Divisibility of codes meeting the Griesmer bound
The Griesmer bound is a well-known bound on the length of a linear \([n,k,d]\) code, and codes which meet this bound are optimal. A divisor of a linear code is an integer dividing the weights of all its codewords, and a code is called divisible if it has a divisor larger than 1. Optimal codes are often divisible. This paper proves the following result: Theorem 1. Let \(C\) be an \([n,k,d]\) code over \(\text{GF}(p)\), \(p\) a prime, meeting the Griesmer bound. If \(p^e | d\), then \(p^e\) is a divisor of \(C\). Note that for large enough \(d\), codes exist which meet the Griesmer bound. A number of examples of divisible codes are given.
scientific article; zbMATH DE number 1212912
scientific article