pinecone/core-2020-05-10-deduplication huggingface dataset:
On the Spectra of Certain Graphs Arising from Finite Fields

AbstractCayley graphs on a subgroup ofGL(3,p),p>3 a prime, are defined and their properties, particularly their spectra, studied. It is shown that these graphs are connected, vertex-transitive, nonbipartite, and regular, and their degrees are computed. The eigenvalues of the corresponding adjacency matrices depend on the representations of the group of vertices. The “1-dimensional” eigenvalues can be completely described, while a portion of the “higher dimensional” eigenfunctions are discrete analogs of Bessel functions. A particular subset of these graphs is conjectured to be Ramanujan and this is verified for over 2000 graphs. These graphs follow a construction used by Terras on a subgroup ofGL(2,p). This method can be extended further to construct graphs using a subgroup ofGL(n, p) forn≥4. The 1-dimensional eigenvalues in this case can be expressed in terms of the 1-dimensional eigenvalues of graphs fromGL(2,p) andGL(3,p); this part of the spectra alone is sufficient to show that forn≥4, the graphs fromGL(n, p) are not in general Ramanujan
Polynomials with Odd Orthogonal Multiplicity

AbstractLet theorthogonal multiplicityof a monic polynomialgover a field F be the number of polynomialsfover F, coprime togand of degree less than that ofg, such that all the partial quotients of the continued fraction expansion off/gare of degree 1. Polynomials with positive orthogonal multiplicity arise in stream cipher theory, part of cryptography, as the minimal polynomials of the initial segments of sequences which have perfect linear complexity profiles. This paper focuses on polynomials which have odd orthogonal multiplicity; such polynomials are characterized and a lower bound on their orthogonal multiplicity is given. A special case of a conjecture on rational functions over the finite field of two elements with partial quotients of degree 1 or 2 in their continued fraction expansion is also proved
The Construction of Cameron–Liebler Line Classes inPG(3,q)

AbstractThis paper is concerned with fundamental questions lying at the boundary of combinatorics, geometry, and group theory. For example, letAbe the point–line incidence matrix of Σ=PG(3,q). ThenAis a (0, 1)-matrix, with the columns (rows) corresponding to points (lines) of Σ. Working over Q, say, the following question is natural: “Which linear combinations of the columns are themselves (0, 1)-vectors?” There are some obvious examples. In fact, in 1982, Cameron and Liebler conjectured that only the obvious examples exist. Here we settle this 16-year-old conjecture in the negative.This question turns out to be intimately related to geometrical properties of certain line sets in Σ, and to studies of the orbit structure of the collineation groupPΓL(n+1,q) of Σ
Power Sums over Finite Subspaces of a Field

AbstractLetVbe a finite additive subgroup of a fieldKof characteristicp>0. We consider sums of the formSh(V:α)=∑v∈V(v+α)hforh≥0 and α∈K. In particular, we give necessary and sufficient conditions for the vanishing ofSh(V; α), in terms of the digit sum in the base-pexpansion ofh, in the case thatVhas indexpinK. The proof involves the polynomialfV(x)=∏v∈V(x−v). We defineV†=fV(K). In the case thatKis finite, the polynomialfV†can be viewed as a generalised trace TrV:K→Vwhich coincides with the usual trace ifVis a subfield ofK
On Zeta Functions of Arithmetically Defined Graphs

AbstractWe study the graphX(n) that is defined as the finite part of the quotient Γ(n)\T, with T the Bruhat–Tits tree over Fq((1/T)) and Γ(n) the principal congruence subgroup of Γ=GL2(Fq[T]) of leveln∈Fq[T]. We give concrete realizations of theL-functions of the finite part of the halfline Γ\T for finite unitary representations of Γ that factor over Γ(n),nprime. This allows us to give explicit formulae for the zeta function ofX(n) for smalln. As an application, we show that these graphs are very good concentrators. Moreover, we construct a new unbounded family of Ramanujan graphs, considering regularizations ofX(n)
Perfect Codes and Balanced Generalized Weighing Matrices

AbstractIt is proved that any set of representatives of the distinct one-dimensional subspaces in the dual code of the unique linear perfect single-error-correcting code of length (qd−1)/(q−1) overGF(q) is a balanced generalized weighing matrix over the multiplicative group ofGF(q). Moreover, this matrix is characterized as the unique (up to equivalence) wieghing matrix for the given parameters with minimumq-rank. The classical, more involved construction for this type of BGW-matrices is discussed for comparison, and a few monomially inequivalent examples are included
Bounds for Self-Dual Codes Over Z4

