Some results on the minimum weight of primitive BCH codes (Corresp.)

Gröbner Basis Techniques to Compute Weight Distributions of Shortened Cyclic Codes

“…Extensive work has been done to obtain the distance of cyclic codes and their weight distribution (see for example [1,2,5,8,10,[14][15][16]20,25]). The use of shortened cyclic codes in many applications has also pushed a parallel study on their properties (see for example [7,9,12,13]).…”

Narrow-Sense BCH Codes Over $ {\mathrm {GF}}(q)$ With Length $n=\frac {q^{m}-1}{q-1}$

“…The cyclic codes C (n,q,m,δ) are treated in almost every book on coding theory. When n = q m − 1, the codes C (n,q,m,δ) are called narrow-sense primitive BCH codes and have been extensively studied in the literature [2], [3], [4], [5], [8], [9], [10], [12], [17], [19], [20], [21], [26], [27], [29], [31], [32]. The reader is referred to [17] for a recent summary of various results on narrow-sense primitive BCH codes.…”

A new algorithm for finding minimum-weight words in a linear code: application to McEliece's cryptosystem and to narrow-sense BCH codes of length 511

“…• Using the inclusion relations between BCH codes and punctured Reed-Muller codes, Kasami and Lin [27] proved that for all 1 i m 0 s 0 2, where 0 s m 0 2i, the intersection of the BCH code of length 2 m 0 1 and designed distance = 2 m0s01 02 m0s0i01 01 with the punctured Reed-Muller code of order s + 2 contains a word of weight . • Helgert and Stinaff [28] exhibited a codeword of weight or + 1 in some shortened codes of B(n; ).…”