Finite Field Fp Research Articles

Seminormal forms for the Temperley-Lieb algebra

Let TLnQ be the rational Temperley-Lieb algebra, with loop parameter 2. In the first part of the paper we study the seminormal idempotents Et for TLnQ for t running over two-column standard tableaux. Our main result is here a concrete combinatorial construction of Et using Jones-Wenzl idempotents JWk for TLkQ where k≤n.In the second part of the paper we consider the Temperley-Lieb algebra TLnFp over the finite field Fp, where p>2. The KLR-approach to TLnFp gives rise to an action of a symmetric group Sm on TLnFp, for some m<n. We show that the Et's from the first part of the paper are simultaneous eigenvectors for the associated Jucys-Murphy elements for Sm. This leads to a KLR-interpretation of the p-Jones-Wenzl idempotent JWnp for TLnFp, that was introduced recently by Burull, Libedinsky and Sentinelli.

Simulation limitations of affine cellular automata

Cellular automata are a famous model of computation, yet it is still a challenging task to assess the computational capacity of a given automaton; especially when it comes to showing negative results. In this paper, we focus on studying this problem via the notion of CA intrinsic simulation. We say that automaton A is simulated by B if each space-time diagram of A can be, after suitable transformations, reproduced by B.We study affine automata – i.e., automata whose local rules are affine mappings of vector spaces. This broad class contains the well-studied cases of linear automata. The main result of this paper shows that (almost) every automaton affine over a finite field Fp can only simulate affine automata over Fp. We discuss how this general result implies, and widely surpasses, limitations of linear and additive automata previously proved in the literature.We provide a formalization of the simulation notions into algebraic language and discuss how this opens a new path to showing negative results about the computational power of cellular automata using deeper algebraic theorems.

A unified Fourier slice method to derive ridgelet transform for a variety of depth-2 neural networks

To investigate neural network parameters, it is easier to study the distribution of parameters than to study the parameters in each neuron. The ridgelet transform is a pseudo-inverse operator that maps a given function f to the parameter distribution γ so that a network NN[γ] reproduces f, i.e. NN[γ]=f. For depth-2 fully-connected networks on a Euclidean space, the ridgelet transform has been discovered up to the closed-form expression, thus we could describe how the parameters are distributed. However, for a variety of modern neural network architectures, the closed-form expression has not been known. In this paper, we explain a systematic method using Fourier expressions to derive ridgelet transforms for a variety of modern networks such as networks on finite fields Fp, group convolutional networks on abstract Hilbert space H, fully-connected networks on noncompact symmetric spaces G/K, and pooling layers, or the d-plane ridgelet transform.

Certain diagonal equations and conflict-avoiding codes of prime lengths

We study the construction of optimal conflict-avoiding codes (CAC) from a number theoretical point of view. The determination of the size of optimal CAC of prime length p and weight 3 is formulated in terms of the solvability of certain twisted Fermat equations of the form g2Xℓ+gYℓ+1=0 over the finite field Fp for some primitive root g modulo p. We treat the problem of solving the twisted Fermat equations in a more general situation by allowing the base field to be any finite extension field Fq of Fp. We show that for q greater than a lower bound of the order of magnitude O(ℓ2) there exists a generator g of Fq× such that the equation in question is solvable over Fq. Using our results we are able to contribute new results to the construction of optimal CAC of prime lengths and weight 3.

Adjacency matrices over a finite prime field and their direct sum decompositions

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field Fp. We apply symmetric (row and column) elementary transformations to the adjacency matrix over Fp in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over Fp for any odd prime p. Our key tool is quadratic residues of Fp.

Leader-following consensus of finite-field networks with time-delays

In this paper, we investigate the leader-following consensus problem for high-ordered finite-field networks (FFNs) with time-delays. It is supposed that the dynamics of each agent is described by a linear equation over a finite field Fp, a distributed control is exerted on the followers to track the leader, and the communication delays are also considered. For such time-delayed FFNs, criteria established on system matrices and interaction graphs are obtained for the leader-following consensus under both fixed and switching topologies. Basing on these criteria, the matrix for the distributed control law is designed. At last, an illustrative example is demonstrated.

Polynomial equations for matrices over integers modulo a prime power and the cokernel of a random matrix

Given a prime p and a positive integer k, let Mn(Z/pkZ) be the ring of n×n matrices over Z/pkZ. We consider the number of solutions X∈Mn(Z/pkZ) to the polynomial equation P(X)=0, where P(t) is a monic polynomial in (Z/pkZ)[t] whose reduction modulo p is square-free over the finite field Fp of p elements. Noting that P(X)=0 if and only if cok(P(X))≃(Z/pkZ)n, we give a conjectural generalization of counting solutions to P(X)=0 as the distribution of the cokernel cok(P(X)) of P(X) up to isomorphisms, where X is a uniform random matrix in Mn(Z/pkZ). This distribution involves an explicit formula when we fix the residue class of X modulo p. We prove this conjecture for the special case when the image of P(t) in Fp[t] modulo p is irreducible. We explain how the distribution we obtain is closely related to the Cohen-Lenstra distribution in number theory. Our proof involves algebraic and combinatorial arguments in linear algebra over Z/pkZ and builds upon a previous work of Cheong and Kaplan.

