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Whereas the design and properties of bent and plateaued functions have been frequently addressed during the past few decades, there are only a few design methods of the so-called five-valued spectra Boolean functions whose Walsh spectra take the values in $\{0, \pm 2^{\lambda _{1}}, \pm 2^{\lambda _{2}}\}$. Moreover, these design methods mainly regard the specification of these functions in their algebraic normal form (ANF) domain. In this paper, we give a precise characterization of this class of functions in their spectral domain using the concept of a dual of plateaued functions. Both necessary and sufficient conditions on the Walsh support of these functions are given, which then connects their design (in the spectral domain) to a family of the so-called totally (non-overlap) disjoint spectra plateaued functions. We identify some suitable families of plateaued functions having this property, thus providing some generic methods in the spectral domain. Furthermore, we also provide an extensive analysis of their constructions in the ANF domain and provide several generic design methods. The importance of this class of functions is manifolded, where apart from being suitable for some cryptographic applications, we emphasize their property of being constituent functions in the so-called four-bent decomposition. [ABSTRACT FROM AUTHOR]

Interleaved Reed–Solomon codes admit efficient decoding algorithms which correct burst errors far beyond half the minimum distance in the random errors regime, e.g., by computing a common solution to the Key Equation for each Reed–Solomon code, as described by Schmidt et al. If this decoder does not succeed, it may either fail to return a codeword or miscorrect to an incorrect codeword, and good upper bounds on the fraction of error matrices for which these events occur are known. The decoding algorithm immediately applies to interleaved alternant codes as well, i.e., the subfield subcodes of interleaved Reed–Solomon codes, but the fraction of decodable error matrices differs, since the error is now restricted to a subfield. In this paper, we present new general lower and upper bounds on the fraction of error matrices decodable by Schmidt et al.’s decoding algorithm, thereby making it the only decoding algorithm for interleaved alternant codes for which such bounds are known. [ABSTRACT FROM AUTHOR]

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 $\{W(u):u\in \mathbb {F}^{n}_{2}\}$ , where $W(u)\in \left\{{0, \pm 2^{\frac {n+s_{1}}{2}}, \pm 2^{\frac {n+s_{2}}{2}}}\right\}$ , so that the sequence $\{W(u):u\in \mathbb {F}^{n}_{2}\}$ is a valid spectrum of a Boolean function (recovered using the inverse Walsh transform). Technically, this is done by allocating a suitable Walsh support $S=S^{[{1}]}\cup S^{[{2}]}\subset \mathbb {F}^{n}_{2}$ , where $S^{[i]}$ corresponds to those $u \in \mathbb {F} _{2}^{n}$ for which $W(u)=\pm 2^{\frac {n+s_{i}}{2}}$. In addition, two dual functions $g_{[i]}:S^{[i]}\rightarrow \mathbb {F}_{2}$ (with $\#S^{[i]}=2^{\lambda _{i}}$) are employed to specify the signs through $W(u)=2^{\frac {n+s_{i}}{2}}(-1)^{g_{[i]}(u)}$ for $u\in S^{[i]}$ whereas $W(u)=0$ for $u\not \in S$. In this work, two closely related problems are considered. Firstly, the specification of plateaued functions (duals) $g_{[i]}$ , which additionally satisfy the so-called totally disjoint spectra property, is fully characterized (so that $W(u)$ is a spectrum of a Boolean function) when the Walsh support $S$ is given as a union of two disjoint affine subspaces $S^{[i]}$. Especially, when plateaued dual functions $g_{[i]}$ themselves have affine Walsh supports, an efficient spectral design that utilizes arbitrary bent functions (as duals of $g_{[i]}$) on the corresponding ambient spaces is given. The problem of specifying affine inequivalent 5-value spectra functions is also addressed and an efficient construction method that ensures the inequivalence property is derived (sufficient condition being a selection of affine inequivalent duals). In the second part of this work, we investigate duals of plateaued functions with affine Walsh supports. For a given such plateaued function, we show that different orderings of its Walsh support which are employing the Sylvester-Hadamard recursion actually induce bent duals which are affine equivalent. [ABSTRACT FROM AUTHOR]