AbstractNew bounds are given for the minimal Hamming and Lee weights of self-dual codes over Z4. For a self-dual code of length n, the Hamming weight is bounded above by 4[n/24]+f (n mod 24), for an explicitly given function f; the Lee weight is bounded above by 8[n/24]+g(n mod 24), for a different function g. These bounds appear to agree with the full linear programming bound for a wide range of lengths. The proof of these bounds relies on a reduction to a problem of binary codes, namely that of bounding the minimum dual distance of a doubly even binary code
Arcs and Curves over a Finite Field

AbstractIn [11], a new bound for the number of points on an algebraic curve over a finite field of odd order was obtained, and applied to improve previous bounds on the size of a complete arc not contained in a conic. Here, a similar approach is used to show that a complete arc in a plane of even order q has size q+2 or q−q+1 or less than q−2q+6. To obtain this result, first a new characterization of a Hermitian curve for any square q is given; more precisely, it is shown that a curve of sufficiently low degree has a certain upper bound for the number of its rational points with equality occurring in this bound only when the curve is Hermitian. Finally, another application is given concerning the degree of the curve on which a unital can lie
On the Density of Normal Bases in Finite Fields

AbstractLet Fqn denote the finite field with qn elements, for q a prime power. Fqn may be regarded as an n-dimensional vector space over Fq. α∈Fqn generates a normal basis for this vector space (Fqn:Fq), if {α, αq, αq2 , … , αqn−1} are linearly independent over Fq. Let Nq(n) denote the number of elements in Fqn that generate a normal basis for Fqn:Fq, and let νq(n)=Nq(n)/qn denote the frequency of such elements. We show that there exists a constant c>0 such that and this is optimal up to a constant factor in that we showWe also obtain an explicit lower bound
Cryptanalysis of the Wu–Dawson Public Key Cryptosystem

AbstractA new public key cryptosystem was introduced by Wu and Dawson at the Fourth International Conference on Finite Fields (Fq4). This scheme is similar to the McEliece public key cryptosystem, in the sense that it also can be described in terms of linear error-correcting codes over finite fields. However, in contrast to the McEliece scheme, the security of the Wu–Dawson system is not based on a decoding problem which is assumed to be intractable but on the theory of generalized inverses of matrices over finite fields. The authors compare their scheme with the McEliece scheme and claim that the same level of security can be obtained using smaller codes, therefore reducing the key size. In this note it will be shown that the Wu–Dawson scheme is insecure, i.e., a trapdoor can be computed efficiently from the knowledge of the public key
On Trellis Structures for Reed–Muller Codes

AbstractWe study trellises of Reed–Muller codes from first principles. Our approach to local trellis behaviour seems to be new and yields amongst other things another proof of a result of Berger and Be'ery on the state complexity of Reed–Muller codes. We give a general form of a minimal-span generator matrix for the family of Reed–Muller codes with their standard bit-order. We apply this to determining the number of parallel subtrellises in any uniform sectionalisation of a Reed–Muller code and to designing trellises for Reed–Muller codes with more parallel subtrellises than the minimal trellis, but with the same state complexity
Computing Weierstrass Semigroups and the Feng–Rao Distance from Singular Plane Models

AbstractWe present an algorithm to compute the Weierstrass semigroup at a point P together with functions for each value in the semigroup, provided P is the only branch at infinity of a singular plane model for the curve. As a byproduct, the method also provides us with a basis for the spaces L(mP) and the computation of the Feng–Rao distance for the corresponding array of geometric Goppa codes. A general computation of the Feng–Rao distance is also obtained. Everything can be applied to the decoding problem by using the majority scheme of Feng and Rao
New Techniques for the Computation of Linear Recurrence Coefficients

AbstractThe n coefficients of a fixed linear recurrence can be expressed through its m≤2n terms or, equivalently, the coefficients of a polynomial of a degree n can be expressed via the power sums of its zeros—by means of a polynomial equation known as the key equation for decoding the BCH error-correcting codes. The same problem arises in sparse multivariate polynomial interpolation and in various fundamental computations with sparse matrices in finite fields. Berlekamp's algorithm of 1968 solves the key equation by using order of n2 operations in a fixed field. Several algorithms of 1975–1980 rely on the extended Euclidean algorithm and computing Padé approximation, which yields a solution in O(n(log n)2 log log n) operations, though a considerable overhead constant is hidden in the “O” notation. We show algorithms (depending on the characteristic c of the ground field of the allowed constants) that simplify the solution and lead to further improvements of the latter bound, by factors ranging from order of log n, for c=0 and c>n (in which case the overhead constant drops dramatically), to order of min (c, log n), for 2≤c≤n; the algorithms use Las Vegas type randomization in the case of 2<c≤n
Rings of Low Multiplicative Complexity

