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Let q = 2 m. In this paper, we investigate permutation pentanomials over F q 2 of the form f (x) = x t + x r 1 (q − 1) + t + x r 2 (q − 1) + t + x r 3 (q − 1) + t + x r 4 (q − 1) + t with gcd (x r 4 + x r 3 + x r 2 + x r 1 + 1 , x t + x t − r 1 + x t − r 2 + x t − r 3 + x t − r 4 ) = 1. We transform the problem concerning permutation property of f (x) into demonstrating that the corresponding fractional polynomial permutes the unit circle U of F q 2 with order q + 1 via a well-known lemma, and then into showing that there are no certain solution in F q for some high-degree equations over F q associated with the fractional polynomial. According to numerical data, we have found all such permutations with 4 ≤ t < 100 , 1 ≤ r i ≤ t , i ∈ [ 1 , 4 ]. Several permutation polynomials are also investigated from the fractional polynomials permuting the unit circle U found in this paper. [ABSTRACT FROM AUTHOR]

Linear complementary dual codes (LCD codes for short) are an important subclass of linear codes which have nice applications in communication systems, cryptography, consumer electronics and information protection. In the literature, it has been proved that an [ n , k , d ] Euclidean LCD code over F q with q > 3 exists if there is an [ n , k , d ] linear code over F q , where q is a prime power. However, the existence of binary and ternary Euclidean LCD codes has not been totally characterized. Hence it is interesting to construct binary and ternary Euclidean LCD codes with new parameters. In this paper, we construct new families of binary and ternary leading-systematic Euclidean LCD codes from some special functions including semibent functions, quadratic functions, almost bent functions, and planar functions. These LCD codes are not constructed directly from such functions, but come from some self-orthogonal codes constructed with such functions. Compared with known binary and ternary LCD codes, the LCD codes in this paper have new parameters. [ABSTRACT FROM AUTHOR]

This paper corrects an error in the proof of Theorem 1.4 (3) of our earlier paper, Further Improvements to the Chevalley-Warning Theorems. The error originally appeared in Heath-Brown's paper, On Chevalley-Warning Theorems , which invalidates the proof of Theorem 2 (iii) in that paper. In this paper, we use a new method to give a correct proof of Theorem 1.4 (3). The correction in this paper also fixes the proof of Theorem 2 (iii) in Heath-Brown's paper. The proof in this paper provides slightly stronger estimates for some of the inequalities that were used in Further Improvements to the Chevalley-Warning Theorems. [ABSTRACT FROM AUTHOR]

The classical way of extending an [ n , k , d ] linear code C is to add an overall parity-check coordinate to each codeword of the linear code C. This extended code, denoted by C ‾ (− 1) and called the standardly extended code of C , is a linear code with parameters [ n + 1 , k , d ¯ ] , where d ¯ = d or d ¯ = d + 1. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper. [ABSTRACT FROM AUTHOR]

Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this paper, we consider the recently introduced linear version of such sets, which is, in turn, related to the covering problem in the rank metric. The main questions in this context are how small the rank of a saturating linear set can be and how to construct saturating linear sets of small rank. Recently, Bonini, Borello, and Byrne provided a lower bound on the rank of saturating linear sets in a given projective space, which is shown to be tight in some cases. In this paper, we provide construction of saturating linear sets meeting the lower bound and we develop a link between the saturating property and the scatteredness of linear sets. The last part of the paper is devoted to show some parameters for which the bound is not tight. [ABSTRACT FROM AUTHOR]

Over the past several years, there are numerous papers about permutation polynomials of the form x r h (x q − 1) over F q 2 . A bijection between the multiplicative subgroup μ q + 1 of F q 2 and the projective line PG (1 , q) = F q ∪ { ∞ } plays a very important role in the research. In this paper, we mainly construct permutation polynomials of the form x r h (x q − 1) over F q 3 from bijections of the projective plane PG (2 , q). A bijection from the multiplicative subgroup μ q 2 + q + 1 of F q 3 to PG (2 , q) is studied, which is a key theorem of this paper. On this basis, some explicit permutation polynomials of the form x r h (x q − 1) over F q 3 are constructed from the collineation of PG (2 , q) , d -homogeneous monomials, 2-homogeneous permutations. It is worth noting that although the bijections of PG (2 , q) are simple, the corresponding permutation polynomials over F q 3 are usually complex. [ABSTRACT FROM AUTHOR]

