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Finding functions, particularly permutations, with good differential properties has received a lot of attention due to their varied applications. For instance, in combinatorial design theory, a correspondence of perfect $c$-nonlinear functions and difference sets in some quasigroups was recently shown by Anbar et al. (J. Comb. Des.31(12):1–24, 2023). Additionally, in a recent manuscript by Pal et al. (Adv. Math. Communications, to appear), a very interesting connection between the $c$-differential uniformity and boomerang uniformity, when $c=-1$, was pointed out, showing that they are the same for an odd APN permutation, sparking yet more interest in the construction of functions with low $c$-differential uniformity. We investigate the $c$-differential uniformity of some classes of permutation polynomials. As a result, we add four more classes of permutation polynomials to the family of functions that only contains a few (non-trivial) perfect $c$-nonlinear functions over finite fields of even characteristic. Moreover, we include a class of permutation polynomials with low $c$-differential uniformity over the field of characteristic $3$. To solve the involved equations over finite fields, we use various number theoretical techniques, in particular, we find explicitly many Walsh transform coefficients and Weil sums that may be of an independent interest.

The Feistel Boomerang Connectivity Table and the related notion of $F$-Boomerang uniformity (also known as the second-order zero differential uniformity) have been recently introduced by [H. Boukerrou, P. Huynh, V. Lallemand, B. Mandal and M. Minier, On the Feistel counterpart of the boomerang connectivity table, IACR Trans. Symmetric Cryptol.1 (2020) 331–362]. In the same paper, a characterization of almost perfect nonlinear (APN) functions over fields of even characteristic in terms of second-order zero differential uniformity was also given. Here, we find a sufficient condition for an odd or even function over fields of odd characteristic to be an APN function, in terms of second-order zero differential uniformity. Moreover, we compute the second-order zero differential spectra of several APN or other low differential uniform functions, and show that our considered functions also have low second-order zero differential uniformity, though it may vary widely, unlike the case for even characteristic, when it is always zero.

Determining the weight distributions of the projective Reed–Muller codes is a very hard problem and has been studied extensively in the literature. In this paper, we provide an alternative proof of the second weight of the projective Reed–Muller codes ${\mathrm{\text{PRM}}}_q(d,m)$ where $m\ge 3$ and $3\le d\le \frac{q+3}{2}$. We show that the second weight is attained by codewords that correspond to hypersurfaces containing a hyperplane under the hypothesis on $d$. Furthermore, we compute the second weight of ${\mathrm{\text{PRM}}}_q(d,2)$ for $3\le d\le q$. Furthermore, we give an upper bound for the third weight of ${\mathrm{\text{PRM}}}_q(d,2)$.

An upper bound for the maximum number of rational points on a hypersurface in a projective space over a finite field has been conjectured by Tsfasman and proved by Serre in 1989. The analogue question for hypersurfaces on weighted projective spaces has been considered by Aubry, Castryck, Ghorpade, Lachaud, O’Sullivan and Ram in 2017. A conjecture has been proposed there under the assumption that the first weight is equal to one and proved in the particular case of the dimension 2. We prove here the conjecture in any dimension provided the second weight is also equal to one.

In this paper, we characterize and classify the orbits of the fixed-point group on the unipotent elements of the generalized symmetric spaces for inner involutions of ${\mathrm{\text{SL}}}_3(k)$ and ${\mathrm{\text{SL}}}_4(k)$ where $k$ is a finite field of even characteristic. We also provide some generalized results for ${\mathrm{\text{SL}}}_n(k)$. These results, together with our previous work that determined similar orbits when $k$ is a finite field of odd characteristic, complete the classification of the orbits of the fixed-point group on the unipotent elements of the generalized symmetric spaces for inner involutions of ${\mathrm{\text{SL}}}_3(k)$ and ${\mathrm{\text{SL}}}_4(k)$ for any finite field $k$ of any characteristic.

