We show that many infinite classes of permutations over finite fields can be constructed via translators with a large choice of parameters. We first charac- terize some functions having linear translators, based on which several families of permutations are then derived. Extending the results of [10], we give in several cases the compositional inverse of these permutations. The connection with complete permutations is also utilized to provide further infinite classes of permutations. Moreover, we propose new tools to study permutations of the form x is mapped to x+(x^(p^m) - x+ lambda)^s and a few infinite classes of permutations of this form are proposed.
We design a heuristic method, a genetic algorithm, for the computation of an upper bound of the minimum distance of a linear code over a finite field. By the use of the row reduced echelon form, we obtain a permutation encoding of the problem, so that its space of solutions does not depend on the size of the base field or the dimension of the code. Actually, the efficiency of our method only grows non-polynomially with respect to the length of the code.
This paper considers a noisy data structure recovery problem. The goal is to investigate the following question: Given a noisy observation of a permuted data set, according to which permutation was the original data sorted? The focus is on scenarios where data is generated according to an isotropic Gaussian distribution, and the noise is additive Gaussian with an arbitrary covariance matrix. This problem is posed within a hypothesis testing framework. The objective is to study the linear regime in which the optimal decoder has a polynomial complexity in the data size, and it declares the permu…
In this paper we prove new lower bounds for the maximal size of permutation codes by connecting the theory of permutation codes with the theory of linear block codes. More specifically, using the columns of a parity check matrix of an $[n,k,d]_q$ linear block code, we are able to prove the existence of a permutation code in the symmetric group of degree $n$, having minimum distance at least $d$ and large cardinality. With our technique, we obtain new lower bounds for permutation codes that enhance the ones in the literature and provide asymptotic improvements in certain regimes of length and d…
All almost perfect nonlinear (APN) permutations that we know to date admit a special kind of linear self-equivalence, i.e., there exists a permutation $G$ in their CCZ-equivalence class and two linear permutations $A$ and $B$, such that $G \circ A = B \circ G$. After providing a survey on the known APN functions with a focus on the existence of self-equivalences, we search for APN permutations in dimension 6, 7, and 8 that admit such a linear self-equivalence. In dimension six, we were able to conduct an exhaustive search and obtain that there is only one such APN permutation up to CCZ-equival…
Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating matrix rows that provides a codeword with minimum weight. Therefore, as the dimension of the code or the size of the underlying finite field increase, so it does exponentially the run time. In this work, we prove that, given a generating matrix, there exists a column permutation which leads to a reduced row echelon form containing a row whose weight is the cod…
Permutation resemblance measures the distance of a function from being a permutation. Here we show how to determine the permutation resemblance through linear integer programming techniques. We also present an algorithm for constructing feasible solutions to this integer program, and use it to prove an upper bound for permutation resemblance for some special functions. Additionally, we present a generalization of the linear integer program that takes a function on a finite group and determines a permutation with the lowest differential uniformity among those most resembling it.
Let $r\geq 3$ be any positive integer which is relatively prime to $p$ and $q^2\equiv 1 \pmod r$. Let $τ_1, τ_2$ be any permutation polynomials over $\mathbb{F}_{q^2},$ $σ_M$ is an invertible linear map over $\mathbb{F}_{q^2}$ and $σ=τ_1\circσ_M\circτ_2$. In this paper, we prove that, for suitable $τ_1, τ_2$ and $σ_M$, the map $σ$ could be $r$-regular complete permutation polynomials over quadratic extension fields.
