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:F2n→F2), 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 ϕ when specifying the bent function g(y1, y2) = ϕ(y2) · y1 in the Maiorana–McFarland class. The weight distribution is given explicitly for any (suitable) f when ϕ is an almost bent permutation.
dr. Enes Pasalic
Full Professor, Head of the center