Locally repairable codes (LRCs) are a class of erasure codes that are widely used in distributed storage systems, which allow for efficient recovery of data in the case of node failures or data loss. In 2014, Tamo and Barg introduced Reed-Solomon-like (RS-like) Singleton-optimal (r , δ) -LRCs based on polynomial evaluation. These constructions rely on the existence of so-called good polynomial that is constant on each of some pairwise disjoint subsets of F q. In this paper, we extend the aforementioned constructions of RS-like LRCs and propose new constructions of (r , δ) -LRCs whose code length can be larger. These new (r , δ) -LRCs are all distance-optimal, namely, they attain an upper bound on the minimum distance that will be established in this paper. This bound is sharper than the Singleton-type bound in some cases owing to the extra conditions, it coincides with the Singleton-type bound for certain cases. Combining our constructions with known explicit good polynomials of special forms, we can get various explicit Singleton-optimal (r , δ) -LRCs with new parameters, whose code lengths are all larger than that constructed by the RS-like (r , δ) -LRCs introduced by Tamo and Barg. Note that the code length of classical RS codes and RS-like LRCs are both bounded by the field size. We explicitly construct the Singleton-optimal (r , δ) -LRCs with length n = q − 1 + δ for any positive integers r , δ ≥ 2 and (r + δ − 1) | (q − 1). We also show the existence of Singleton-optimal (r , δ) -LRCs with length q + δ over F q = F p a (a ≥ 3) provided p 2 | (r + δ − 1) , (r + δ − 1) | p a − 1 and p | δ. When δ is proportional to q , they are asymptotically longer than that constructed via elliptic curves whose length is at most q + 2 q. Besides, they allow more flexibility on the values of r and δ. [ABSTRACT FROM AUTHOR]

We prove two conjectures involving permutation polynomials in a paper of Dmytrenko, Lazebnik, Williford, in a low degree regime, using the theory of algebraic curves over finite fields. More precisely, we prove that Conjecture A holds whenever q ≥ max ⁡ { (2 k − 1) 2 + 1 , 1.823 (4 k 2 − 14 k + 12) } , whereas Conjecture B holds if q ≥ 2.233 (9 k 2 − 21 k + 12). Although one of these conjectures was already proved by Hou without any restriction on the degree of the polynomials, we consider the proof contained in this paper is more direct and less computational. [ABSTRACT FROM AUTHOR]

This paper aims to study linear sets of minimum size on the projective line, that is F q -linear sets of rank k in PG (1 , q n) admitting one point of weight one and having size q k − 1 + 1. Examples of these linear sets have been found by Lunardon and the second author (2000) and, more recently, by Jena and Van de Voorde (2021). However, classification results for minimum size linear sets are known only for k ≤ 5. In this paper we provide classification results for minimum size linear sets admitting two points with complementary weights. We construct new examples (not necessarily with complementary weights) and also study the related Γ L (2 , q n) -equivalence issue. These results solve an open problem posed by Jena and Van de Voorde. The main tool relies on two results by Bachoc, Serra and Zémor (2017 and 2018) on the linear analogues of Kneser's and Vosper's theorems. We then conclude the paper pointing out a connection between critical pairs and linear sets, obtaining also some classification results for critical pairs. [ABSTRACT FROM AUTHOR]

CSIDH is an isogeny-based key exchange protocol proposed by Castryck et al. in 2018. It is based on the ideal class group action on F p -isomorphism classes of Montgomery curves. The original CSIDH algorithm requires a calculation over F p by representing points as x -coordinate over Montgomery curves. There is a special coordinate on Edwards curves (the w -coordinate) to calculate group operations and isogenies. If we try to calculate the class group action on Edwards curves by using the w -coordinate in a similar way on Montgomery curves, we have to consider points defined over F p 4 . Therefore, calculating the class group action on Edwards curves with w -coordinates over only F p is not a trivial task. In this paper, we prove some theorems about the properties of Edwards curves. We construct the new CSIDH algorithm using these theorems on Edwards curves with w -coordinates over F p. This algorithm is as fast as (or a little bit faster than) the algorithm proposed by Meyer and Reith. This paper is an extended version of [29]. We added the construction of a technique similar to Elligator on Edwards curves. This technique contributes to the efficiency of the constant-time CSIDH algorithm. We also added the construction of new formulas to compute isogenies in O ˜ (ℓ) time on Edwards curves. It is based on formulas on Montgomery curves proposed by Bernstein et al. ( élu's formulas). In our analysis, these formulas on Edwards curves are a little bit faster than those on Montgomery curves. We finally implemented CSIDH, élu's formulas, and CTIDH [3] (faster constant-time CSIDH) on Edwards curves. Each result shows the efficiency of algorithms on Edwards curves. [ABSTRACT FROM AUTHOR]

