Quadratic residue code

A quadratic residue code is a type of cyclic code.

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

where $Q$ is the set of quadratic residues of $p$ in the set $\{1,2,\ldots,p-1\}$ and $\zeta$ is a primitive $p$ th root of unity in some finite extension field of $GF(l)$ . The condition that $l$ is a quadratic residue of $p$ ensures that the coefficients of $f$ lie in $GF(l)$ . The dimension of the code is $(p+1)/2$ . Replacing $\zeta$ by another primitive $p$ -th root of unity $\zeta^r$ either results in the same code or an equivalent code, according to whether or not $r$ is a quadratic residue of $p$ .

单词	Quadratic residue code
释义	Quadratic residue code 英语百科 Quadratic residue code A quadratic residue code is a type of cyclic code. There is a quadratic residue code of length $p$ over the finite field $GF(l)$ whenever $p$ and $l$ are primes, $p$ is odd, and $l$ is a quadratic residue modulo $p$ . Its generator polynomial as a cyclic code is given by where $Q$ is the set of quadratic residues of $p$ in the set $\{1,2,\ldots,p-1\}$ and $\zeta$ is a primitive $p$ th root of unity in some finite extension field of $GF(l)$ . The condition that $l$ is a quadratic residue of $p$ ensures that the coefficients of $f$ lie in $GF(l)$ . The dimension of the code is $(p+1)/2$ . Replacing $\zeta$ by another primitive $p$ -th root of unity $\zeta^r$ either results in the same code or an equivalent code, according to whether or not $r$ is a quadratic residue of $p$ .
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