The Algebra Seminar series presents Dr. Xiaoyu Liu of Wright State University discussing “Divisible Codes over Various Algebraic Alphabets” on Tuesday, Oct. 29, from 5 to 6 p.m. in Morton 313.
Abstract: In 1981, Harold N. Ward introduced the concept of divisibility in linear codes and investigated linear codes over finite fields from this point of view. Briefly speaking, divisible codes are simply codes whose codewords all have weights divisible by a nontrivial integer. This concept can be easily generalized to codes over algebraic alphabets other than finite fields. 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. One of Ward’s most important result on divisible codes is 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. With the equivalence, Ward’s bound can be generalized to other alphabets as well. One application of the result is to generalize the Gleason-Pierce-Ward theorem to additive codes. For additive codes over finite fields, only the abelian group structure of the alphabet matters. As a result, we follow this direction to study additive codes over general finite abelian groups.
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