For integer k ≥ 2 and prime power q , the algebraic bipartite graph D (k , q) proposed by Lazebnik and Ustimenko in 1993 is useful not only in extremal graph theory but also in coding theory and cryptography. This graph is q -regular, edge-transitive and of girth at least k + 4. Its exact girth g = g (D (k , q)) was conjectured in 1995 to be k + 5 for odd k and q ≥ 4. This conjecture was shown to be valid in 2016 when k + 5 2 | p (q − 1) , where p is the characteristic of F q and m | p n means that m divides p r n for some nonnegative integer r. In this paper, for t ≥ 1 , we prove that (a) g (D (4 t + 2 , q)) = g (D (4 t + 1 , q)) ; (b) g (D (4 t + 3 , q)) = 4 t + 8 if g (D (2 t , q)) = 2 t + 4 ; (c) g (D (8 t , q)) = 8 t + 4 if g (D (4 t − 2 , q)) = 4 t + 2 ; (d) g (D (2 s + 2 t − 5 , q)) = 2 s + 2 t if p ≥ 3 , 2 s | (q − 1) , 2 s + 1 ∤ (q − 1) , 2 ∤ t and t | p (q − 1). A simple upper bound for the girth of D (k , q) is proposed in the end of this paper. [ABSTRACT FROM AUTHOR]

In the thirties of the last century, I.M. Vinogradov proved that the inequality | | p α | | ≤ p − 1 / 5 + ε has infinitely prime solutions p , where | |. | | denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent 1/5 by 1/4 using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality | | p α | | ≤ p − 1 / 3 + ε. This exponent 1/3 is considered the limit of the current technology. Recently, in [3] , the authors established an analogue of Matomäki's result on imaginary quadratic extensions of the function field k = F q (T). In this paper, we consider the case of real quadratic extensions of k of class number 1, for which we prove a function field analogue of Vaughan's above-mentioned result (exponent θ = 1 / 4). Our method uses versions of Vaughan's identity and the Dirichlet approximation theorem for function fields. The latter was established by Arijit Ganguly in the appendix to our previous paper [3] on the imaginary quadratic case. We also simplify arguments in the paper [1] on the same problem for real quadratic number fields by D. Mazumder and the first-named author. [ABSTRACT FROM AUTHOR]

Permutation polynomials with low c -differential uniformity have important applications in cryptography and combinatorial design. In this paper, we investigate perfect c -nonlinear (PcN) and almost perfect c -nonlinear (APcN) polynomials over finite fields. Based on some known permutation polynomials, we present several classes of PcN or APcN polynomials by using the Akbary-Ghioca-Wang criterion. [ABSTRACT FROM AUTHOR]

In this paper we find the largest automorphism group of a smooth cubic surface over any finite field of characteristic 2. We prove that if the order of the field is a power of 4, then the automorphism group of maximal order of a smooth cubic surface over this field is PSU 4 (F 2). If the order of the field of characteristic 2 is not a power of 4, then we prove that the automorphism group of maximal order of a smooth cubic surface over this field is the symmetric group of degree 6. Moreover, we prove that smooth cubic surfaces with such properties are unique up to isomorphism. [ABSTRACT FROM AUTHOR]

n -to-1 mappings have many applications in cryptography, finite geometry, coding theory and combinatorial design. In this paper, we first use cyclotomic cosets to construct several kinds of n -to-1 mappings over F q ⁎. Then we characterize a new form of AGW-like criterion, and use it to present many classes of n -to-1 polynomials with the form x r h (x q − 1) over F q 2 ⁎. Finally, by using monomials on the cosets of a subgroup of μ q + 1 and another form of AGW-like criterion, we show some n -to-1 trinomials over F q 2 ⁎. [ABSTRACT FROM AUTHOR]

In this paper we completely classify spreads of 2-dimensional subspaces of a 6-dimensional vector space over a finite field of characteristic not two or three upon which a cyclic group acts transitively. This addresses one of the remaining open cases in the classification of flag-transitive linear spaces. We utilise the polynomial approach innovated by Pauley and Bamberg to obtain our results. [ABSTRACT FROM AUTHOR]

