Considerations On Low Complexity Codes Concerning Symmetry And Cyclicity

Abstract

The binary error correcting codes that are examined in this paper are codes which are constructed out of shorter subcodes. The method of construction is well described by a bipartite graph. The one set of nodes of this graph represents the single bits of a codeword of the new code, while the other set of nodes represents the subcodes; there are edges only between subcode-nodes and bit-nodes, having the interpretation that the set of bits that are connected to a specific subcode must be a codeword of this subcode. Cyclically symmetrical structures of such code constructing graphs are studied here and the positive effect of cyclic codes (when used as subcodes on such structures) on the error-correcting capability of the resulting code is demonstrated. A class of codes with error-correcting capability t equals 3 is extensively studied in a rigorous theoretical way.