This paper characterizes Goppa codes of certain maximal curves over finite fields defined by equations of the form $y^n=x^m+x$. We investigate algebraic geometric and quantum stabilizer codes associated with these maximal curves and propose modifications to improve their parameters. The theoretical analysis is complemented by extensive simulation results, which validate the performance of these codes under various error rates. We provide concrete examples of the constructed codes, comparing them with known results to highlight their strengths and trade-offs. The simulation data, presented through detailed graphs and tables, offers insights into the practical behavior of these codes in noisy environments. Our findings demonstrate that while the constructed codes may not always achieve optimal minimum distances, they offer systematic construction methods and interesting parameter trade-offs that could be valuable in specific applications or for further theoretical study.

By embedding a Toeplitz matrix-vector product (MVP) of dimension n into a circulant MVP of dimension $\text{N = 2n +}\delta \text{-1}$, where δ can be any nonnegative integer, we present a GF(2ⁿ) multiplication algorithm. This algorithm leads to a new redundant representation, and it has two merits: (1) The flexible choices of δ make it possible to select a proper N such that the multiplication operation in ring ${\text{GF(2)[x]/(x}}^{\text{N}}\text{ + 1)}$ can be performed using some asymptotically faster algorithms, e.g. the Fast Fourier Transformation (FFT)-based multiplication algorithm; (2) The redundant degrees, which are defined as N/n, are smaller than those of most previous GF(2ⁿ) redundant representations, and in fact they are approximately equal to 2 for all applicable cases.

Although for more than 20 years, Whiteman’s generalized cyclotomic sequences have been thought of as the most important pseudo-random sequences, but, there are only a few papers in which their 2-adic complexities have been discussed. In this paper, we construct a class of binary sequences of order four with odd length (product of two distinct odd primes) from Whiteman’s generalized cyclotomic classes. After that, we determine both 2-adic complexity and linear complexity of these sequences. Our results show that these complexities are greater than half of the period of the sequences, therefore, it may be good pseudo-random sequences.

Linear code with locality $r$ and availability $t$ is that the value at each coordinate $i$ can be recovered from $t$ disjoint repairable sets each containing at most $r$ other coordinates. This property is particularly useful for codes in distributed storage systems because it permits local repair of failed nodes and parallel access of hot data. In this paper, two constructions of $(r,t)$-locally repairable linear codes based on totally isotropic subspaces in symplectic space ${𝔽}_q^{(2\nu )}$ over finite fields ${𝔽}_q$ are presented. Meanwhile, comparisons are made among the $(r,t)$-locally repairable codes we construct, the direct product code in Refs. [8], [11] and the codes in Ref. [9] about the information rate $\frac{k}{n}$ and relative distance $\frac{d}{n}$.

In this short note, we present two classes of (almost) perfect $c$-nonlinear permutations over finite fields of even characteristic.

Cryptography in current digital world offers integrity and confidentiality of data being transferred over a communication network. Modern cryptography provides such security through cryptographic algorithms which mainly involve multiplication operation in finite fields. Since the finite field multiplication operation is a computationally intensive operation, various algorithms and architectures are proposed in the literature to obtain efficient finite field multiplications in both hardware and software. In this paper, a modified interleaved multiplication algorithm to perform multiplication of two finite field elements over GF(2^m) for pentanomials is derived from a conventional interleaved multiplication algorithm using a novel Pre-Computation (PC) technique to reduce the computational complexity of the algorithm. Consequently, an m-bit systolic multiplier for pentanomials (SMP) is designed by employing the proposed algorithm. Hardware and delay complexity analysis is performed and comparison of the proposed SMP structure with similar multipliers available in the literature is presented. The SMP structure achieves about 28% improvement in hardware for $m=283$ when compared with the best multiplier available in the literature. The functionality of the proposed SMP structure is verified by implementing on a field-programmable gate array (FPGA) Virtex-7 (XC7V2000TFLG1925-2) prototype board and synthesizing in application specific integrated circuit (ASIC) using Synopsys Design Vision compiler with 90nm generic library. It can be observed from FPGA and ASIC implementation results that the proposed SMP structure shows improvement in area and power consumption when compared with similar multipliers available in the literature.

