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Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for every positive even integer ℓ. In this paper, we study the case where $ x^m-1 $ x m − 1 has an arbitrary number of irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] ; in the previous studies, only some special cases where $ x^m-1 $ x m − 1 has exactly two or three irreducible factors in $ {\mathbb F}_q [x] $ F q [ x ] , were studied. Firstly, the binary code case is completed: for any even positive integer ℓ, every binary ℓ-quasi-cyclic self-dual code can be obtained by our construction. Secondly, we work on the q-ary code cases for an odd prime power q. We find an explicit method for construction of all ℓ-quasi-cyclic self-dual codes over $ {\mathbb F}_q $ F q of length $ m\ell $ mℓ for any even positive integer ℓ, where we require that $ q \equiv 1 \pmod {4} $ q ≡ 1 (mod 4) if the index $ \ell \ge 6 $ ℓ ≥ 6. By implementation of our method, we obtain a new optimal binary self-dual code $ [172, 86, 24] $ [ 172 , 86 , 24 ] , which is also a quasi-cyclic code of index 4. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of order q. In this paper, we investigate the permutation property of the quadrinomial f (x) = x + a 1 x q (q − 1) + 1 + a 2 x q + a 3 x 2 (q − 1) + 1 over F q 2 , where q is odd and a 1 , a 2 , a 3 ∈ F q * . More precisely, we obtain the necessary and sufficient conditions for f (x) to be a permutation polynomial over F q 2 . [ABSTRACT FROM AUTHOR]

Let R be a noncommutative ring with unity and $ Z(R) $ Z (R) be its centre. The commuting graph of R denoted by $ \Gamma (R) $ Γ (R) is a graph whose vertices are noncentral elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let F be a finite field. In this paper, we show that if $ \Gamma (R)\cong \Gamma (M_3(F)) $ Γ (R) ≅ Γ (M 3 (F)) and $ Z(R) $ Z (R) is a field, then $ R\cong M_3(F) $ R ≅ M 3 (F). In particular, if $ \Gamma (R)\cong \Gamma (M_3(F_p)) $ Γ (R) ≅ Γ (M 3 (F p)) , then $ R\cong M_3(F_p) $ R ≅ M 3 (F p). [ABSTRACT FROM AUTHOR]

Let R be a noncommutative ring with unity and $ Z(R) $ Z (R) be its centre. The commuting graph of R denoted by $ \Gamma (R) $ Γ (R) is a graph whose vertices are noncentral elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let F be a finite field. In this paper, we show that if $ \Gamma (R)\cong \Gamma (M_3(F)) $ Γ (R) ≅ Γ (M 3 (F)) and $ Z(R) $ Z (R) is a field, then $ R\cong M_3(F) $ R ≅ M 3 (F). In particular, if $ \Gamma (R)\cong \Gamma (M_3(F_p)) $ Γ (R) ≅ Γ (M 3 (F p)) , then $ R\cong M_3(F_p) $ R ≅ M 3 (F p). [ABSTRACT FROM AUTHOR]

We determine the minimal bi-degree(s) of an irreducible filling curve over F q for ℙ 1 × ℙ 1 by constructing examples. It is (q + 1 , q + 1) if q ≠ 2 , and they are (4, 3) and (3, 4) if q = 2. [ABSTRACT FROM AUTHOR]

–Let F be a finite field of characteristic 2. In this article, we have looked into the existence of normal complement of G in V (F G) , where G is either the alternating group A4 or the dihedral group D 4 m of order 4m, for an odd integer m ≥ 3 . Also, we have explicitly found a normal complement of the symmetric group S4 in V (F S 4) over the field F containing 2 elements. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field with q elements. In this paper, we study the permutation property of the class of quadrinomials x 3 + a x q + 2 + b x 2 q + 1 + c x 3 q ∈ F q 2 [ x ] , where char (F q) ∈ { 3 , 5 }. With some additional assumption on the coefficients, we propose several new necessary and sufficient conditions on the coefficients that yield permutation quadrinomials. [ABSTRACT FROM AUTHOR]

Let Sn be the symmetric group and An be the alternating group on n symbols. In this article, we have proved that if F is a finite field of characteristic p > n, then there does not exist a normal complement of Sn (n is even) and An (n ≥ 4) in their corresponding unit groups U (F S n) and U (F A n). Moreover, if F is a finite field of characteristic 3, then A4 does not have normal complement in the unit group U (F A 4). [ABSTRACT FROM AUTHOR]

