Complete Weight enumerators and probability of undetected error for some Reed-Solomon codes
Kai‐Ming Ho · cIRcle (University of British Columbia) · 2009
The Hamming weight enumerators for Reed-Solomon codes are known, but do not often adequately describe the structure of the codes. Complete weight enumerators, on the other hand, describe the code in more detail, but are more difficult to determine. A procedure described by Blake and Kith [1] is used to derive the complete weight enumerators for Reed-Solomon and extended Reed- Solomon codes of dimensions one, two and three. Binary codes may be obtained from Reed-Solomon codes over a field with q = 2m elements. Many different bases may be used to obtain these binary codes. We analyse conditions under which two bases will yield binary codes with the same weight distribution. We also consider the probability of undetected error of these binary codes. It has been shown by Kasami and Lin that (n,k) Reed-Solomon codes used over a q-ary symmetric channel are proper. In this thesis, it is shown that the binary expansions of these codes and their extensions, when used on the binary symmetric channel, are not necessarily proper. In particular, the codes of rate less than [l — log[sub]2 m + m-1/m log[sub]2(m — 1)] are not proper.