Explicit Sato–Tate type distribution for a family of K ⁢ 3 K3 surfaces

Abstract In the 1960s, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual Sato–Tate for non-CM elliptic curves). In analogy with Birch’s result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces A λ ⁢ ( p ) {A_{\lambda}(p)} of a certain family of K ⁢ 3 {K3} surfaces X λ {X_{\lambda}} with generic Picard rank 19 is the O ⁢ ( 3 ) {O(3)} distribution. This distribution, which we denote by 1 4 ⁢ π ⁢ f ⁢ ( t ) {\frac{1}{4\pi}f(t)} , is quite different from the semicircular distribution. It is supported on [ - 3 , 3 ] {[-3,3]} and has vertical asymptotes at t = ± 1 {t=\pm 1} . Here we make this result explicit. We prove that if p ≥ 5 {p\geq 5} is prime and - 3 ≤ a &lt; b ≤ 3 {-3\leq a&lt;b\leq 3} , then | # ⁢ { λ ∈ 𝔽 p : A λ ⁢ ( p ) ∈ [ a , b ] } p - 1 4 ⁢ π ⁢ ∫ a b f ⁢ ( t ) ⁢ 𝑑 t | ≤ 98.28 p 1 / 4 . \biggl{\lvert}\frac{\#\{\lambda\in\mathbb{F}_{p}:A_{\lambda}(p)\in[a,b]\}}{p}-% \frac{1}{4\pi}\int_{a}^{b}f(t)\,dt\biggr{\rvert}\leq\frac{98.28}{p^{1/4}}. As a consequence, we are able to determine when a finite field 𝔽 p {\mathbb{F}_{p}} is large enough for the discrete histograms to reach any given height near t = ± 1 {t=\pm 1} . To obtain these results, we make use of the theory of Rankin–Cohen brackets in the theory of harmonic Maass forms.

Two new classes of projective two-weight linear codes

In this paper, for an odd prime p, several classes of two-weight linear codes over the finite field Fp are constructed from the defining sets, and then their complete weight enumerators are determined by employing character sums. These codes can be suitable for applications in secret sharing schemes. Furthermore, two new classes of projective two-weight codes are obtained, and then two new classes of strongly regular graphs are given.

An Introduction to the Theory of Field Extensions

This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field Fp using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.

Narain CFTs and error-correcting codes on finite fields

We construct Narain CFTs from self-dual codes on the finite field Fp through even self-dual lattices for any prime p > 2. Using this correspondence, we can relate the spectral gap and the partition function of the CFT to the error correction capability and the extended enumerator polynomial of the code. In particular, we calculate specific spectral gaps of CFTs constructed from codes and compare them with the largest spectral gap among all Narain CFTs.

Twisted conjugacy in classical Chevalley groups over certain domains of positive characteristic

Let F be a subfield of the algebraic closure of a finite field Fp of characteristic p≠2, and let R be any ring such that F[t]⊆R⊊F(t). Let G(R) be a classical Chevalley group of adjoint type defined over R. We prove that the group G(R) has the R∞-property.

Invertibility of adjacency matrices for random d-regular graphs

Let d≥3 be a fixed integer, and let A be the adjacency matrix of a random d-regular directed or undirected graph on n vertices. We show that there exists a constant d>0 such that P(A is singular in R)≤n−d, for n sufficiently large. This answers an open problem by Frieze and Vu. The key idea is to study the singularity probability over a finite field Fp. The proof combines a local central limit theorem and a large deviation estimate.

Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D

We study locality preserving automorphisms of operator algebras on D-dimensional uniform lattices of prime p-dimensional qudits quantum cellular automata (QCAs), specializing in those that are translation invariant (TI), and map every prime p-dimensional Pauli matrix to a tensor product of Pauli matrices (Clifford). We associate antihermitian forms of the unit determinant over Laurent polynomial rings to TI Clifford QCA with lattice boundaries and prove that the form determines the QCA up to Clifford circuits and shifts (trivial). It follows that every 2D TI Clifford QCA is trivial since the antihermitian form in this case is always trivial. Furthermore, we prove that for any D, the fourth power of any TI Clifford QCA is trivial. We present explicit examples of nontrivial TI Clifford QCA for D = 3 and any odd prime p and show that the Witt group of the finite field Fp is a subgroup of the group C(D=3,p) of all TI Clifford QCA modulo trivial ones. That is, C(D=3,p≡1mod4)⊇Z2×Z2 and C(D=3,p≡3mod4)⊇Z4. The examples are found by disentangling the ground state of a commuting Pauli Hamiltonian, which is constructed by coupling layers of prime dimensional toric codes such that an exposed surface has an anomalous topological order that is not realizable by commuting Pauli Hamiltonians strictly in two dimensions. In an appendix independent of the main body of this paper, we revisit a recent theorem of Freedman and Hastings that any two-dimensional QCA, which is not necessarily Clifford or translation invariant, is a constant depth quantum circuit followed by a shift. We give a more direct proof of the theorem without using any ancillas.

