Nonbinary random error-correcting codes (Corresp.)

Primitive BCH codes with symbols from
GF(q)
and designed distance
d
have parameter values \begin{align} \text{block length} &= n = q^m - 1 \\ \text{check symbols/block} &= r \leq m(d - 1) \end{align} where
m
is any positive integer. For many nonbinary BCH codes (called maximally redundant codes), the maximum number of check symbols per block is required, i.e.
r = m(d - 1)
. Conditions whereby a primitive nonbinary BCH code is maximally redundant are discussed. It is shown that a class of codes exists, with symbols from
GF(q)
, based upon doubly lengthened Reed-Solomon codes over
GF(q^m)
, having parameter values \begin{align} \text{block length} &= n = m(q^m + 1) \\ \text{check symbols/block} &= r = m(d - 1) \\ \text{designed distance} &= d \end{align} where again
m
is any positive integer. Thus this class of codes extends the block length of maximally redundant codes by a multiplicative factor exceeding
m
, while retaining the same designed distance and same number of check symbols.