On the non-minimal codewords in binary Reed–Muller codes

Secret sharing schemes from binary linear codes

“…Then (6) implies that h i = h j and then i = j. This is contrary to the assumption that i and j are distinct.…”

“…The code C ? m has parameters [15,6,6] and the following weight distribution: 1 þ 30x 6 þ 15x 8 þ 18x 10 : Table 1 Weight distribution of C ? m for gcd(s, n) = 1.…”

“…This was done only for a small number of classes of special linear codes. Several authors have investigated the minimal codewords for certain linear codes and characterized the access structures of the secret sharing schemes based on their dual codes [8,19,17,18,[1][2][3]6,[11][12][13]22]. Although the determination of the set of all minimal codewords for linear codes over arbitrary finite fields is very hard, one may anticipate it could be relatively easier for binary codes.…”

Minimal codewords in Reed–Muller codes

“…The smallest non-trivial case is wt(c) = 2d. This was solved by Borissov, Manev, and Nikova for RM(r, m) [3], by interpreting the non-minimal codewords of weight 2d geometrically as a union of two disjoint affine spaces AG(m − r, 2). To state their result, we first introduce some notations.…”

“…Since we are interested in counting the number of non-minimal codewords of weight less than 3 · 2 m−r , we take c 1 to be a non-zero codeword of smallest weight, namely 2 m−r , and c 2 to be a codeword of first or second type with small µ. We don't take weight 2 m−r for both codewords since this case has already been solved by Borissov, Manev, and Nikova [3]; their result is stated here in Theorem 1.7. So a non-minimal codeword corresponds to a pair (c 1 , c 2 ) of geometric objects having no affine intersection points, where c 1 is an (m − r)-dimensional space, and where c 2 is a quadric or a symmetric difference.…”