Name
Abstract | ||
|---|---|---|
Minimal codes are characterized by the property that none of the codewords is covered by some other linearly independent codeword. We first show that the use of a bent function g in the so-called direct sum of Boolean functions
$$h(x,y)=f(x)+g(y)$$
, where f is arbitrary, induces minimal codes. This approach gives an infinite class of minimal codes of length
$$2^n$$
and dimension
$$n+1$$
(assuming that
$$h: {\mathbb {F}}_2^n \rightarrow {\mathbb {F}}_2$$
), whose weight distribution is exactly specified for certain choices of f. To increase the dimension of these codes with respect to their length, we introduce the concept of non-covering permutations (referring to the property of minimality) used to construct a bent function g in s variables, which allows us to employ a suitable subspace of derivatives of g and generate minimal codes of dimension
$$s+s/2+1$$
instead. Their exact weight distribution is also determined. In the second part of this article, we first provide an efficient method (with easily satisfied initial conditions) of generating minimal
$$[2^n,n+1]$$
linear codes that cross the so-called Ashikhmin–Barg bound. This method is further extended for the purpose of generating minimal codes of larger dimension
$$n+s/2+2$$
, through the use of suitable derivatives along with the employment of non-covering permutations. To the best of our knowledge, the latter method is the most general framework for designing binary minimal linear codes that violate the Ashikhmin–Barg bound. More precisely, for a suitable choice of derivatives of
$$h(x,y)=f(x) + g(y)$$
, where g is a bent function and f satisfies certain minimality requirements, for any fixed f, one can derive a huge class of non-equivalent wide binary linear codes of the same length by varying the permutation
$$\phi $$
when specifying the bent function
$$g(y_1,y_2)=\phi (y_2)\cdot y_1$$
in the Maiorana–McFarland class. The weight distribution is given explicitly for any (suitable) f when
$$\phi $$
is an almost bent permutation. |
Year | DOI | Venue |
|---|---|---|
2022 | 10.1007/s10623-022-01037-z | Designs, Codes and Cryptography |
Keywords | DocType | Volume |
Minimal linear codes, Ashikhmin–Barg’s bound, Derivatives, Direct sum, 94C10, 06E30 | Journal | 90 |
Issue | ISSN | Citations |
5 | 0925-1022 | 0 |
PageRank | References | Authors |
0.34 | 23 | 4 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Fengrong Zhang | 1 | 41 | 11.72 |
| Enes Pasalic | 2 | 0 | 0.68 |
| Rene Rodriguez | 3 | 0 | 0.34 |
| Yongzhuang Wei | 4 | 69 | 16.94 |