Quadratic residue code

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A quadratic residue code is a type of cyclic code.

There is a quadratic residue code of length over the finite field whenever and are primes, is odd, and is a quadratic residue modulo . Its generator polynomial as a cyclic code is given by

where is the set of quadratic residues of in the set and is a primitive th root of unity in some finite extension field of . The condition that is a quadratic residue of ensures that the coefficients of lie in . The dimension of the code is . Replacing by another primitive -th root of unity either results in the same code or an equivalent code, according to whether or not is a quadratic residue of .

An alternative construction avoids roots of unity. Define

for a suitable . When choose to ensure that . If is odd, choose , where or according to whether is congruent to or modulo . Then also generates a quadratic residue code; more precisely the ideal of generated by corresponds to the quadratic residue code.

The minimum weight of a quadratic residue code of length is greater than ; this is the square root bound.

Adding an overall parity-check digit to a quadratic residue code gives an extended quadratic residue code. When (mod ) 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), the automorphism group of an extended quadratic residue code has a subgroup which is isomorphic to either or .

Examples of quadratic residue codes include the Hamming code over , the binary Golay code over and the ternary Golay code over .

References[edit]

  • F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
  • Blahut, R. E. (September 2006), "The Gleason-Prange theorem", IEEE Trans. Inf. Theory, Piscataway, NJ, USA: IEEE Press, 37 (5): 1269–1273, doi:10.1109/18.133245 .