Applied and Computational Mathematics Seminar

Abstract: In 1981, Harold N. Ward introduced the concept of divisibility in linear codes and investigated linear codes from this point of view. Briefly speaking, divisible codes are simply codes whose codewords all have weights divisible by a nontrivial integer, say \Delta (called a divisor of the code). The reason that these codes are interesting is that people observe nontrivial divisibility in most of the optimal codes, as well as in many other good codes. Ward proved a bound on dimension of a linear divisible codes over a finite field. In this talk, we will show an equivalence of this Ward's bound. Using the equivalence, we generalize Ward's bound to some nonlinear codes. Another application of the result is to generalize the Gleason-Pierce-Ward theorem to additive codes.