An improvement of Griesmer bound for some classes of distances (Q1095108)

Let n(k,d) be the smallest integer such that there exists a linear binary (n,k,d) code. Let \[ g(k,d)=\sum^{k-1}_{j=0}\lceil \frac{d}{2^ j}\rceil \] where \(\lceil x\rceil\) is the smallest integer greater than or equal to x. The Griesmer bound states that n(k,d)\(\geq g(k,d)\). In this paper it is shown that for \(d=2^{k-2}-2^ a-2^ b\), \(0\leq b<a\leq k- 3\), \(2\leq a\), \(9\leq k\), n(k,d)\(\geq 2+g(k,d)\). In addition, four particular cases are shown: n(8,44)\(\geq 92\), n(8,58)\(\geq 120\), n(8,60)\(\leq 123\) and n(9,122)\(\geq 248\).