↑ arvau

Wikipedia / Quadratic residue code

A quadratic residue code is a type of cyclic code.

Examples

Hamming code binary Golay code ternary Golay code.

Constructions

over the finite field is a quadratic residue-th root of unity.

corresponds to the quadratic residue code.

Weight

The minimum weight; this is the square root bound.

Extended code

Adding an overall parity-check digit to a quadratic residue code gives an extended quadratic residue code) an extended quadratic residue code is self-dual; otherwise it is equivalent but not equal to its dual. By the Gleason–Prange theorem (named for Andrew Gleason and Eugene Prange.

Decoding method

of the quadratic residue codes with the code length up to 113. However, decoding of long binary quadratic residue codes and non-binary quadratic residue codes continue to be a challenge. Currently, decoding quadratic residue codes is still an active research area in the theory of error-correcting code.

References

  • F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

  • .

  • M. Elia, Algebraic decoding of the (23,12,7) Golay code, IEEE Transactions on Information Theory, Volume: 33, Issue: 1, p.150–151, January 1987.

  • Reed, I. S., Yin, X., Truong, T.K., Algebraic decoding of the (32, 16, 8) quadratic residue code. IEEE Trans. Inf. Theory 36(4), 876–880 (1990)

  • Reed, I. S., Truong, T.K., Chen, X., Yin, X., The algebraic decoding of the (41, 21, 9) quadratic residue code. IEEE Trans. Inf. Theory 38(3), 974–986 (1992).

  • Humphreys, J. F. Algebraic decoding of the ternary (13, 7, 5) quadratic-residue code. IEEE Trans. Inf. Theory 38(3), 1122–1125 (May 1992).

  • Chen, X., Reed, I. S., Truong, T. K., Decoding the (73, 37, 13) quadratic-residue code. IEE Proc., Comput. Digit. Tech. 141(5), 253–258 (1994).

  • Higgs, R. J., Humphreys, J. F.: Decoding the ternary (23, 12, 8) quadratic-residue code. IEE Proc., Comm. 142(3), 129–134 (June 1995).

  • He, R., Reed, I. S., Truong, T. K., Chen, X., Decoding the (47, 24, 11) quadratic residue code. IEEE Trans. Inf. Theory 47(3), 1181–1186 (2001).

  • Y. Li, Y. Duan, H. C. Chang, H. Liu, T. K. Truong, Using the difference of syndromes to decode quadratic residue codes, IEEE Trans. Inf. Theory 64(7), 5179–5190 (2018).