Abstract
It is well-known that there exists a constant-weight <TEX>$[s{\theta}_{k-1},k,sq^{k-1}]_q$</TEX> code for any positive integer s, which is an s-fold simplex code, where <TEX>${\theta}_j=(q^{j+1}-1)/(q-1)$</TEX>. This gives an upper bound <TEX>$n_q(k,sq^{k-1}+d){\leq}s{\theta}_{k-1}+n_q(k,d)$</TEX> for any positive integer d, where <TEX>$n_q(k,d)$</TEX> is the minimum length n for which an <TEX>$[n,k,d]_q$</TEX> code exists. We construct a two-weight <TEX>$[s{\theta}_{k-1}+1,k,sq^{k-1}]_q$</TEX> code for <TEX>$1{\leq}s{\leq}k-3$</TEX>, which gives a better upper bound <TEX>$n_q(k,sq^{k-1}+d){\leq}s{\theta}_{k-1}+1+n_q(k-1,d)$</TEX> for <TEX>$1{\leq}d{\leq}q^s$</TEX>. As another application, we prove that <TEX>$n_q(5,d)={\sum_{i=0}^{4}}{\lceil}d/q^i{\rceil}$</TEX> for <TEX>$q^4+1{\leq}d{\leq}q^4+q$</TEX> for any prime power q.
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