AbstractThe complexity of the multiplication operation in finite fields is of interest for both theoretical and practical reasons. For example, an optimal normal basis for F2N has complexity 2N−1. A construction described in J. H. Silverman, (“Cryptographic Hardware and Embedded Systems,” Lecture Notes in Computer Science, Vol. 1717, pp. 122–134, Springer–Verlag, Berlin, 1999.) allows multiplication of complexity N+1 to be performed in F2N by working in a larger ring R of dimension N+1 over F2. In this paper we give a complete classification of all such rings and show that this construction is the only one which also has a certain useful permutability property
Constructions of Low Rank Relative Difference Sets in 2-Groups Using Galois Rings

AbstractIn this paper, we first determine the exact values of s(2, t), a function defined by X.-D. Hou [preprint], which depends on the structure of GR(4, t). This result gives the exact range of parameters of the partial difference sets in Zt4 constructed by Chen et al. (J. Combin. Theory Ser. A76 (1996), 179–196) and Hou (preprint). Second, we construct a new family of (22t, 2t, 22t, 2t) relative difference sets in abelian 2-groups of rank as low as t+⌈t2⌉. This is a significant reduction over the previous minimum of 2t found in Davis (Discrete Math.103, (1992), 7–15)
On the Cyclicity of Goppa Codes, Parity-Check Subcodes of Goppa Codes, and Extended Goppa Codes

AbstractClassical Goppa codes are a special case of Alternant codes. First we prove that the parity-check subcodes of Goppa codes and the extended Goppa codes are both Alternant codes. Before this paper, all known cyclic Goppa codes were some particular BCH codes. Many families of Goppa codes with a cyclic extension have been found. All these cyclic codes are in fact Alternant codes associated to a cyclic Generalized Reed–Solomon code. In (1989, J. Combin. Theory Ser. A 51, 205–220) H. Stichtenoth determined all cyclic extended Goppa codes with this property. In a recent paper (T. P. Berger, 1999, in “Finite Fields: Theory, Applications and Algorithms (R. Mullin and G. Mullen, Eds.), pp. 143–154, Amer. Math. Soc., Providence), we used some semi-linear transformations on GRS codes to construct cyclic Alternant codes that are not associated to cyclic GRS codes. In this paper, we use these results to construct cyclic Goppa codes that are not BCH codes, new families of Goppa codes with a cyclic extension, and some families of non-cyclic Goppa codes with a cyclic parity-check subcode
A Proof of the Welch and Niho Conjectures on Cross-Correlations of Binary m-Sequences

AbstractBinary m-sequences are widely applied in navigation, radar, and communication systems because of their nice autocorrelation and cross-correlation properties. In this paper, we consider the cross-correlation between a binary m-sequence of length 2m−1 and a decimation of that sequence by an integer t. We will be interested in the number of values attained by such cross-correlations. As is well known, this number equals the number of nonzero weights in the dual of the binary cyclic code C1,t of length 2m−1 with defining zeros α and αt, where α is a primitive element in GF(2m). There are many pairs (m, t) for which C⊥1,t is known or conjectured to have only few nonzero weights. The three-weight examples include the following cases:We present a method of proving many of these known or conjectured results, including all of the above cases, in a unified way
New Codes for the Berlekamp–Massey–Sakata Algorithm

AbstractThe key concept for generalizing the Berlekamp–Massey algorithm is the existence of an order function, a map from a ring to the nonnegative integers which determines a filtration of the ring with one-dimensional quotients. In this article I show that an order function determines a unique valuation on the function field, which has a residue field equal to the base field. The geometry of these valuations for several monomial orderings on a polynomial ring is discussed, and an ordering is constructed which does not correspond to any monomial ordering. The geometric description allows us to define order functions on general surfaces and higher dimensional varieties
Primitive Elements with Zero Traces

AbstractLet Fq denote the finite field of order q, a power of a prime p, and n be a positive integer. We resolve completely the question of whether there exists a primitive element of Fqn which is such that it and its reciprocal both have zero trace over Fq. Trivially, there is no such element when n<5: we establish existence for all pairs (q, n) (n≥5) except (4, 5), (2, 6), and (3, 6). Equivalently, with the same exceptions, there is always a primitive polynomial P(x) of degree n over Fq whose coefficients of x and of xn-1 are both zero. The method employs Kloosterman sums and a sieving technique
Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