An explicit formula for the number of solutions of the equation in the title is given when a certain condition, depending only on the exponent and the characteristic of the field, holds. This formula improves the one given by the authors in an earlier paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the (2 k + 1) -th moment to the zeroeth moment as the size of the finite field F p r goes to infinity. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. As an application, we prove that the distribution of the trace of Frobenius in arithmetic progressions is equidistributed with respect to the Sato–Tate measure. [ABSTRACT FROM AUTHOR]

A (projective, geometrically irreducible, non-singular) curve X defined over a finite field F q 2 is maximal if the number N q 2 of its F q 2 -rational points attains the Hasse-Weil upper bound, that is N q 2 = q 2 + 2 g q + 1 where g is the genus of X. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order p 2 where p is the characteristic of F q 2 . Doing so we also determine the F q 2 -isomorphism classes of such curves and describe their full F q 2 -automorphism groups. [ABSTRACT FROM AUTHOR]

The notion of a Heffter array, which received much attention in the last decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we study the existence problem of a set of r mutually orthogonal Heffter systems for any r. Such a set is equivalent to a resolvable partial linear space of degree r whose parallel classes are Heffter systems: this is a new combinatorial design that we call a Heffter space. We present a series of direct constructions of Heffter spaces with odd block size and arbitrarily large degree r obtained with the crucial use of finite fields. Among the applications we establish, in particular, that if q = 2 k w + 1 is a prime power with kw odd and k ≥ 3 , then there are at least ⌈ w 4 k 4 ⌉ mutually orthogonal k -cycle systems of order q. [ABSTRACT FROM AUTHOR]

It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in P 4 (F q 2 ) has at most d (q 5 + q 2) + q 3 + 1 points in common with a threefold of degree d defined over F q 2 . He proved the conjecture for d = 2. In this paper, we show that the conjecture is true for d = 3 and q ≥ 7. [ABSTRACT FROM AUTHOR]

Let F p n be a finite field with p n elements. Ness and Helleseth in [29] first studied a class of functions over F p n with the form f (x) = u x p n − 3 2 + x p n − 2 , u ∈ F p n ⁎ , which is called the Ness-Helleseth function. The f (x) has been proved to be an almost perfect nonlinear (APN) function by Ness and Helleseth for p = 3 in [29] and by Zeng et al. for any odd prime p in [43] under the condition p n ≡ 3 (mod 4) and η (1 + u) = η (u − 1). In this paper, we continue to study the Ness-Helleseth functions under the condition that p n ≡ 3 (mod 4) and η (1 + u) ≠ η (u − 1). Firstly, we prove that f (x) is a permutation polynomial with differential uniformity not more than 4 if η (1 + u) = η (1 − u). Moreover, for some more special u , f is an involution with differential uniformity at most 3. Secondly, we show that f (x) is a locally-APN function for u = ± 1. In addition, the differential spectrum and boomerang spectrum of f (x) are obtained via judging the number of solutions of some special equations. We obtain the first non-PN function that its boomerang uniformity can attain 0 or 1. [ABSTRACT FROM AUTHOR]

In this paper we introduce a multistep generalization of the guess-and-determine or hybrid strategy for solving a system of multivariate polynomial equations over a finite field. In particular, we propose performing the exhaustive evaluation of a subset of variables stepwise, that is, by incrementing the size of such subset each time that an evaluation leads to a polynomial system which is possibly unfeasible to solve. The decision about which evaluation to extend is based on a preprocessing consisting in computing an incomplete Gröbner basis after the current evaluation, which possibly generates linear polynomials that are used to eliminate further variables. If the number of remaining variables in the system is deemed still too high, the evaluation is extended and the preprocessing is iterated. Otherwise, we solve the system by a complete Gröbner basis computation. Having in mind cryptanalytic applications, we present an implementation of this strategy in an algorithm called MultiSolve which is designed for polynomial systems having at most one solution. We prove explicit formulas for its complexity which are based on probability distributions that can be easily estimated by performing the proposed preprocessing on a testset of evaluations for different subsets of variables. We prove that an optimal complexity of MultiSolve is achieved by using a full multistep strategy with a maximum number of steps and in turn the standard guess-and-determine strategy, which essentially is a strategy consisting of a single step, is the worst choice. Finally, we extensively study the behaviour of MultiSolve when performing an algebraic attack on the well-known stream cipher Trivium. [ABSTRACT FROM AUTHOR]