Rescaled evolution sets of linear cellular automata on a lattice ℤ ⊕ G, where G is a finite Abelian group, with states in the finite field are defined and investigated.

We determine the integers a, b ≥ 1 and the prime powers q for which the word map w(x, y) = x^ay^b is surjective on the group PSL(2, q) (and SL(2, q)). We moreover show that this map is almost equidistributed for the family of groups PSL(2, q) (and SL(2, q)). Our proof is based on the investigation of the trace map of positive words.

We give an LU-decomposition of the supercharacter table of the group of n × n unipotent upper triangular matrices over 𝔽_q, into a lower-triangular matrix with entries in ℤ[q] and an upper-triangular matrix with entries in ℤ[q^-1]. To this end, we introduce a q deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommuting variables. The decomposition is obtained from the transition matrices between the supercharacter basis, the q-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group S_n given by the transition matrices between Schur functions, monomials and power-sums. We deduce some combinatorial results associated to this decomposition. In particular, we compute the determinant of the supercharacter table.

We study the nilpotency degree of a relatively free finitely generated associative algebra with the identity xⁿ = 0 over a finite field 𝔽 with q elements. In the case of q ≥ n the nilpotency degree is proven to be the same as in the case of an infinite field of the same characteristic. In the case of q = n - 1 it is shown that the nilpotency degree differs from the nilpotency degree for an infinite field of the same characteristic by at most one. The nilpotency degree is explicitly computed for n = 3.

Let $F$ be a finite field with the characteristic $p>2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis for ${\mathbb{Z}}_2$-graded polynomial identities for any ${\mathbb{Z}}_2$-grading such that its underlying vector space is homogeneous.

Let $M$ be a closed oriented 3-manifold and let $G$ be a discrete group. We consider a representation $\rho :{\pi }_1(M)\to G$. For a 3-cocycle $\alpha$, the Dijkgraaf–Witten invariant is given by $({\rho }^{\ast }\alpha )[M]$, where ${\rho }^{\ast }:H^3(G)\to H^3(M)$ is the map induced by $\rho$, and $[M]$ denotes the fundamental class of $M$. Note that $({\rho }^{\ast }\alpha )[M]=\alpha ({\rho }_{\ast }[M])$, where ${\rho }_{\ast }:H_3(M)\to H_3(G)$ is the map induced by $\rho$, we consider an equivalent invariant ${\rho }_{\ast }[M]\in H_3(G)$, and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the complex hyperbolic volume of $M$ in terms of the image of the Dijkgraaf–Witten invariant for $G={\mathrm{\text{SL}}}_2ℂ$ by the Bloch–Wigner map from $H_3({\mathrm{\text{SL}}}_2ℂ)$ to the Bloch group of $ℂ$.

In this paper, by replacing $ℂ$ with a finite field ${𝔽}_p$, we calculate the reduced Dijkgraaf–Witten invariants of the complements of twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL${}_2{𝔽}_p$ by the Bloch–Wigner map from $H_3({\mathrm{\text{SL}}}_2{𝔽}_p)$ to the Bloch group of ${𝔽}_p$.

In 2004, Neumann showed that the complex hyperbolic volume of a hyperbolic 3-manifold $M$ can be obtained as the image of the Dijkgraaf–Witten invariant of $M$ by a certain 3-cocycle. After that, Zickert gave an analogue of Neumann’s work for free fields containing finite fields. The author formulated a geometric method to calculate a weaker version of Zickert’s analogue, called the reduced Dijkgraaf–Witten invariant, for finite fields and gave a formula for twist knot complements and ${𝔽}_p$ in his previous work. In this paper, we show concretely how to calculate the reduced Dijkgraaf–Witten invariants of double twist knot complements and ${𝔽}_p$, and give a formula of them for $p=7$.

An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo $p$. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of ${𝔽}_3$. As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over ${𝔽}_q$ which attains Perret's and Weil's upper bounds.

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.