Nonbinary random error-correcting codes (Corresp.)

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