In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [9,24] , the authors proved that the trinomial f c (X) = X q + X q 3 + c X q 5 of F q 6 [ X ] is scattered under the assumptions that q is odd and c 2 + c = 1. They also explicitly observed that this is false when q is even. In this paper, we provide a different set of conditions on c for which this trinomial is scattered in the case of even q. Using tools of algebraic geometry in positive characteristic, we show that when q is even and sufficiently large, there are roughly q 3 elements c ∈ F q 6 such that f c (X) is scattered. Also, we prove that the corresponding MRD-codes and F q -linear sets of PG (1 , q 6) are not equivalent to the previously known ones. [ABSTRACT FROM AUTHOR]

Bonini, Borello and Byrne started the study of saturating linear sets in Desarguesian projective spaces, in connection with the covering problem in the rank metric. In this paper we study 1 -saturating linear sets in PG (2 , q 4) , that is F q -linear sets in PG (2 , q 4) with the property that their secant lines cover the entire plane. By making use of a characterization of generalized Gabidulin codes, we prove that the rank of such a linear set is at least 5. This answers to a recent question posed by Bartoli, Borello and Marino. [ABSTRACT FROM AUTHOR]

In this paper, as an analogue of the integer case, we study detailedly the period and the rank of the generalized Fibonacci sequences of polynomials over a finite field modulo an arbitrary polynomial. We establish some formulas to compute them, and we also obtain some properties about their quotient. We find that the polynomial case is much more complicated than the integer case. [ABSTRACT FROM AUTHOR]

In this paper, we examine superspecial genus-2 curves C : y 2 = x (x − 1) (x − λ) (x − μ) (x − ν) in odd characteristic p. As a main result, we show that the difference between any two elements in { 0 , 1 , λ , μ , ν } is a square in F p 2 . Moreover, we show that C is maximal or minimal over F p 2 without taking its F p 2 -form (we give an explicit criterion in terms of p that tells whether C is maximal or minimal). As an application, we also study the maximality of superspecial hyperelliptic curves of genera 3 and 4 whose automorphism groups contain Z / 2 Z × Z / 2 Z. [ABSTRACT FROM AUTHOR]

A normal element in a finite field extension F q n / F q is characterized by having linearly independent conjugates over F q. We consider the generalization of normal elements known as k -normal elements, where a subset of the conjugates are required to be linearly independent. In this paper, we provide an explicit combinatorial formula for counting the number of k -normal elements in a finite field extension motivated by an open problem proposed by Huczynska, Mullen, Panario, and Thomson in 2013. Furthermore, we use these results to establish new insights about F q -practical numbers. [ABSTRACT FROM AUTHOR]

In this paper we develop further studies on nilpotent linearized polynomials (NLP's) over finite fields, a class of polynomials introduced by the second author. We characterize certain NLP's that are binomials and show that, in general, NLP's are also nilpotent over a particular tower of finite fields. We also develop results on the construction of permutation polynomials from NLP's, extending some past results. In particular, the latter yields polynomials that permutes certain infinite subfields of F ‾ q and have a very particular cycle structure. Finally, we provide a nice correspondence between certain NLP's and involutions in binary fields and, in particular, we discuss a general method to produce affine involutions over binary fields without fixed points. [ABSTRACT FROM AUTHOR]

Ahlswede, Khachatrian, Mauduit and Sárközy introduced the notion of family complexity, Gyarmati, Mauduit and Sárközy introduced the cross-correlation measure for families of binary sequences. It is a challenging problem to find families of binary sequences with both small cross-correlation measure and large family complexity. In this paper we present a family of binary sequences with both small cross-correlation measure and large family complexity by using the properties of character sums and primitive normal elements in finite fields. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic p > 0. A pressing problem in the theory of algebraic curves is the determination of the p -rank of a (nonsingular, projective, irreducible) curve X over K. This birational invariant affects arithmetic and geometric properties of X , and its fundamental role in the study of the automorphism group Aut (X) has been noted by many authors in the past few decades. In this paper, we provide an extensive study of the p -rank of curves of Fermat type y m = x n + 1 over K = F ¯ p. We determine a combinatorial formula for this invariant in the general case and show how this leads to explicit formulas of the p -rank of several such curves. By way of illustration, we present explicit formulas for more than twenty subfamilies of such curves, where m and n are generally given in terms of p. We also show how the approach can be used to compute the p -rank of other types of curves. [ABSTRACT FROM AUTHOR]

