Divisibility of Codes Meeting the Griesmer Bound

The brackets of ``[n, k, d]'' signal that C is linear, and n is the length, k the dimension, and d the minimum weight of C. The bound was proved by Griesmer in 1960 for q=2 and generalized by Solomon and Stiffler in 1965. Over the years, much effort has gone into constructing codes meeting the bound or showing, for selected parameter values, that they do not exist. This effort is part of the more comprehensive program of finding optimal linear codes over GF(q), those having the smallest n for given k and d. The recent paper [14] of Hill provides a thorough survey of this research and a sizeable bibliography. We shall abbreviate the right side of the bound by gq(k, d) and call a code meeting the bound a Griesmer code. A divisor of a linear code is an integer dividing the weights of all its words, and a code is called divisible if it has a divisor larger than 1 [20]. Optimal codes are often divisible, and Dodunekov and Manev showed that for a binary code meeting the Griesmer bound, the power of 2 dividing the Article No. TA972864