Nonbinary double-error-correcting codes designed by means of algebraic varieties

Linear q-ary codes of growing length n --> infinity and designed distance delta are studied. At first, we examine cyclic codes defined by the sets of code zeros {g(i)\i = q(s) + 1, q(s+1)+1,...,g(s+delta-2) + 1} over a primitive element g of GF(q(m)). Then special cubic varieties are designed and employed in order to attain distances delta = 5, 6. The resulting double-error-correcting codes of length n = q(m) have r less than or equal to 2 m + [m/3] + 1 parity check symbols, and reduce the best known redundancy by [2m/3] symbols. A decoding procedure of complexity O(rn) operations is also considered.