Let p be a prime and let q = p n , with n ≥ 1 an odd integer. We describe a method for the construction of characteristic polynomials of simple non-ordinary abelian varieties of dimension g and p-rank g − 1 or g − 2 over the finite field F q , when 2 g = ρ b − 1 (ρ − 1) for some prime ρ ≥ 3 and integer b ≥ 1. [ABSTRACT FROM AUTHOR]

Recently, rational functions of degree three that permute the projective line P 1 (F q) over a finite field F q were determined by Ferraguti and Micheli. In the present paper, using a different method, we determine all rational functions of degree four that permute P 1 (F q). [ABSTRACT FROM AUTHOR]

Let (A, M) be any finite local (nonzero commutative unital) ring. Then the set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that A ⊂ B is a decomposed (minimal ring) extension is in one-to-one correspondence with the collection of sets { I 1 , I 2 } of ideals I 1 and I 2 of A such that M = I 1 ⊕ I 2 (internal direct sum), with any such { I 1 , I 2 } corresponding to the class represented by the ring B = A / I 1 × A / I 2. If A is not a field, then the (finite) set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that A ⊂ B is a ramified (minimal ring) extension has cardinality at least 2; known examples show that this bound cannot be improved. If X denotes an indeterminate over the field F 4 , then R : = F 2 + X F 4 [ X ] / (X 2) is of minimal cardinality in the class of finite local rings A such that A is not a field and the A-algebra isomorphism classes represented by a ramified extension of A constitute fewer than half of the (finitely many) A-algebra isomorphism classes represented by a minimal ring extension of A. [ABSTRACT FROM AUTHOR]

In this article we introduce the abstract notion of generalized wavelet (affine) groups over finite fields as the finite group of generalized dilations, and translations. We then present a unified theoretical linear algebra approach to the theory of generalized wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as a finite coherent sum of generalized wavelet coefficients as well. [ABSTRACT FROM AUTHOR]

Let F q (x) be the field of rational functions over F q and treat PGL (2 , F q) as the group of degree one rational functions in F q (x) equipped with composition. PGL (2 , F q) acts on F q (x) from the right through composition. The Galois correspondence and Lüroth's theorem imply that every subgroup H of PGL (2 , F q) is the stabilizer of some rational function π H (x) ∈ F q (x) with deg π H = | H | under this action, where π H (x) is uniquely determined by H up to a left composition by an element of PGL (2 , F q). In this article, we determine the rational function π H (x) explicitly for every H < PGL (2 , F q). [ABSTRACT FROM AUTHOR]

Let A be a finite local (commutative unital) ring. If A is not a field, there exists a positive integer N such that | B | ≤ N for each ring B such that A ⊂ B is an inert (minimal ring) extension. It follows that A is a (finite) field ⇔ the inert extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes ⇔ the minimal ring extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes ⇔ there exists a minimal ring extension A ⊂ B such that B is a flat A-module ⇔ there exists a minimal ring extension A ⊂ B such that B is a local ring and A ⊂ B is a Galois ring extension. [ABSTRACT FROM AUTHOR]

Given a quiver Q, let MQ(α,q) be the number of isomorphism classes of representations of Q over the finite field 픽q with dimension vector α, and let be the canonical decomposition of α. We establish a relationship between polynomials MQ(α,q) and when Q is a tame quiver. This is based on some methods in the representation theory of quivers and algebraic combinatorics. [ABSTRACT FROM AUTHOR]

We classify the maximal subrings of the ring of n×n matrices over a finite field, and show that these subrings may be divided into three types. We also describe all of the maximal subrings of a finite semisimple ring, and categorize them into two classes. As an application of these results, we calculate the covering number of a finite semisimple ring. [ABSTRACT FROM AUTHOR]

Let R be an integral domain with quotient field F, let S be a non-empty subset of R and let n ≥ 2 be an integer. If there exists a rational function ϕ: S → F such that ϕ(a) n = a for all a ∈ S, then S is finite. As a consequence, if F is an ordered field (for instance, ) and S is an open interval in F, no such rational function ϕ exists. Applications to finite fields and additional examples are given. The methods used are algebraic. A closing remark indicates how this note could be used as enrichment material in courses ranging from precalculus to undergraduate courses on abstract algebra or analysis. [ABSTRACT FROM AUTHOR]