Abstract Hyperovals in PG (2 , GF (q)) with even q are maximal arcs and an interesting research topic in finite geometries and combinatorics. Hyperovals in PG (2 , GF (q)) are equivalent to [ q + 2 , 3 , q ] MDS codes over GF (q) , called hyperoval codes, in the sense that one can be constructed from the other. Ovals in PG (2 , GF (q)) for odd q are equivalent to [ q + 1 , 3 , q − 1 ] MDS codes over GF (q) , which are called oval codes. In this paper, we investigate the binary subfield codes of two families of hyperoval codes and the p -ary subfield codes of the conic codes. The weight distributions of these subfield codes and the parameters of their duals are determined. As a byproduct, we generalize one family of the binary subfield codes to the p -ary case and obtain its weight distribution. The codes presented in this paper are optimal or almost optimal in many cases. In addition, the parameters of these binary codes and p -ary codes seem new. [ABSTRACT FROM AUTHOR]

Abstract This paper presents design of T - Direct codes over GF (2 N) with increased users. The class of T - Direct codes are an extension to the class of linear codes with complementary duals (LCD codes) and can be effectively used over a multiple access channel environment. In this paper, a construction procedure that constructs an n 3 - Direct code from an n - Direct code is described. An n n - Direct codes are also constructed. Further, a recursive construction procedure is presented that constructs n 3 m - Direct codes for m ≥ 0. Finally, T T - Direct codes with constituent codes having variable code rates are obtained. The advantage of these constructions is that it increases the number of constituent codes (users) of an existing T - Direct code, thereby supporting more users in a multi-user environment. [ABSTRACT FROM AUTHOR]

Abstract As a special class of linear codes, minimal linear codes have important applications in secret sharing and secure two-party computation. Constructing minimal linear codes with new and desirable parameters has been an interesting research topic in coding theory and cryptography. Ashikhmin and Barg showed that w min / w max > (q − 1) / q is a sufficient condition for a linear code over the finite field GF (q) to be minimal, where q is a prime power, w min and w max denote the minimum and maximum nonzero weights in the code, respectively. The first objective of this paper is to present a sufficient and necessary condition for linear codes over finite fields to be minimal. The second objective of this paper is to construct an infinite family of ternary minimal linear codes satisfying w min / w max ≤ 2 / 3. To the best of our knowledge, this is the first infinite family of nonbinary minimal linear codes violating Ashikhmin and Barg's condition. [ABSTRACT FROM AUTHOR]

Denniston constructed partial difference sets (PDSs) with the parameters (2 3 m , (2 m + r − 2 m + 2 r) (2 m − 1) , 2 m − 2 r + (2 m + r − 2 m + 2 r) (2 r − 2) , (2 m + r − 2 m + 2 r) (2 r − 1)) in elementary abelian groups of order 2 3 m for all m ≥ 2 , 1 ≤ r < m. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters (p 3 m , (p m + r − p m + p r) (p m − 1) , p m − p r + (p m + r − p m + p r) (p r − 2) , (p m + r − p m + p r) (p r − 1)) exist in all elementary abelian groups of order p 3 m for all m ≥ 2 , r ∈ { 1 , m − 1 } where p is an odd prime, and present a construction. Our approach uses PDSs formed as unions of cyclotomic classes. [ABSTRACT FROM AUTHOR]

In this paper, we determine the dual codes of cyclic codes of length p s over R 3 = F p m [ u ] 〈 u 3 〉 , where p is a prime number and u 3 = 0. Also, we improve and give correction of the results stated by B. Kim and J. Lee (2020) in [11]. Finally, we provide some examples of optimal and near-MDS cyclic codes of length p s over R 3 and compute dual of them. [ABSTRACT FROM AUTHOR]

For an [ n , k , d ] q linear code C , the singleton defect of C is defined by S (C) = n − k + 1 − d. When S (C) = S (C ⊥) = 1 , the code C is called a near maximum distance separable (NMDS) code, where C ⊥ is the dual code of C. NMDS codes have important applications in finite projective geometries, designs and secret sharing schemes. In this paper, we present four new constructions of infinite families of NMDS codes with dimension 4 and completely determine their weight enumerators. As an application, we also determine the locality of the dual codes of these NMDS codes and obtain four families of distance-optimal and dimension-optimal locally recoverable codes. [ABSTRACT FROM AUTHOR]

Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise from a subfield. In this paper, we investigate the existence of a clique of size q with a subspace structure in pseudo-Paley graphs of order q from unions of semi-primitive cyclotomic classes. We show that such a clique must have an equal contribution from each cyclotomic class and that most such pseudo-Paley graphs do not admit such cliques, suggesting that the Delsarte bound q on the clique number can be improved in general. We also prove that generalized Peisert graphs are not isomorphic to Paley graphs or Peisert graphs, confirming a conjecture of Mullin. [ABSTRACT FROM AUTHOR]

