Finite Field Fq Research Articles

Treewidth of the [formula omitted]-Kneser graphs

Let V be an n-dimensional vector space over a finite field Fq, where q is a prime power. Define the generalizedq-Kneser graphKq(n,k,t) to be the graph whose vertices are the k-dimensional subspaces of V and two vertices F1 and F2 are adjacent if dim(F1∩F2)<t. Then Kq(n,k,1) is the well-known q-Kneser graph. In this paper, we determine the treewidth of Kq(n,k,t) for n≥2t(k−t+1)+k+1 and t≥1 exactly. Especially, for any possible n, k and q we also determine the treewidth of Kq(n,k,k−1), which is the complement of the Grassmann graph Gq(n,k).

On the lower bound for the length of minimal codes

In recent years, many connections have been made between minimal codes, a classical object in coding theory, and other remarkable structures in finite geometry and combinatorics. One of the main problems related to minimal codes is to give lower and upper bounds on the length m(k,q) of the shortest minimal codes of a given dimension k over the finite field Fq. It has been recently proved that m(k,q)≥(q+1)(k−1).In this note, we prove that liminfk→∞m(k,q)k≥(q+ε(q)), where ε is an increasing function such that 1.52<ε(2)≤ε(q)≤2+12. Hence, the previously known lower bound is not tight for large enough k. We then focus on the binary case and prove some structural results on minimal codes of length 3(k−1). As a byproduct, we are able to show that, if k=5(mod8) and for other small values of k, the bound is not tight.

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.

Existence of pencils with nonblocking hypersurfaces

We prove that there is a pencil of hypersurfaces in Pn of any given degree over a finite field Fq such that every Fq-member of the pencil is not blocking with respect to Fq-lines.

Complete set of pure gaps in function fields

In this work, we provide a way to completely determine the set of pure gaps G0(P1,P2) at two rational places P1,P2 in a function field F over a finite field Fq, and its cardinality. Furthermore, we give a bound for the cardinality of the set G0(P1,P2) which is better, in some cases, than the generic bound given by Homma and Kim in [11]. As a consequence, we completely determine the set of pure gaps and its cardinality for two families of function fields: the GK function field and Kummer extensions.

Mutually orthogonal frequency rectangles

A frequency rectangle of type FR(m,n;q) is an m×n matrix such that each symbol from a set of size q appears n/q times in each row and m/q times in each column. Two frequency rectangles of the same type are said to be orthogonal if, upon superimposition, each possible ordered pair of symbols appear the same number of times. A set of k frequency rectangles in which every pair is orthogonal is called a set of mutually orthogonal frequency rectangles, denoted by k–MOFR(m,n;q). We show that a k–MOFR(2,2n;2) and an orthogonal array OA(2n,k,2,2) are equivalent. We also show that an OA(mn,k,2,2) implies the existence of a k–MOFR(2m,2n;2). We construct (4a−2)–MOFR(4,2a;2) assuming the existence of a Hadamard matrix of order 4a.A k–MOFR(m,n;q) is said to be t–orthogonal, if each subset of size t, when superimposed, contains each of the qt possible ordered t-tuples of entries exactly mn/qt times. A set of vectors over a finite field Fq is said to be t-independent if each subset of size t is linearly independent. We describe a method to obtain a set of t–orthogonal k–MOFR(qM,qN,q) corresponding to a set of t–independent vectors in (Fq)M+N. We also discuss upper and lower bounds on the sizes of sets of t–independent vectors and give a table of values for binary vectors of length N⩽16.A frequency rectangle of type FR(n,n;q) is called a frequency square and a set of k mutually orthogonal frequency squares is denoted by k–MOFS(n;q) or k–MOFS(n) when there is no ambiguity about the symbol set. For p an odd prime, we show that there exists a set of (p−1) binary MOFS(2p), hence improving the lower bounds in (Britz et al. 2020) for p⩾19.

Maximal curves and Tate-Shafarevich results for quartic and sextic twists

We study elliptic surfaces corresponding to an equation of the specific type y2=x3+f(t)x, defined over the finite field Fq for a prime power q≡3mod4. It is shown that if s4=f(t) defines a curve that is maximal over Fq2 then the rank of the group of sections defined over Fq on the elliptic surface is determined in terms of elementary properties of the rational function f(t). Similar results are shown for elliptic surfaces given by y2=x3+g(t) using prime powers q≡5mod6 and curves s6=g(t). Finally, for each of the forms used here, existence of curves with the property that they are maximal over Fq2 is discussed, as well as various examples.

Divisibility on point counting over finite Witt rings

Let Fq denote the finite field of q elements with characteristic p. Let Zq denote the unramified extension of the p-adic integers Zp with residue field Fq. In this paper, we investigate the q-divisibility for the number of solutions of a polynomial system in n variables over the finite Witt ring Zq/pmZq, where the n variables of the polynomials are restricted to run through a box lifting Fqn. It turns out that in general the answers do depend upon the box chosen. Based on the addition operation of Witt vectors, we prove a q-divisibility theorem for any box of low algebraic complexity, including the simplest Teichmüller box. This extends the classical Ax-Katz theorem over finite field Fq (the case m=1). Taking q=p to be a prime, our result extends and improves a recent theorem of Grynkiewicz for the unweighted case.

