A study of multifold Euclidean geometry codes

Abstract

A class of majority-logic decodable codes recently introduced by Lin, the multifold Euclidean geometry codes, is considered in this dissertation. An improvement of the multifold Euclidean geometry codes is presented. The improved multifold Euclidean geometry codes are shown to be more efficient than the original multifold Euclidean geometry codes while retaining the same majority-logic error-correcting capabilities. Based on the new definition of the improved multifold Euclidean geometry codes, it is easy to show that certain restrictions on the construction of multifold Euclidean geometry codes can be removed. Thus, the class of improved multifold Euclidean geometry codes is larger than the class of multifold Euclidean geometry codes defined by Lin. A simple decoding algorithm of the improved multifold Euclidean geometry codes is described. In some cases, each step of decoding can be made orthogonal. The improved multifold Euclidean geometry codes are shown to be maximal in the sense that they are the largest linear codes whose null spaces are spanned by all the (µ,d+1) -frames in EG(m,q). The extensions of the improved multifold Euclidean geometry codes are shown to be invariant under the affine group of permutations. The algebraic structures of the improved multifold Euclidean geometry codes based on primitive BCH codes are investigated. It is shown that this class of improved multifold Euclidean geometry codes is more efficient than the dual of polynomial codes with the same majority-logic error-correcting capabilities. In the case where the minimum distance of the base code divides the length of the base code, the corresponding improved multifold Euclidean geometry codes are shown to be identical to a subclass of regular generalized Euclidean geometry codes. Finally, a generalization of the improved multifold Euclidean geometry codes is briefly discussed. The multifold generalized Euclidean geometry codes contain the regular generalized Euclidean geometry codes as a subclass.