Search

In this paper, we propose a new class of error-detecting codes based on quasigroups using automorphisms of the finite field 픽pn and the additive group (픽p, +). We demonstrate that these codes are effective in detecting burst errors caused by hardware damage or noisy transmission. We also explore the codes ability to detect various types of double errors, including transposition errors, twin errors, and phonetic errors. Our analysis extends beyond bit-level errors to include block-level errors. Finally, we provide a comparative analysis and demonstrate the real-world applications of our code. [ABSTRACT FROM AUTHOR]

Permutation polynomials with low boomerang uniformity have wide applications in cryptography. In this paper, by utilizing the Weil sums technique and solving some certain equations over 픽2n, we determine the boomerang uniformity of these permutation polynomials: (1) f1(x) = (x2m + x + δ)22m+1 + x, where n = 3m, δ ∈ 픽2n with Trmn(δ) = 1; (2) f2(x) = (x2m + x + δ)22m−1+2m−1 + x, where n = 3m, δ ∈ 픽2n with Trmn(δ) = 0; (3) f3(x) = (x2m + x + δ)23m−1+2m−1 + x, where n = 3m, δ ∈ 픽2n with Trmn(δ) = 0. The results show that the boomerang uniformity of f1(x), f2(x) and f3(x) can attain 2n. [ABSTRACT FROM AUTHOR]

By further exploring some ideas in Ding and Yin, the results on preimage distributions of a class of zero-difference balanced functions are extended and the parameters of a class of partitioned difference families are determined by means of self-conjugate Gauss sums over finite fields. [ABSTRACT FROM AUTHOR]

For each prime power q, we determine all a 휖 픽q2 for which X + Xq + X2q−1 + aXq2−q+1 permutes 픽q2. [ABSTRACT FROM AUTHOR]

Let q denote the finite field with q elements. In this paper, we investigate the trinomial f (x) = x 4 q + 1 + α x 5 q + x q + 4 over the finite field 5 2 k , where α ∈ 5 k * with k being a positive integer. We prove that the trinomial f (x) permutes 5 2 k if and only if α = − 1 and k is even. This work is a continuation of the previous work of Bai and Xia [A new class of permutation trinomials constructed from Niho exponents, Cryptogr. Commun. 10 (2018) 1023–1036]. [ABSTRACT FROM AUTHOR]

Given F = 픽pt, a field with pt elements, where p is a prime power and t ≥ 7 is a positive integer. Let n ∈ ℕ and f = f1/f2 is a rational function, where f1 and f2 are distinct irreducible polynomials with deg(f1) +deg(f2) = n ≤ pt in F[x]. We construct a sufficient condition on (p,t) which guarantees primitive pairing (휖,f(휖)) exists in F such that Tr픽 pt/픽p(휖) = a and Tr픽 pt/픽p(f(휖)) = b for any prescribed a,b ∈ 픽p. Further, we demonstrate for any positive integer n ≤ pt, such a pair definitely exists for large t. The scenario when n = 2 is handled separately and we verified that such a pair exists for all (p,t) except from possible 71 values of (p,t). A result for the case n = 3 is given as well. [ABSTRACT FROM AUTHOR]

Bateman and Duquette have initiated the study of Salem elements in positive characteristic. Later, Kerada introduced and studied j-Salem numbers (j ≥ 2). To complete this circle of ideas we introduce 2-Salem elements over the field of formal Laurent series over a finite field q . We extend the above-cited studies and we provide some analog results. The main result of this note is Theorem 1.2 which characterizes the set T 2 ′ ∗ of 2-Salem series when q ≠ 2 r . [ABSTRACT FROM AUTHOR]

Boolean functions are fundamental bricks in the development of various applications in Cryptography and Coding theory by making benefit from the weights of related Boolean functions (Walsh spectrum). Towards this, the discrete Fourier transform (Walsh–Hadamard) plays a pivotal tool. The work in this paper is dedicated towards the algebraic and numerical degrees, together with the relationship between weights of Boolean function and their Walsh transforms. We introduce Walsh matrices and generalize them to any arbitrary Boolean function. This improves the complexity in computation of Walsh–Hadamard and Fourier transform in certain cases. We also discuss some useful results related to the degree of the algebraic normal form using Walsh–Hadamard transform. [ABSTRACT FROM AUTHOR]