In this work, a new construction based on companion matrices of primitive polynomials is provided. Given two primitive polynomials over the finite fields Fq and Fqb, we construct a ring isomorphism that transforms the companion matrix of the primitive polynomial over Fqb into amatrixwith elements in Fq whose characteristic polynomial is another primitive polynomial over Fq. [ABSTRACT FROM AUTHOR]

The set of functionsis linearly independent over(with respect to any open subinterval of (0, ∞)). The titular result is a corollary for any integern≥ 2 (and the domain [0, ∞)). Some more accessible proofs of that result are also given. LetFbe a finite field of characteristicpand cardinalitypk. Then thepth-root functionF→Fis a polynomial function of degree at mostpk− 2 ifpk≠ 2 (resp., the identity function ifpk= 2). Also, for any integern≥ 2, every element ofFhas annthroot inFif and only if, for each prime numberqdividingn,qis not a factor ofpk− 1. Various parts of this note could find classroom use in courses at various levels, on precalculus, calculus or abstract algebra. A final section addresses educational benefits of such coverage and offers some recommendations to practitioners. [ABSTRACT FROM AUTHOR]

In this paper, we completely characterize when a group algebraFGof a finite abelian groupGover a finite fieldFis a ∗-clean ring. The main result states that, for a finite abelian groupGof ordernand a finite fieldFqwithchar(Fq) = p, the group algebraFqGis ∗-clean under the classical involution if and only if there exists a positive integerwsuch thatqdw≡−1(modm), whereG(p) is the Sylowp-subgroup ofG,m = exp(G∕G(p)) is the exponent ofG∕G(p),m2is the second exponent ofG∕G(p) anddis the order ofqmodulom2. Particularly, ifG = Cnis the cyclic group of ordernwithgcd(p,n) = 1, thenis ∗-clean if and only if there exists a positive integerwsuch thatqw≡−1(modn). Our results improve several known results and give an answer to an existing question in the literature. [ABSTRACT FROM AUTHOR]

We characterize certain maximal curves over finite fields whose plane models are of Hurwitz type, namely. We also consider maximal hyperelliptic curves of maximal genus. Finally, we discuss maximal curves of typethrough class field theory. [ABSTRACT FROM PUBLISHER]

We investigate the open problem of Li and Wang [On EA-equivalence of certain permutations to power mappings, Des. Codes Cryptogr. 58(2011), pp. 259–269] that whether the polynomialonis a permutation for. Several classes of polynomials of the formare proven to be permutations by decomposing the finite fieldwhen. Some relationships among this type of permutation polynomials, which allow certain secondary constructions, are also proposed. [ABSTRACT FROM PUBLISHER]

Letbe the-dimensional unitary space over finite fieldand letdenote the orbit of subspaces ofunder the unitary group. Denote bythe set of subspaces generated by. By orderingby ordinary inclusion, the poset denotedis obtained. In this paper, we construct the subposet of, show that this subposet is strongly uniform and construct Leonard pairs from it. [ABSTRACT FROM PUBLISHER]

Fix a real tensorof format. Here we prove that its real tensor rank is at mosttimes its complex tensor rank. A similar bound is true for symmetric tensors and for any perfect field (e.g. a finite field). [ABSTRACT FROM PUBLISHER]

A matrixover a field is minimal if for every matrixthat commutes with, the centralizer ofis a subset of the centralizer of. In this paper, we study the minimal matrices over the field with two elements. We characterize the minimal matrices with their minimal polynomial of the form, whereis an irreducible polynomial and. We also characterize all minimal matrices with spectrum in. [ABSTRACT FROM AUTHOR]

An n × n zero–nonzero pattern 𝒜 is spectrally arbitrary over a field 𝔽 provided that for each monic polynomial r(x)∈𝔽[x] of degree n, there exists a matrix A over 𝔽 with zero–nonzero pattern 𝒜 such that the characteristic polynomial p A (x) = r(x). In this article, we investigate several classes of zero–nonzero patterns over finite fields and algebraic extensions of ℚ. We prove that there are no spectrally arbitrary patterns over 𝔽2 and show that the full 2 × 2 pattern is spectrally arbitrary over 𝔽 if and only if F contains at least five elements. We explore an n × n pattern with precisely 2n nonzero entries that is spectrally arbitrary over finite fields 𝔽 q with , as well as ℚ. We also investigate an interesting 3 × 3 pattern for which the algebraic structure of the finite field rather than just the size of the field is a critical factor in determining whether or not it is spectrally arbitrary. This pattern turns out to be spectrally arbitrary over [ABSTRACT FROM PUBLISHER]