Linear codes associated to symmetric determinantal varieties: Even rank case

We consider linear codes over a finite field Fq, for odd q, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a codeword is derived. Using this formula, we have computed the minimum distance for the codes corresponding to matrices upperbounded by any fixed, even rank. A conjecture is proposed for the cases where the upper bound is odd. At the end of the article, tables for the weights of these codes, for spaces of symmetric matrices up to order 5, are given.

Hybrid character sums and near-optimal partial Hadamard codebooks

In this paper, we first study exponential sums of the form∑x∈Fqψ(f(x))χ(g(x)) called hybrid character sums, where f,g are functions over finite field Fq, ψ is a nontrivial multiplicative character and χ is a nontrivial additive character of Fq. The absolute values or explicit values of them are determined under certain conditions. We then present several constructions of complex codebooks which are also called frames. The proposed codebooks are constructed from Hadamard matrices with binary row selection sequences. The hybrid character sums play an important role in determining the maximum magnitude of the inner products between vectors in these codebooks. It is proved that the magnitude of the inner products between distinct code vectors of each codebook is two-valued, and the maximum magnitude of each codebook asymptotically achieves the Welch bound equality. The constructed codebooks are equivalent to nearly equiangular tight frames and have nice application in compressed sensing matrices.

Duality for certain multi-Frobenius nonclassical curves in higher dimensional spaces

We show how a type of multi-Frobenius nonclassicality of a curve defined over a finite field Fq of characteristic p reflects on the geometry of its strict dual curve. In particular, in such cases we may describe all the possible intersection multiplicities of its strict dual curve with the linear system of hyperplanes. Among other consequence, using a result by Homma, we are able to construct nonreflexive space curves such that their tangent surfaces are nonreflexive as well, and the image of a generic point by a Frobenius map is in its osculating hyperplane. We also obtain generalizations and improvements of some known results of the literature.

Index calculus algorithm for non-planar curves

In this paper, we develop a variation of the index calculus algorithm using non-planar models of non-hyperelliptic curves of genus g. Using canonical model of degree 2g−2 in the projective space of dimension g−1, intersections with hyperplanes and following similar ideas to those of Diem (who used intersections with lines on planar models), we obtain an upper bound of O(q2−2g−1+ε) for the computation of discrete logarithms for all non-hyperelliptic curves of genus g defined over the finite field Fq. This asymptotic cost is essentially the same as Diem's, but our algorithm offers several advantages over Diem's, including a constant speed-up.

Suboptimal s-union families and s-union antichains for vector spaces

Let V be an n-dimensional vector space over the finite field Fq, and let L(V)=⋃0≤k≤n[Vk] be the set of all subspaces of V. A family of subspaces F⊆L(V) is s-union if dim(F+F′)≤s holds for all F, F′∈F. A family F⊆L(V) is an antichain if F≰F′ holds for any two distinct F,F′∈F. An s-union family (resp. s-union antichain) with the largest possible size is called optimal. The optimal s-union families in L(V) have been determined by Frankl and Tokushige in 2013. An upper bound on cardinalities of s-union (s<n) antichains in L(V) has been established by Frankl recently, while the structures of optimal ones have not been displayed. The largest s-union families (resp. s-union antichains) that are not contained in any optimal one are called suboptimal. The present paper determines all suboptimal s-union families for vector spaces and then investigates s-union antichains. For s=n or s=2d<n, we determine all optimal and suboptimal s-union antichains completely. For s=2d+1<n, we prove that an optimal antichain is either [Vd] or contained in [Vd]∪[Vd+1] and satisfies an equation related to shadows.

New MDS self-dual codes from GRS codes and extended GRS codes

Maximum-distance-separable (MDS) codes and self-dual codes are important families of classical codes in coding theory. It is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field Fq has been completely solved for q is even. In this paper, for finite field with odd characteristic, we obtain some new results on the existence of MDS self-dual codes by generalized Reed-Solomon (GRS) codes and extended GRS codes.

Two families of few-weight codes over a finite chain ring

Linear codes over finite chain rings meeting the Griesmer bound are particularly interesting since they are closely related to minihyper in projective Hjelmslev spaces. In this paper, motivated by the nice article (Li et al., 2018 [23]), two classes of linear codes with few non-zero Hamming weights over the finite chain ring R=Fq+uFq are constructed, and one of them meets the Griesmer bound over finite commutative local rings. By a Gray map, we obtain that the Gray images of the constructed codes over the ring R are linear codes with two weights over the finite field Fq with q elements, and these codes are optimal with respect to the well-known Griesmer bound under certain conditions. Our results give an affirmative answer to two (related) open questions raised in the article mentioned above. Remarkably, some of the constructed linear codes over R presented in this paper correspond to minihypers in projective Hjelmslev spaces over finite chain rings. Besides, their Gray images can be used to construct secret sharing schemes with nice access structures and other applications thanks to the minimality property that possesses our constructed codes.