For q = 3 r (r ∈ ℕ), denote by q the finite field of order q and for a positive integer m ≥ 2 , let q m be its extension field of degree m. We establish a sufficient condition for existence of a primitive normal element α such that f (α) is a primitive element, where f (x) = a x 2 + b x + c , with a , b , c ∈ q m satisfying b 2 ≠ a c in q m . [ABSTRACT FROM AUTHOR]

Let q be an even prime power and m ≥ 2 an integer. By q , we denote the finite field of order q and by q m its extension of degree m. In this paper, we investigate the existence of a primitive normal pair (α , f (α)) , with f (x) = a x 2 + b x + c d x + e ∈ q m (x) where the rank of the matrix F = a b c 0 d e ∈ M 2 × 3 ( q m ) is 2. Namely, we establish sufficient conditions to show that nearly all fields of even characteristic possess such elements, except for 1 1 0 0 1 0 if q = 2 and m is odd, and then we provide an explicit small list of possible and genuine exceptional pairs (q , m). [ABSTRACT FROM AUTHOR]

In this paper, we give the characterization of the unit groups of semisimple group algebras of some non-metabelian groups of order 1 2 0. This study completes the study of unit groups of semisimple group algebras of all groups up to order 1 2 0 , except that of the symmetric group S 5 and groups of order 9 6. [ABSTRACT FROM AUTHOR]

Let q be a prime power, q ≠ 2 , 3 , and let A be an n × n matrix over q 2 . We prove that the numerical range of A B and B A is the same for every rank 1 matrix B if and only if A is a multiple of a unitary matrix. The corresponding result for complex matrices was proved by Chien, Gau, Li, Tsai and Wang [Products of operators and numerical range, Linear Multilinear Algebra 64(1) (2016) 58–67]. [ABSTRACT FROM AUTHOR]

For a prime p and a positive integer k , let q = p k and F q n be the extension field of F q. We derive a sufficient condition for the existence of a primitive element α in F q n such that α 3 − α + 1 is also a primitive element of F q n , a sufficient condition for the existence of a primitive normal element α in F q n over F q such that α 3 − α + 1 is a primitive element of F q n , and a sufficient condition for the existence of a primitive normal element α in F q n over F q such that α 3 − α + 1 is also a primitive normal element of F q n over F q. [ABSTRACT FROM AUTHOR]

Reed–Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm. Usually we use the maximum likelihood decoding (MLD) algorithm in the decoding process of Reed–Solomon codes. MLD algorithm relies on determining the error distance of received word. Dür, Guruswami, Wan, Li, Hong, Wu, Yue and Zhu et al. got some results on the error distance. For the Reed–Solomon code C , the received word u is called an ordinary word of C if the error distance d (u , C) = n − deg ⁡ u (x) with u (x) being the Lagrange interpolation polynomial of u. We introduce a new method of studying the ordinary words. In fact, we make use of the result obtained by Y.C. Xu and S.F. Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed–Solomon codes over the finite field of q elements. This completely answers an open problem raised by Li and Wan in [On the subset sum problem over finite fields, Finite Fields Appl.14 (2008) 911–929]. [ABSTRACT FROM AUTHOR]

In this paper, we give a largely self-contained proof that the quartic extension q 4 of the finite field q contains a primitive element α such that the element α + α − 1 is also a primitive element of q 4 , and Tr q 4 | q (α) = a for any prescribed a ∈ q . The corresponding result has already been established for finite field extensions of degrees exceeding 4 in [Primitive element pairs with one prescribed trace over a finite field, Finite Fields Appl.54 (2018) 1–14.]. [ABSTRACT FROM AUTHOR]

In this paper, we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [Footprints or generalized Bezout's theorem, IEEE Trans. Inform. Theory46(2) (2000) 635–641, On (or in) Dick Blahut's 'footprint', Codes , Curves Signals (1998) 3–9] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes irreducible polynomials. The purpose of the paper is to determine a large class of polynomials for which only the mentioned trivial polynomials can attain the bound, implying that to search for other polynomials with the maximal number of roots one must look outside this class. [ABSTRACT FROM AUTHOR]

