Undetected error probability of q-ary constant weight codes

Abstract

In this paper, we introduce a new combinatorial invariant called q-binomial moment for q-ary constant weight codes. We derive a lower bound on the q-binomial moments and introduce a new combinatorial structure called generalized (s, t)-designs which could achieve the lower bounds. Moreover, we employ the q-binomial moments to study the undetected error probability of q-ary constant weight codes. A lower bound on the undetected error probability for q-ary constant weight codes is obtained. This lower bound extends and unifies the related results of Abdel-Ghaffar for q-ary codes and Xia-Fu-Ling for binary constant weight codes. Finally, some q-ary constant weight codes which achieve the lower bounds are found.

Bounds on the Probability of Undetected Error for q-Ary Codes

“…In particular, constant weight codes are attractive and many bounds are developed, for example, binary constant weight codes (see [ 14 , 15 ]) and q -ary constant weight codes (see [ 16 ]). In fact, the probability of an undetected error for binary constant weight codes has been studied and can be given explicitly (see [ 14 , 16 ]).…”