An improved sphere covering bound for the codes with n=3 R+2

Let C be a binary code (not necessarily linear) with covering radius R and length n=3R+2. The sphere covering bound on the cardinality of C is improved considerably provided C has minimal distance d>2. Some new results on the function t[n,k] (the smallest covering radius of any binary linear code with length n and dimension k): t[38.6]⩾13, t[47.7]⩾16, t[59.8]⩾20 are given