Let F be a field, and let e , k be integers such that 1 ≤ e ≤ | F ∖ { 0 } | and k ≥ 0. We show that for any subset { a 1 , ... , a e } ⊆ F ∖ { 0 } , the curious identity ∑ (i 1 , ... , i e) ∈ Z ≥ 0 e , i 1 + ⋯ + i e = k a 1 i 1 ⋯ a e i e = ∑ i = 1 e a i k + e − 1 ∏ i ≠ j = 1 e (a i − a j) holds with Z ≥ 0 being the set of nonnegative integers. As an application, we prove that for any subset { a 1 , ... , a e } ⊆ F q ∖ { 0 } with F q being the finite field of q elements and e , l being integers such that 2 ≤ e ≤ q − 1 and 0 ≤ l ≤ e − 2 , ∑ (i 1 , ... , i e) ∈ Z ≥ 0 e , i 1 + ⋯ + i e = q − e + l a 1 i 1 ⋯ a e i e = 0 Using this identity and providing an extension of the principle of cross-classification that slightly generalizes the one obtained by Hong in 1996, we show that if r is an integer with 1 ≤ r ≤ q − 2 , then for any subset { a 1 , ... , a r } ⊂ F q ∗ we have x q − 1 − 1 ∏ i = 1 r (x − a i) = ∑ i = 0 q − 1 − r ∑ i 1 + ⋯ + i r = q − 1 − r − i a 1 i 1 ⋯ a r i r x i. This implies # x ∈ F q ∗ | ∑ i = 0 q − 1 − r ∑ i 1 + ⋯ + i r = q − 1 − r − i a 1 i 1 ⋯ a r i r x i = 0 = q − 1 − r. [ABSTRACT FROM AUTHOR]

Some results on the k -normal elements and k -normal polynomials over finite fields are given in the recent literature. In this paper, we show that a transformation x → (x p − x p − 1 + 1) / (− x p − 1 + 1) can be used to produce an infinite sequence of irreducible polynomials over a finite field 𝔽 q of characteristic p. By iteration of this transformation, we construct the k -normal polynomials of degree n p u in 𝔽 q [ x ] starting from a suitable initial k -normal polynomial of degree n. We also construct an infinite sequence of k -normal polynomials using a certain quadratic transformation over 𝔽 2 s . [ABSTRACT FROM AUTHOR]

There are two types of Belyi's Theorems for curves defined over finite fields of characteristic p , namely the Wild and the Tame p -Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi's theorem in characteristic two, Compos. Math. 156(2) (2020) 325–339], which leads to a self-contained proof for the Tame 2 -Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi's Theorems unlike the known ones that use technical results from Algebraic Geometry. [ABSTRACT FROM AUTHOR]

Generalized almost perfect nonlinear (GAPN) functions were defined to satisfy some generalizations of basic properties of almost perfect nonlinear (APN) functions for even characteristic. In particular, on finite fields of even characteristic, GAPN functions coincide with APN functions. In this paper, we study monomial GAPN functions for odd characteristic. We give monomial GAPN functions whose algebraic degree are maximum or minimum on a finite field of odd characteristic. Moreover, we define a generalization of exceptional APN functions and give typical examples. [ABSTRACT FROM AUTHOR]

We show that under certain technical conditions that simple (q , k , i λ min ) balanced incomplete block designs (BIBDs) exist for all allowable values of i , where q is an odd prime power. Our primary technique is to argue for the existence of difference families in finite fields, in the flavor of Wilson [J. Number Theory 4 (1972) 17–47]. We provide an extensive analysis in the cases, where q ≡ 1 (mod k (k − 1)) and also for k = 9 , 1 6. [ABSTRACT FROM AUTHOR]

In this paper, as an extension of the integer case, we define polynomial functions over the residue class rings of Dedekind domains, and then we give canonical representations and counting formulas for such polynomial functions. In particular, we give an explicit formula for the number of polynomial functions over the residue class rings of polynomials over finite fields. [ABSTRACT FROM AUTHOR]

