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 : F n 2 → F 2 ), whose weight distribution is exactly specified for certain choices of f . To

more »

... rease 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(y 1 , y 2 ) = φ(y 2 ) • y 1 in the Maiorana-McFarland class. The weight distribution is given explicitly for any (suitable) f when φ is an almost bent permutation.