Generating Idempotents of Sixth Residue Codes over the Binary Field

Abstract

Abstract. Higher power residue codes over finite fields are generated by factors of the polynomial 1 n x − . Unfortunately, to decompose the polynomial 1 n x − over finite fields is difficult. Generating idempotents can also generate higher power residue codes. Thus it is important to get generating idempotents of cyclic codes. We find precise expressions of generating idempotents of some sixth residue codes of length over the binary field, where p is a prime such that ( ) 1 mod 24 p ≡ .

Generating Idempotents of Sixth Residue Codes over the Binary Field (II)

“…Charters [4] provided a generalization of binary quadratic residue codes to the cases of higher power prime residues over the finite field of the same order. Higher power residue codes and forms of generating polynomials of these codes were proposed in [5][6][7][8][9].Generating polynomials of higher power residue codes are factors of 1 n x  . Generally speaking, it is difficult to factor the polynomial 1 n x  over finite fields.…”

“…In [9], explicit expressions of generating idempotents of some sixth residue codes of length p over the binary field were given, where p is a prime with   1 mod 24 p  . This paper gives explicit expressions of generating idempotents of sixth residue codes of length p over the binary field, where p is a prime with   7 mod 24 p  .…”