An improvement of the Griesmer bound for some small minimum distances (Q1083427)

Let n(k,d) be the smallest integer n for which there exists a binary [n,k,d] code and let \(g(k,d)=\sum^{k-1}_{j=0}\lceil d/2^ j\rceil\) where \(\lceil x\rceil\) is the smallest integer greater than or equal to x. The Griesmer bound states that n(k,d)\(\geq g(k,d)\). A technique is developed and used to give improved upper and lower bounds on n(k,d). The quantity n(8,d) is evaluated for a variety of values of d.