A class of minimal cyclic codes over finite fields

Primitive Idempotents of Irreducible Cyclic Codes of Length n

“…Every cyclic code of length over a finite field F is identified with exactly one ideal of the quotient algebra F [ ]/⟨ − 1⟩. Some explicit factorizations of − 1 can be found in [7][8][9][10][11][13][14][15][16]. We need the following results about the irreducible factorization of − 1 over F .…”

Weight distributions of cyclic codes of length lm

“…This is quite useful in practice. The weight distributions of many important families of cyclic codes have been studied extensively in the literature [2,[4][5][6][7][8][10][11][12][13][14][15]. We describe the known results as follows.…”

“…Further, if l is a prime satisfying that l v ||(q − 1) and v is a positive integer, Chen et al [1] gave the irreducible factorization of x l m − 1 over F q , all primitive idempotent of a ring F q [x]/(x l m − 1), and the minimum Hamming distances of irreducible cyclic codes of length l m clearly in the cases: m = v + r, 2v + r, 0 ≤ r < v.…”

Weight distributions of cyclic codes of length tlm

“…On the other hand, several papers investigated the primitive idempotent elements of F q [x]/(x n − 1) to determine the minimum Hamming weights of cyclic codes of length n = tl m in [1,3,4,8]. Sharma and Bakshi [21] gave the weight distributions of irreducible cyclic codes of length l m , where the multiplicative order of q modulo l m is one among φ(l m ), l i , and 2l i .…”