Complete weight enumerators for several classes of two-weight and three-weight linear codes

In this paper, for an odd prime p, by extending Li et al.'s construction [17], several classes of two-weight and three-weight linear codes over the finite field Fp are constructed from a defining set, and then their complete weight enumerators are determined by using Weil sums. Furthermore, we show that some examples of these codes are optimal or almost optimal with respect to the Griesmer bound. Our results generalize the corresponding results in [15,17].

The Hurwitz curve over a finite field and its Weierstrass points for the morphism of lines

For any smooth Hurwitz curve Hn:XYn+YZn+XnZ=0 over the finite field Fp, an explicit description of its Weierstrass points for the morphism of lines is presented. As a consequence, bounds on the number of rational points of Hn are obtained via Stöhr-Voloch Theory. Further, the full automorphism group Aut(Hn), as well as the genera of all Galois subcovers of Hn, with n≠3,pr, are computed. Finally, a question by F. Torres on plane nonsingular maximal curves is answered.

Two hardware implementations for modular multiplication in the AMNS: Sequential and semi-parallel

We propose two hardware architectures for the modular multiplication over a finite field Fp using the Adapted Modular Number System (AMNS). We present their optimized implementations, with low latency.

The Order of Edwards and Montgomery Curves

The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of the Digital Signature Algorithm (DSA) [2]. It is well known that the problem of discrete logarithm is NP-hard on group on elliptic curve (EC) [5]. The orders of groups of an algebraic affine and projective curves of Edwards [3, 9] over the finite field Fpn is studied by us. We research Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve [F ] d p E over a finite field Fp . It should be noted that this method can be applied to the order of elliptic curves due to the birational equivalence between elliptic curves and Edwards curves. The method we have proposed has much less complexity 22 O p log p at not large values p in comparison with the best Schoof basic algorithm with complexity 8 2 O(log pn ) , as well as a variant of the Schoof algorithm that uses fast arithmetic, which has complexity 42O(log pn ) , but works only for Elkis or Atkin primes. We not only find a specific set of coefficients with corresponding field characteristics for which these curves are supersingular, but we additionally find a general formula by which one can determine whether a curve [F ] d p E is supersingular over this field or not. The symmetric of the Edwards curve form and the parity of all degrees made it possible to represent the shape curves and apply the method of calculating the residual coincidences. A birational isomorphism between the Montgomery curve and the Edwards curve is also constructed. A oneto- one correspondence between the Edwards supersingular curves and Montgomery supersingular curves is established. The criterion of supersingularity for Edwards curves is found over F pn .

Study the Linear Equivalent of Nonlinear Sequences over &amp;lt;i&amp;gt;Fp&amp;lt;/i&amp;gt; Where p Is Larger Than Two

Linear orthogonal binary sequences, special M-Sequences, are used widely in the systems communication channels as in the forward links for mixing the information on connection and as in the backward links of these channels to sift this information which transmitted and the receivers get the information in a correct form In current research trying study the construction of the linear equivalent of a product sequence on two, three, and four degrees over a linear sequences from the field Fp, where p is larger than two, and answering on the request, how is the maximum length of the linear equivalent of a product sequence (on a linear sequence {an} over the field Fp), is it less than rNh as the binary sequences?, or can reach it ?, or the length is exceed this value rNh? And is the product sequences are orthogonal? And we show that in some cases, the maximum length rNh for the binary sequences is not correct for the linear sequences in the finite field Fp for p larger than two and the result product sequences are not orthogonal, also trying study the product sequence on two different LFSRs, and how can use one shift feedback shift register LFSR as a monitor register of other p registers. In the current time, I think, there is no coders or decoders using the sequences over finite fields Fp where p is larger than 2 and from this idea this article showing very need for using in the future.

On the Hamming Distances of Constacyclic Codes of Length 5pS

Let p be a prime, s, m be positive integers, and λ be a nonzero element of the finite field F p(m) . In this paper, the algebraic structures of constacyclic codes of length 5 p s (p ≠ 5) are obtained, which provide all self-dual, self-orthogonal and dual containing codes. Moreover, the exact values of the Hamming distances of all such codes are completely determined. Among other results, we obtain the degrees of the generator polynomials of all MDS repeated-root constacyclic codes of arbitrary length. As applications, several new and optimal codes are provided.