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- Quadratic_residue_code abstract "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 andis a quadratic residue modulo .Its generator polynomial as a cyclic code is given bywhere is the set of quadratic residues ofin the set andis a primitive th root ofunity in some finite extension field of .The condition that is a quadratic residueof ensures that the coefficients of lie in . The dimension of the code isReplacing by another primitive -throot of unity either results in the same codeor an equivalent code, according to whether or not is a quadratic residue of .An alternative construction avoids roots of unity. Definefor a suitable . When choose to ensure that while if is odd where or according to whetheris congruent to or modulo . Then also generatesa quadratic residue code; more precisely the ideal ofgenerated 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 codegives an extended quadratic residue code. When(mod ) an extended quadraticresidue code is self-dual; otherwise it is equivalent but notequal to its dual. By the Gleason–Prange theorem (named for Andrew Gleason and Eugene Prange), the automorphism group of an extended quadratic residuecode has a subgroup which is isomorphic toeither or .Examples of quadraticresidue codes include the Hamming codeover , the binary Golay codeover and the ternary Golay codeover .".
- Quadratic_residue_code wikiPageID "14433598".
- Quadratic_residue_code wikiPageRevisionID "586846336".
- Quadratic_residue_code hasPhotoCollection Quadratic_residue_code.
- Quadratic_residue_code subject Category:Coding_theory.
- Quadratic_residue_code subject Category:Quadratic_residue.
- Quadratic_residue_code comment "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 andis a quadratic residue modulo .Its generator polynomial as a cyclic code is given bywhere is the set of quadratic residues ofin the set andis a primitive th root ofunity in some finite extension field of .The condition that is a quadratic residueof ensures that the coefficients of lie in .".
- Quadratic_residue_code label "Quadratic residue code".
- Quadratic_residue_code sameAs m.03d3dh1.
- Quadratic_residue_code sameAs Q7268368.
- Quadratic_residue_code sameAs Q7268368.
- Quadratic_residue_code wasDerivedFrom Quadratic_residue_code?oldid=586846336.
- Quadratic_residue_code isPrimaryTopicOf Quadratic_residue_code.