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Title:
Minimal binary linear codes - a general framework based on bent concatenation
Authors: Fengrong Zhang, Enes Pasalic, René Rodríguez, Yongzhuang Wei
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: \F_2^n \rightarrow \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 \textit{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.
ePrint: https://eprint.iacr.org/2020/1398
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