AbstractLet Fq be the finite field of q elements with characteristic p and Fqm its extension of degree m. Fix a nontrivial additive character Ψ of Fp. If f(x1,…, xn)∈Fq[x1,…, xn] is a polynomial, then one forms the exponential sum Sm(f)=∑(x1,…,xn)∈(Fqm)nΨ(TrFqm/Fp(f(x1,…,xn))). The corresponding L functions are defined by L(f, t)=exp(∑∞m=0Sm(f)tm/m). In this paper, we apply Dwork's method to determine the Newton polygon for the L function L(f(x), t) associated with one variable polynomial f(x) when deg f(x)=4. As an application, we also give an affirmative answer to Wan's conjecture for the case deg f(x)=4
Partial Zeta Functions of Algebraic Varieties over Finite Fields

AbstractMotivated by arithmetic applications, we introduce the notion of a partial zeta function which generalizes the classical zeta function of an algebraic variety defined over a finite field. We then explain two approaches to the general structural properties of the partial zeta function in the direction of the Weil-type conjectures. The first approach, using an inductive fibred variety point of view, shows that the partial zeta function is rational in an interesting case, generalizing Dwork's rationality theorem. The second approach, due to Faltings, shows that the partial zeta function is always nearly rational
Classical Congruences for Parameters in Binary Quadratic Forms

AbstractLet Q(−k) be an imaginary quadratic field with discriminant −k and class number h, with k≠3, 4, or 8. Let p be a prime such that (−kp)=1. There are integers C, D, unique up to sign, such that 4ph=C2+kD2, p∤C. Stickelberger gave a congruence for C modulo p which extends congruences of Gauss, Jacobi, and Eisenstein. Stickelberger also gave a simultaneous congruence for C modulo k, but only for prime k. We prove an extension of his result that holds for all k, giving along the way an exposition of his work
All Two-Weight Irreducible Cyclic Codes?

AbstractThe aim of this paper is the classification of two-weight irreducible cyclic codes. Using Fourier transforms and Gauss sums, we obtain necessary and sufficient numerical conditions for an irreducible cyclic code to have at most two weights. This gives a unified explanation for all two-weight irreducible cyclic codes and allows a conjecturally complete classification. Aside from the two known infinite families of two-weight irreducible cyclic codes, a computer search reveals 11 sporadic examples. We conjecture that these are already all two-weight irreducible cyclic codes and give a partial proof of our conjecture conditionally on GRH
Hyperplane Sections of Grassmannians and the Number of MDS Linear Codes

AbstractWe obtain some effective lower and upper bounds for the number of (n,k)-MDS linear codes over Fq. As a consequence, one obtains an asymptotic formula for this number. These results also apply for the number of inequivalent representations over Fq of the uniform matroid or, alternately, the number of Fq-rational points of certain open strata of Grassmannians. The techniques used in the determination of bounds for the number of MDS codes are applied to deduce several geometric properties of certain sections of Grassmannians by coordinate hyperplanes
Galois Groups of Generalized Iterates of Generic Vectorial Polynomials

AbstractLet q=pu>1 be a power of a prime p, and let kq be an overfield of GF(q). Let m>0 be an integer, let J* be a subset of {1,…,m}, and let E*m, q(Y)=Yqm+∑j∈J*XjYqm−j where the Xj are indeterminates. Let J‡ be the set of all m−ν where ν is either 0 or a divisor of m different from m. Let s(T)=∑0≤i≤nsiTi be an irreducible polynomial of degree n>0 in T with coefficients si in GF(q). Let E*[s]m, q(Y) be the generalized sth iterate of E*m, q(Y); i.e., E*[s]m, q(Y)=∑0≤i≤nsiE*[i]m, q(Y), where E*[i]m, q(Y), is the ordinary ith iterate. We prove that if J‡⊂J*, m is square-free, and GCD(m,n)=1=GCD(mnu,2p), then Gal(E*[s]m, q,kq({Xj:j∈j*})=GL(m, qn). The proof is based on CT (=the Classification Theorem of Finite Simple Groups) in its incarnation as CPT (=the Classification of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors)
Factoring Polynomials over Special Finite Fields