Linear equations on Drinfeld modules

Let L be a finite extension of the rational function field in one variable over a finite field Fq and E be a Drinfeld module defined over L. Given finitely many elements in E(L), this paper aims to prove that linear relations among these points can be characterized by solutions of an explicitly constructed system of homogeneous linear equations over Fq[t]. As a consequence, we show that there is an explicit upper bound for the size of the generators of linear relations among these points. This result can be regarded as an analogue of a theorem of Masser for finitely many K-rational points on an elliptic curve defined over a number field K.

Constructions of optimum distance full flag codes

The full flag codes of maximum distance and size on vector space Fq2ν are studied in this paper. We start to construct the subspace codes of maximum distance by making uses of the companion matrix of a primitive polynomial and the cosets of a subgroup in the general linear group over the finite field Fq. And a spread code is given. Then we focus on flag codes having maximum distance (optimum distance flag codes). Based on the constructed spread code, an optimum distance full flag code attaining the largest size is given. Then we consider the structure of flag codes on the symplectic space Fq2ν. We show that for any symmetric matrix A in GLν(Fq), there is a totally isotropic flag orbit code with optimum distance on the symplectic space Fq2ν. An optimum distance flag orbit code with larger size is presented on symplectic space Fq4k. And based on a partial spread code that are constructed, another optimum distance flag orbit code on the symplectic space Fq2ν is given.

Algebraic groups over finite fields: Connections between subgroups and isogenies

Abstract Let 𝐺 be a linear algebraic group defined over a finite field F q \mathbb{F}_{q} . We present several connections between the isogenies of 𝐺 and the finite groups of rational points ( G ⁢ ( F q n ) ) n ≥ 1 (G(\mathbb{F}_{\smash{q^{n}}}))_{n\geq 1} . We show that an isogeny ϕ : G ′ → G \phi\colon G^{\prime}\to G over F q \mathbb{F}_{q} gives rise to a subgroup of fixed index in G ⁢ ( F q n ) G(\mathbb{F}_{\smash{q^{n}}}) for infinitely many 𝑛. Conversely, we show that if 𝐺 is reductive, the existence of a subgroup H n H_{n} of fixed index 𝑘 for infinitely many 𝑛 implies the existence of an isogeny of order 𝑘. In particular, we show that the infinite sequence H n H_{n} is covered by a finite number of isogenies. This result applies to classical groups GL m \mathrm{GL}_{m} , SL m \mathrm{SL}_{m} , SO m \mathrm{SO}_{m} , SU m \mathrm{SU}_{m} , Sp 2 ⁢ m \mathrm{Sp}_{2m} and can be extended to non-reductive groups if 𝑘 is prime to the characteristic. As a special case, we see that if 𝐺 is simply connected, the minimal indices of proper subgroups of G ⁢ ( F q n ) G(\mathbb{F}_{\smash{q^{n}}}) diverge to infinity. Similar results are investigated regarding the sequence ( G ⁢ ( F p ) ) p (G(\mathbb{F}_{p}))_{p} by varying the characteristic 𝑝.

Efficient Private Circuits with Precomputation

At CHES 2022, Wang et al. described a new paradigm for masked implementations using private circuits, where most intermediates can be precomputed before the input shares are accessed, significantly accelerating the online execution of masked functions. However, the masking scheme they proposed mainly featured (and was designed for) the cost amortization, leaving its (limited) suitability in the above precomputation-based paradigm just as a bonus. This paper aims to provide an efficient, reliable, easy-to-use, and precomputation-compatible masking scheme. We propose a new masked multiplication over the finite field Fq suitable for the precomputation, and prove its security in the composable notion called Probing-Isolating Non-Inference (PINI). Particularly, the operations (e.g., AND and XOR) in the binary field can be achieved by assigning q = 2, allowing the bitsliced implementation that has been shown to be quite efficient for the software implementations. The new masking scheme is applied to leverage the masking of AES and SKINNY block ciphers on ARM Cortex M architecture. The performance results show that the new scheme contributes to a significant speed-up compared with the state-of-the-art implementations. For SKINNY with block size 64, the speed and RAM requirement can be significantly improved (saving around 45% cycles in the online-computation and 60% RAM space for precomputed values) from AES-128, thanks to its smaller number of AND gates. Besides the security proof by hand, we provide formal verifications for the multiplication and T-test evaluations for the masked implementations of AES and SKINNY. Because of the structure of the new masked multiplication, our formal verification can be performed for security orders up to 16.

An open problem of k-Galois hulls and its application

The k-Galois hull of a linear code C over the finite field Fq is defined as Hullk(C)=C∩C⊥k, where q=pe, p is prime and C⊥k is the k-Galois dual code of C with 0≤k<e. In this paper, by considering an equation of the matrix rank over finite fields, one of the two problems proposed by Cao about k-Galois hull of a linear code in (arXiv:2002.12892v2, 2020) is settled. Then we construct nine new classes of MDS EAQECCs with flexible parameters. In addition, our obtained MDS EAQECCs have parameters better than the ones available in the literature.