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Several classes of minimal binary linear codes violating the Ashikhmin-Barg bound

Minimal binary linear codes are a special class of binary codes with important applications in secret sharing and secure two-party computation. These codes are characterized by the property that none of the nonzero codewords is covered by any other …

Wide minimal binary linear codes from the general Maiorana–McFarland class

Minimal linear codes form a special class of linear codes that have important applications in secret sharing and secure two-party computation. These codes are characterized by the property that linearly independent codewords do not cover each other. …

Characterization of Basic 5-Value Spectrum Functions Through Walsh-Hadamard Transform

The first and the third authors recently introduced a spectral construction of plateaued and of 5-value spectrum functions. In particular, the design of the latter class requires a specification of integers $\\{W(u)\\colon u \\in {F}_2^n\\}$, where …

Proving the conjecture of O’Donnell in certain cases and disproving its general validity

For a function $f:\\{−1,1\\}^n\\to \\{−1,1\\}$ the relationship between the sum of its linear Fourier coefficients $\\hat{f}(i)$ (defined by $\\hat{f}(i)≔\\frac{1}{2^n}\\sum_{x\in \\{−1,1\\}^n} f(x)x_i$ for $i=1,2,\\ldots,n$ and …