Gaps in the binary weight distributions of Reed-Solomon codes

摘要

When a Reed-Solomon code is expanded to form a binary code, certain combinations of the spectrum of the Reed-Solomon code and the basis used for the expansion result in large gaps in the weight distribution of the binary code. It is shown that the size of these gaps can be bounded by computing the sums of various powers of the basis elements and applying a theorem normally used for cyclic codes. This explains why codes obtained by using a polymomial basis often have smaller gaps in their weight distributions than those obtained by using a normal basis. Similar results apply to the number of intersections between codewords, which can be used to show that the codewords are orthogonal.