Let n(k,d) denote the length of the shortest binary linear code of dimension k and minimum distance d. The well known Griesmer bound states that \(n(k,d)\geq g(k,d)=\sum^{k-1}_{j=0}\lceil d/2^ j\rceil,\) where \(\lceil x\rceil\) denotes the smallest integer not less than x. In this paper it is proved, applying the weight enumeration method, that if \(d=2^{k-i}-2^{k-i-1}-2^ i\) or \(d=2^{k-i}-2^{k-i-1}-2^ i-2\) with \(k\geq 2i+2\) then \(n(k,d)=g(k,d)+1.\)