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

Linear q-ary codes of growing length n→∞ and designed distance δ are studied. At first, we examine cyclic codes defined by the sets of code zeros {gⁱ|i=q^s+1, q^s+1 +1, ···, q^s+δ-2+1} over a primitive element g of GF(q^m). Then special cubic varieties are designed and employed in order to attain distances δ=5, 6. The resulting double-error-correcting codes of length n=q^m have r⩽2m+[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