Maximum distance q -nary codes

t immaru-A q-nary error-correcting code with N = qk code words of length n = k + r can have no greater miniium distance d than r + 1. The class of codes for which d = r + 1 is studied first in general, then with the restriction that the codes be linear. Examples and construction methods are given to show that these codes exist for a number of values of q, k, and r. Proof: Pick any k position. There are qk possible assignments of q-nary symbols to these positions. Since d = r + 1, no two among the qk code words can agree in all k ,, of these positions. Thus each of the qk possible assignments occurs in exactly one code word. _