In this paper, the Sphere-packing bound, Wang-Xing-Safavi-Naini bound, Johnson bound and Gilbert-Varshamov bound on the subspace code of length 2 ν + δ , size M , minimum subspace distance 2 j based on m -dimensional totally singular subspace in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over finite fields 𝔽 q of characteristic 2, denoted by (2 ν + δ , M , 2 j , m) q , are presented, where ν is a positive integer, δ = 0 , 1 , 2 , 0 ≤ m ≤ ν , 0 ≤ j ≤ m. Then, we prove that (2 ν + δ , M , 2 j , m) q codes attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in ℳ (m , 0 , 0 ; 2 ν + δ) , where ℳ (m , 0 , 0 ; 2 ν + δ) denotes the collection of all the m -dimensional totally singular subspaces in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over 𝔽 q of characteristic 2. Finally, Gilbert-Varshamov bound and linear programming bound on the subspace code (2 ν + δ , M , d) q in ℳ (2 ν + δ) are provided, where ℳ (2 ν + δ) denotes the collection of all the totally singular subspaces in the (2 ν + δ) -dimensional orthogonal space 𝔽 q (2 ν + δ) over 𝔽 q of characteristic 2. [ABSTRACT FROM AUTHOR]

Let 𝔽q be the finite field of q elements and f be a nonzero polynomial over 𝔽q. For each b ϵ 𝔽q, let Nq(f = b) denote the number of 𝔽q-rational points on the affine hypersurface f = b. We obtain the formula of Nq(f = b) for a class of hypersurfaces over 𝔽q by using the greatest invariant factors of degree matrices under certain cases, which generalizes the previously known results. We also give another simple direct proof to the known results. [ABSTRACT FROM AUTHOR]

Let be an extension of the field of degree where for some positive integer and prime In this paper, we establish a sufficient condition for the existence of a primitive element such that is also primitive as well as a primitive normal element of over such that is primitive. [ABSTRACT FROM AUTHOR]

We introduce some sequences of binary constant weight codes which are constructed from the affine constructions of balanced incomplete block designs. They give the values for the function of constant weight codes, and they are optimal since those values reach the Johnson Bound. We focus on the values obtained by this method. [ABSTRACT FROM AUTHOR]

Let 픽 q stand for the finite field of odd characteristic p with q elements ( q = p n, n ∈ ℕ) and denote the set of all the nonzero elements of 픽 q. Let m and t be positive integers. By using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over where the integers t > 0, r0 = 0 < r1 < r2 < ⋯ < r t, 1 ≤ n1 < n2 <, ⋯ < n t and 0 ≤ j ≤ t − 1, b k ∊ 픽 q, a k, i ∊ ( k = 1, ..., m, i = 1, ..., r t), and the exponent of each variable is a positive integer. Further, under some natural conditions, we arrive at an explicit formula for the number of 픽 q-rational points on the above variety. It extends the results obtained previously by Wolfmann, Sun, Wang, Hong et al. Our result gives a partial answer to an open problem raised in [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015) 135-153]. [ABSTRACT FROM AUTHOR]

In this paper, we initiate a study on Gauss factorials of polynomials over finite fields, which are the analogues of Gauss factorials of positive integers. [ABSTRACT FROM AUTHOR]

If is a polynomial of degree in , let be the number of cycles of length in the directed graph on with edges For random polynomials, the numbers have asymptotic behavior resembling that for the cycle lengths of random functions However random polynomials differ from random functions in important ways. For example, given the set of cyclic (periodic) points, it is not necessarily true that all permutations of those cyclic points are equally likely to occur as the restriction of . This, and the limitations of Lagrange interpolation, together complicate research on the ultimate period of under compositional iteration. We prove a lower bound for the average value of : if , but , then the expected value of is where the sum is over all polynomials of degree in . Similar results are proved for rational functions. [ABSTRACT FROM AUTHOR]

Let be the -rational point on the Fermat curve with . It has recently been proved that if then each is a cube in . It is natural to wonder whether there is a generalization to . In this paper, we show that the result cannot be extended to in general and conjecture that each is a cube in if and only if . [ABSTRACT FROM AUTHOR]

Let be a finite field with elements, where is a prime power. denotes the -dimensional row linear space over . In this paper, we construct a series of LDPC codes based on the subspaces of singular linear space over , and calculate their parameters. [ABSTRACT FROM AUTHOR]

Let denotes the finite field of order where A permutation polynomial over with and such that for each is a permutation polynomial satisfying is called a o-polynomial. In this paper, we determine all o-polynomials up to degree . [ABSTRACT FROM AUTHOR]

