to be more flexible and the parameter variability is enhanced. We propose that the current development cost should be lower with the progress of electronic technology and we need research on more practical Low-Density Parity-Check Codes (LDPC). Combined with the application of the quantum distribution key, we urgently need to explore the research direction of relevant theories and technologies of LDPC codes in other fields of quantum information in the future. 

Low-Density Parity-Check Codes: Research Status and Development Direction

It is shown that directed graphs can be used to generate a class of moderately efficient error-correcting codes, which are easily decodable.

J. Bredeson

Papers

Ternary graph theoretic error-correcting codes (Corresp.)