Using codes for error correction and detection (Corresp.)

A linear code
C
over GF
(q)
is good for
t
-error-correction and error detection if
P(C,t;\epsilon) \leq P(C,t;(q - 1)/q)
for all
\epsilon, 0 \leq \epsilon \leq (q - 1)/q
, where
P(C, t; \epsilon)
is the probability of an undetected error after a codeword in
C
is transmitted over a
q
-ary symmetric channel with error probability
\epsilon
and correction is performed for all error patterns with
t
or fewer errors. A sufficient condition for a code to be good is derived. This sufficient condition is easy to check, and examples to illustrate the method are given.