In order to extend the results of in [P. Das, The number of permutation polynomials of a given degree over a finite field, Finite Fields Appl. 8(4) (2002) 478-490], where is a prime, to arbitrary finite fields , we find a formula for the number of permutation polynomials of degree over a finite field , which has elements, in terms of the permanent of a matrix. We write down an expression for the number of permutation polynomials of degree over a finite field , using the permanent of a matrix whose entries are th roots of unity and using this we obtain a nontrivial bound for the number. Finally, we provide a formula for the number of permutation polynomials of degree less than . [ABSTRACT FROM AUTHOR]

Several classes of permutation polynomials with given form over were recently proposed by Tu, Zeng, Li and Helleseth. In this paper, continuing their work, we present more permutation polynomials of the form over the finite field where is a linearized polynomial with coefficients in . [ABSTRACT FROM AUTHOR]

Let denote the -dimensional singular linear space over a finite field . For a fixed integer , denote by the set of all subspaces of type , where . Partially ordered by ordinary inclusion, one family of quasi-regular semilattices is obtained. Moreover, we compute its all parameters. [ABSTRACT FROM AUTHOR]

We indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in where n corresponds to the extension degree, and is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in n. Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field 픽357. [ABSTRACT FROM AUTHOR]

In this note we first introduce the finite weak Herstein condition, which is a property that holds for every finite field. For a field F satisfying the finite weak Herstein condition, we then characterize whether all the finite-dimensional division F-algebras are commutative. This gives an alternate proof of Wedderburn's Theorem. [ABSTRACT FROM AUTHOR]

For cryptographic systems the method of confusion and diffusion is used as a fundamental technique to achieve security. Confusion is reflected in nonlinearity of certain Boolean functions describing the cryptographic transformation. In this paper, we present two Boolean functions which have low Walsh spectra and high nonlinearity. In the proof of the nonlinearity, a new method for evaluating some exponential sums over finite fields is provided. [ABSTRACT FROM AUTHOR]

A method for constructing piecewise differentially 4-uniform permutations has been recently introduced by Zha, Hu and Sun. By using this method, we provide two new infinite families of differentially 4-uniform permutations from the known APN functions. The CCZ inequivalence between these constructed functions and the known differentially 4-uniform permutations are also investigated by computation. [ABSTRACT FROM AUTHOR]

We present a method for constructing almost perfect nonlinear (APN) functions in odd characteristic by modifying some values of known perfect nonlinear functions. As a consequence, new APN polynomial functions such as ones over 픽13 which are CCZ-inequivalent to known APN functions are obtained. [ABSTRACT FROM AUTHOR]

Let f be a multivariate polynomial over a finite field and its degree matrix be composed of the degree vectors appearing in f. In this paper, we provide an elementary approach to estimating the exponential sums of the polynomials with positive square degree matrices in terms of the elementary divisors of the degree matrices. [ABSTRACT FROM AUTHOR]

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well. [ABSTRACT FROM AUTHOR]

We use the Smith normal form of the augmented degree matrix to estimate the number of rational points on a toric hypersurface over a finite field. This is the continuation of a previous work by Cao in 2009. [ABSTRACT FROM AUTHOR]

We estimate multiplicative character sums along the orbits of a class of nonlinear recurrence vector sequences. In the one-dimensional case, only much weaker estimates are known and our results have no one-dimensional analogs. [ABSTRACT FROM AUTHOR]

Let f be a polynomial in n variables over the finite field 픽q and Nq(f) denote the number of 픽q-rational points on the affine hypersurface f = 0 in 픸n(픽q). A φ-reduction of f is defined to be a transformation σ : 픽q[x1, ..., xn] → 픽q[x1, ..., xn] such that Nq(f) = Nq(σ(f)) and deg f ≥ deg σ(f). In this paper, we investigate φ-reduction by using the degree matrix which is formed by the exponents of the variables of f. With φ-reduction, we may improve various estimates on Nq(f) and utilize the known results for polynomials with low degree. Furthermore, it can be used to find the explicit formula for Nq(f). [ABSTRACT FROM AUTHOR]

In this paper, we discuss the relationship between the generation polynomial of a linear sequence and its decimation sequence over the finite field 픽2. Then we propose an upper bound of the number of terms in the generation polynomial of a decimation sequence of a linear sequence whose generation polynomial is trinomial. Finally, we suggest a method to calculate the terms of a decimation polynomial and their number directly, which can be used in the construction of irreducible polynomials with a controlled number of terms, when the source trinomial and the decimation distance satisfy a certain condition. [ABSTRACT FROM AUTHOR]