We describe the T-ideal of identities and the T-space of central polynomials for the unitary finite dimensional Grassmann algebra over a finite field. [ABSTRACT FROM AUTHOR]

Let Fq be a finite field of characteristic two, S be a nonsingular non-alternate symmetric matrix over Fq and Psn(Fq, S) be the associated pseudo-symplectic group. Let Psn(Fq, S) act linearly on the polynomial ring Fq[x1,..., xn]. In this note, we find an explicit set of generators of the ring of invariants of Psn(Fq, S) for n = 2, 4 and 2ν +1. In particular, the results assert that the ring of invariants of Ps4(Fq, S) is not a polynomial algebra but is an example of hypersurface and the ring of invariants of Ps2ν+1(Fq, S) is a complete intersection. [ABSTRACT FROM AUTHOR]

We describe the T-ideal of identities and the T-space of central polynomials for the infinite dimensional unitary Grassmann algebra over a finite field. [ABSTRACT FROM AUTHOR]

This investigation presents an effective algorithm for computing multiplication over a class of GF(2m) based on both irreducible all one polynomials (AOPs) and equally spaced polynomials (ESPs). The proposed AOP-based multiplier uses the modified Booth's algorithm to develop a new multiplexer-based bit-parallel multiplier that is simple and modular and such properties are important for VLSI hardware implementation. The multiplier requires [m/4](m + 1) MUX4x1 and (1.5m² + 0.5m - 1) XOR gates. Its time delay is not greater than TM + (3 + log2[m/4])TX, where TM and TX are the time delays of MUX4x1 and 2-input XOR gate, respectively. For a certain degree, an irreducible ESP with a high degree can be obtained from a corresponding irreducible AOP with a relatively very low degree. Using the subword parallel processing, the proposed AOP-based multiplier with low degree can also be adopted to realize ESP-based multipliers with high degrees. [ABSTRACT FROM AUTHOR]

We derive explicit equations for the maximal function fields F over q2n given by F = q2n(X, Y) with the relation A(Y) = f(X), where A(Y) and f(X) are polynomials with coefficients in the finite field q2n, and where A(Y) is q-additive and deg(f) = qn + 1. We prove in particular that such maximal function fields F are Galois subfields of the Hermitian function field H over q2n (i.e., the extension H/F is Galois). [ABSTRACT FROM AUTHOR]

In this paper, we report efficient implementations of the linear sieve and the cubic sieve methods for computing discrete logarithms over prime fields. We demonstrate through empirical performance measures that for a special class of primes the cubic sieve method runs about two times faster than the linear sieve method even in cases of small prime fields of the size about 150 bits. We also provide a heuristic estimate of the number of solutions of the congruence X 3 ?=? Y 2 Z (mod p ) that is of central importance in the cubic sieve method. [ABSTRACT FROM AUTHOR]

In this article we study an analogue of the classical Bessel function over finite fields. The definition of the this function is associated with the explicit form of matrix elements for the irreducible representation of the group of motions on a plane over finite fields. For the Bessel function, we present the addition theorem, multiplication theorem, Hanson theorem and some others. [ABSTRACT FROM AUTHOR]

We extend L. Rédei's definition of polynomials with a restricted range to include multivariable polynomial functions. Given a proper subfield Fs of the finite field Fq, we identify all ƒ ε F q [×1, ..., × k] (with degx1 ƒ < q for 1 ≤ i ≤ k) such that for all γ ε Fkq we have ƒ(γ) ε Fs. [ABSTRACT FROM AUTHOR]

We prove that for a smooth projective variety X of dimension d defined over a finite field k, the structure map σ : X [xrarr] Spec k induces an isomorphism σ∗ : CHd+1(X, 1) ≅ CH1(k, 1) = k*. We also prove that the higher Chow groups CHd+s−1(X, s) of one-dimensional cycles are torsion for s ≥ 2. [ABSTRACT FROM AUTHOR]