关于三次剩余码

On Cubic Residue Codes

设p、q是两个不同的素数且p≡ 1(mod3) ,qp - 13 ≡ 1(modp) ,β是Fp 中一个本原元素 ,α是Fq 的某个扩域中的一个本原p次单位根 ,R0 =β3i(modp) |1≤i≤ p -13,go(x) =Πj∈Ro(x -αj) .Fq 上长度为p由go(x)生成的循环码称为三次剩余码 .证明了这样码的极小距离d≥3 p .

Let p and q be two different primes, p ≡1(mod3), q p-13 ≡1 (mod p ), β a primitive element of field F p , α a primitive pth root of unity in an extension field of F q , R 0=β 3i (mod) p)|1≤i≤p-13,g 0(x) =Π j∈R 0(x-α j). The cyclic code of length p over F q with generator g 0(x) is called cubic residue code. In this paper, it is shown that the minimal distance of a cubic residue code is not smaller than 3 p .