In this paper we compute the automorphism group of the curves X a , b , n , s and Y n , s introduced in Tafazolian et al. [27] as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained. [ABSTRACT FROM AUTHOR]

A complete complementary code (CCC) consists of M sequence sets with size M. The sum of the auto-correlation functions of each sequence set is an impulse function, and the sum of cross-correlation functions of the different sequence sets is equal to zero. Thanks to their excellent correlation, CCCs received extensive use in engineering. In addition, they are strongly connected to orthogonal matrices. In some application scenarios, additional requirements are made for CCCs, such as recently proposed for concatenative CCC (CCCC) division multiple access (CCC-CDMA) technologies. In fact, CCCCs are a special kind of CCCs which requires that each sequence set in CCC be concatenated to form a zero-correlation-zone (ZCZ) sequence set. However, this requirement is challenging, and the literature is thin since there is only one construction in this context. We propose to go beyond the literature through this contribution to reduce the gap between their interest and our limited knowledge of CCCCs. This paper will employ novel methods for designing CCCCs and precisely derive two constructions of these objects. The first is based on perfect cross Z-complementary pair and Hadamard matrices, and the second relies on extended Boolean functions. Specifically, we highlight that optimal and asymptotic optimal CCCCs could be obtained through the proposed constructions. Besides, we shall present a comparison analysis with former structures in the literature and examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the algorithm for computing chains of (2 , 2) -isogenies between products of elliptic curves via theta coordinates proposed by Dartois et al. For each fundamental block of this algorithm, we provide an explicit inversion-free version. Besides, we exploit the technique of x -only ladder to speed up the computation of gluing isogeny. Finally, we present a mixed optimal strategy, which combines the inversion-elimination tool with the original methods together to execute a chain of (2 , 2) -isogenies. We make a cost analysis and present a concrete comparison between ours and the previously known methods for inversion elimination. Furthermore, we implement the mixed optimal strategy for benchmark. The results show that when computing (2 , 2) -isogeny chains with lengths of 126, 208 and 632, compared to Dartois, Maino, Pope and Robert's latest implementation, utilizing our techniques can reduce 9.7%, 9.5% and 9.6% multiplications over the base field F p , respectively. Therefore, even for the updated version that employs their inversion-free algorithms, our tools still possess an advantage. [ABSTRACT FROM AUTHOR]

BCH codes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of BCH codes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of BCH codes, the concept of dually-BCH codes was proposed. In this paper, we study BCH codes of lengths q m + 1 q + 1 and q m + 1 over the finite field F q , both of which are LCD codes. The dimensions of narrow-sense BCH codes of length q m + 1 q + 1 with designed distance δ = ℓ q m − 1 2 + 1 are determined, where q > 2 and 2 ≤ ℓ ≤ q − 1. Lower bounds on the minimum distances of the dual codes of narrow-sense BCH codes of length q m + 1 are developed for odd q , which are good in some cases. Moreover, sufficient and necessary conditions for the even-like subcodes of narrow-sense BCH codes of length q m + 1 being dually-BCH codes are presented, where q is odd and m ≢ 0 (mod 4). [ABSTRACT FROM AUTHOR]

In 2015, Abatzoglou, Silverberg, Sutherland, and Wong presented a framework for primality proving algorithms for special sequences of integers using an elliptic curve with complex multiplication. They applied their framework to obtain algorithms for elliptic curves with complex multiplication by imaginary quadratic field of class numbers one and two, but, they were not able to obtain primality proving algorithms in cases of higher class number. In this paper, we present a method to apply their framework to imaginary quadratic fields of class number three. In particular, our method provides a more efficient primality proving algorithm for special sequences of integers than the existing algorithms by using an imaginary quadratic field of class number three in which 2 splits. As an application, we give two special sequences of integers derived from Q (− 23) and Q (− 31) , which are all the imaginary quadratic fields of class number three in which 2 splits. Finally, we give a computational result for the primality of these sequences. [ABSTRACT FROM AUTHOR]

