Unknown syndrome calculation in algebraic decoding of a class of cyclic codes

Recently, two novel matrices whose some entries are not syndromes and other entries are known syndromes have been presented to generate weak-locator polynomials needed in decoding the ternary quadratic residue code of length 61. This paper proposes a new unknown syndrome calculation for a class of cyclic codes and a completely algebraic decoding of the (23, 11, 9) ternary quadratic residue code up to actual minimum distance. For exactly four errors, the decoding algorithm developed here has to use the unknown syndrome, which appears in an entry of the above-mentioned matrix. The actual value of such an unknown syndrome can be determined precisely by the ratio of two determinants of matrices obtained from any one of the above-mentioned matrices.