AbstractWe exhibit a deterministic algorithm for factoring polynomials in one variable over finite fields. It is efficient only if a positive integer k is known for which Φk(p) is built up from small prime factors; here Φk denotes the kth cyclotomic polynomial, and p is the characteristic of the field. In the case k=1, when Φk(p)=p−1, such an algorithm was known, and its analysis required the generalized Riemann hypothesis. Our algorithm depends on a similar, but weaker, assumption; specifically, the algorithm requires the availability of an irreducible polynomial of degree r over Z/pZ for each prime number r for which Φk(p) has a prime factor l with l≡1 mod r. An auxiliary procedure is devoted to the construction of roots of unity by means of Gauss sums. We do not claim that our algorithm has any practical value
New Partial Difference Sets in Ztp2 and a Related Problem about Galois Rings

AbstractWe generalize a construction of partial difference sets (PDS) by Chen, Ray-Chaudhuri, and Xiang through a study of the Teichmüller sets of the Galois rings. Let R=GR(p2, t) be the Galois ring of characteristic p2 and rank t with Teichmüller set T and let π:R→R/pR be the natural homomorphism. We give a construction of PDS in R with the parameters ν=p2t, k=r(pt−1), λ=pt+r2−3r, μ=r2−r, where r=lpt−s(p, t), 1≤l≤ps(p, t), and s(p, t) is the largest dimension of a GF(p)-subspace W⊂R/pR such that π−1(W)∩T generates a subgroup of R of rank <t. We prove that s(p, t) is the largest dimension of a GF(p)-subspace W of GF(pt) such that dim Wp<t, where Wp is the GF(p)-space generated by {∏pi=1wi∣wi∈W, 1≤i≤p}. We determine the values of s(p, t) completely and solve a general problem about dimEWr for an E-vector space W in a finite extension of a finite field E. The PDS constructed here contain the family constructed by Chen, Ray-Chaudhuri, and Xiang and have a wider range of parameters
Number of Zeros of Diagonal Polynomials over Finite Fields

AbstractLet F be a finite field with q=pf elements, where p is a prime. Let N be the number of solutions (x1,…,xn) of the equation c1xd11+···+cnxdnn=c over the finite fields, where d1∣q−1, ciϵF*(i=1, 2,…,n), and cϵF. In this paper, we prove that if b1 is the least integer such that b1≥∑ni=1 (f/ri) (Di, p−1)/(p−1), then q[b1/f]−1∣N, where ri is the least integer such that di∣pri−1, Didi=pri−1, the (Di, p−1) denotes the greatest common divisor of Di and p−1, [b1/f] denotes the integer part of b1/f. If di=d, then this result is an improvement of the theorem that pb∣N, where b is an integer less than n/d, obtained by J. Ax (1969, Amer. J. Math.86, 255–261) and D. Wan (1988, Proc. AMS103, 1049–1052), under a certain natural restriction on d and n
Finite Commutative Chain Rings

AbstractA commutative ring with identity is called a chain ring if all its ideals form a chain under inclusion. A finite chain ring, roughly speaking, is an extension over a Galois ring of characteristic pnusing an Eisenstein polynomial of degree k. When p∤k, such rings were classified up to isomorphism by Clark and Liang. However, relatively little is known about finite chain rings when p∣k. In this paper, we allowed p∣k. When n=2 or when p∥k but (p−1)∤k, we classified all pure finite chain rings up to isomorphism. Under the assumption that (p−1)∤k, we also determined the structures of groups of units of all finite chain rings
Subregular Spreads of PG (5, 2e)

AbstractThe theory of subregular spreads of PG(3,q) was developed by R.H. Bruck (1969, in “Combinatorial Mathematics and Its Applications,” Chap. 27, pp. 426–514. Univ. of North Carolina Press, Chapel Hill). An extension of these results was provided to the higher-dimensional case by the author (1998, Finite Fields Appl.4, 362–380); however, examples of such spreads were only constructed in PG(5, q) for q odd. In this paper, we give a construction of subregular spreads for PG(5, q), where q is even
On the Distribution of the Diffie–Hellman Pairs

AbstractLet Fp be a prime field of p elements and let g be an element of Fp of multiplicative order t modulo p. We show that for any ε>0 and t≥p1/3+ε the distribution of the Diffie–Hellman pairs (x, gx) is close to uniform in the Cartesian product Zt×Fp, where x runs through•the residue ring Z modulo (that is, as in the classical Diffie–Hellman scheme);•The all -sums =+···+, 1≤<···<≤, where ,…,∈Z are selected at random (that is, an in the recently introduced Diffie–Hellman scheme with precomputation).These results are new and nontrivial even if t=p−1, that is, if g is a primitive root. The method is based on some bounds of exponential sums