Stability results play an important role in Galois geometries. The famous resultant method, developed by Szőnyi and Weiner [12] , [11] , became very fruitful and resulted in many stability theorems in the last two decades. This method is based on some bivariate polynomials associated to point sets. In this paper we generalize the method for the multidimensional case and show some new applications. We build up the multivariate polynomial machinery and apply it for Rédei polynomials. We can prove a high dimensional analogue of the result of Szőnyi-Weiner [9] , concerning the number of hyperplanes being skew to a point set of the space. We prove general results on "partial blocking sets", using the tools we have developed. [ABSTRACT FROM AUTHOR]

This paper defines a linear representation for nonlinear maps F : F n → F n where F is a finite field, in terms of matrices over F. This linear representation of the map F associates a unique number N and a unique matrix M in F N × N , called the Linear Complexity and the Linear Representation of F respectively, and shows that the compositional powers F (k) are represented by matrix powers M k. It is shown that for a permutation map F with representation M , the inverse map has the linear representation M − 1. This framework of representation is extended to a parameterized family of maps F λ (x) : F → F , defined in terms of a parameter λ ∈ F , leading to the definition of an analogous linear complexity of the map F λ (x) , and a parameter-dependent matrix representation M λ defined over the univariate polynomial ring F [ λ ]. Such a representation leads to the construction of a parametric inverse of such maps where the condition for invertibility is expressed through the unimodularity of this matrix representation M λ. Apart from computing the compositional inverses of permutation polynomials, this linear representation is also used to compute the cycle structures of the permutation map. Lastly, this representation is extended to a representation of the cyclic group generated by a permutation map F , and to the group generated by a finite number of permutation maps over F. [ABSTRACT FROM AUTHOR]

The study of n -dimensional inverted Kloosterman sums was suggested by Katz (1995) [7] who handled the case when n = 1 from complex point of view. For general n ≥ 1 , the n -dimensional inverted Kloosterman sums were studied from both complex and p -adic point of view in our previous paper (2024) [10]. In this note, we study the algebraic degree of the inverted n -dimensional Kloosterman sum as an algebraic integer. [ABSTRACT FROM AUTHOR]

In this paper, we generalize [6] , [1] , [5] and [3] by allowing the distance between two points in a finite field vector space to be defined by a general non-degenerate bilinear form or quadratic form. We prove the same bounds on the sizes of large subsets of F q d for them to contain distance graphs with a given maximal vertex degree, under the more general notion of distance. We also prove the same results for embedding paths, trees and cycles in the general setting. [ABSTRACT FROM AUTHOR]

Let q be a power of a prime such that q ≡ 1 (mod r). Let c be an r -th power residue over F q. In this paper, we present a new r -th root formula which generalizes G.H. Cho et al.'s cube root algorithm, and which provides a refinement of Williams' Cipolla-Lehmer based procedure. Our algorithm which is based on the recurrence relations arising from irreducible polynomial h (x) = x r + (− 1) r + 1 (b + (− 1) r r) (x − 1) with b = c + (− 1) r + 1 r requires only O (r 2 log ⁡ q + r 4) multiplications for r > 1. The multiplications for computation of the main exponentiation of our algorithm are half of that of the Williams' Cipolla-Lehmer type algorithms. [ABSTRACT FROM AUTHOR]

The punctured binary Reed–Muller code is cyclic and was generalised into the punctured generalised Reed–Muller code over GF ( q ) in the literature. The first objective of this paper is to present another generalisation of the punctured binary Reed–Muller code and the binary Reed–Muller code, and analyse these codes. The second objective of this paper is to consider two applications of the new codes in constructing LCD codes and 2-designs. The major motivation of constructing and studying the new codes and their extended codes is the construction of 2-designs, which is an interesting topic in combinatorics. It is remarkable that the family of newly generalised cyclic codes contains a subclass of optimal ternary codes with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] for all m ≥ 2 . Their extended codes have parameters [ 3 m , 3 m − 1 − 2 m , 5 ] for all m ≥ 2 , and are also optimal. [ABSTRACT FROM AUTHOR]

In recent years several papers have appeared that investigate the classical discrete logarithm problem for elliptic curves by means of the multivariate polynomial approach based on the celebrated summation polynomials, introduced by Semaev in 2004. With a notable exception by Petit et al. in 2016, all numerous papers on the subject have investigated only the composite-field case, leaving apart the laborious prime-field case. In this paper we propose a variation of Semaev's original approach that reduces to only one the relations to be found among points of the factor base, thus decreasing drastically the necessary Groebner basis computations. Our proposal holds for any finite field but it is particularly suitable for the prime-field case, where it outperforms both the original Semaev's method and the specialised algorithm by Petit et al.. [ABSTRACT FROM AUTHOR]