A Gap in GRM Code Weight Distributions

The weight distribution of GRM (generalized Reed-Muller) codes is unknown in general. This article describes and applies some new techniques to the codes over F3. Specifically, we decompose GRM codewords into words from smaller codes and use this decomposition, along with a projective geometry technique, to relate weights occurring in one code with weights occurring in simpler codes. In doing so, we discover a new gap in the weight distribution of many codes. In particular, we show there is no word of weight 3^m–2 in GRM₃(4,m) for m>6, and for even-order codes over the ternary field, we show that under certain conditions, there is no word of weight d+, where d is the minimum distance and is the largest integer dividing all weights occurring in the code.     

On certain functional equations related to Jordan triple (q, f){(\theta, \phi)}-derivations on semiprime rings

The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q _s and let q{\theta} and f{\phi} be automorphisms of R. Suppose T:R? R{T:R\rightarrow R} is an additive mapping satisfying the relation T(xyx)=T(x)q(y)q(x)-f(x)T(y)q(x)+f(x)f(y)T(x){T(xyx)=T(x)\theta (y)\theta (x)-\phi (x)T(y)\theta (x)+\phi (x)\phi (y)T(x)}, for all pairs x,y ? R{x,y\in R}. In this case T is of the form 2T(x)=qq(x)+f(x)q{2T(x)=q\theta (x)+\phi (x)q}, for all x ? R{x\in R} and some fixed element q ? Q_s{q\in Q_{s}}.     

Center conditions III: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). Folowing [7], we consider a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1).. We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and generalize a “canonical representation” ofm _k (x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

The commutation relation <Emphasis Type="Italic">xy</Emphasis>=<Emphasis Type="Italic">qyx</Emphasis>+<Emphasis Type="Italic">hf</Emphasis>(<Emphasis Type="Italic">y</Emphasis>) and Newton’s binomial formula

Let x and y be two variables satisfying the commutation relation xy=qyx+hf(y), where f(y) is a polynomial. In this paper, using Young diagrams and generating functions techniques, we study the binomial formula (x+y) ⁿ and we present an identity for x ^m y. The connection to Operator Calculus is discussed and several special cases are treated explicitly.     

On cyclic caps in 4-dimensional projective spaces

For any divisor k of q ⁴−1, the elements of a group of k th-roots of unity can be viewed as a cyclic point set C _k in PG(4,q). An interesting problem, connected to the theory of BCH codes, is to determine the spectrum A(q) of maximal divisors k of q ⁴−1 for which C _k is a cap. Recently, Bierbrauer and Edel [Edel and Bierbrauer (2004) Finite Fields Appl 10:168–182] have proved that 3(q ² + 1)∈A(q) provided that q is an even non-square. In this paper, the odd order case is investigated. It is proved that the only integer m for which m(q ² + 1)∈A(q) is m = 2 for q ≡ 3 (mod 4), m = 1 for q ≡ 1 (mod 4). It is also shown that when q ≡ 3 (mod 4), the cap is complete.      

Minimal cyclic codes of length pq

Explicit expressions for all the 3n+2 primitive idempotents in the ring R_pⁿq=GF(ℓ)[x]/(x^pnq−1), where p,q,ℓ are distinct odd primes, ℓ is a primitive root modulo pⁿ and q both, , are obtained. The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length pⁿq over GF(ℓ) are also discussed.     

Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

Let F_q be the finite field of q elements with characteristic p and F_q^m its extension of degree m. Fix a nontrivial additive character Ψ of F_p. If f(x₁,…, x_n)∈F_q[x₁,…, x_n] is a polynomial, then one forms the exponential sum S_m(f)=∑_{(x1,…,xn)∈(Fq^m)ⁿ}Ψ(Tr_Fq^m/Fp(f(x₁,…,x_n))). The corresponding L functions are defined by L(f, t)=exp(∑^∞_m=0S_m(f)t^m/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.     

On the Minimum Distances of Non-Binary Cyclic Codes

We deal with the minimum distances of q-ary cyclic codes of length q ^m - 1 generated by products of two distinct minimal polynomials, give a necessary and sufficient condition for the case that the minimum distance is two, show that the minimum distance is at most three if q > 3, and consider also the case q = 3.     

Weierstrass Semigroups and Codes from a Quotient of the Hermitian Curve

We consider the quotient of the Hermitian curve defined by the equation y^q + y = x^m over where m > 2 is a divisor of q+1. For 2≤ r ≤ q+1, we determine the Weierstrass semigroup of any r-tuple of -rational points on this curve. Using these semigroups, we construct algebraic geometry codes with minimum distance exceeding the designed distance. In addition, we prove that there are r-point codes, that is codes of the form where r ≥ 2, with better parameters than any comparable one-point code on the same curve. Some of these codes have better parameters than comparable one-point Hermitian codes over the same field. All of our results apply to the Hermitian curve itself which is obtained by taking m=q +1 in the above equation Communicated by: J.W.P. Hirschfeld     

A symmetric Roos bound for linear codes

The van Lint-Wilson AB-method yields a short proof of the Roos bound for the minimum distance of a cyclic code. We use the AB-method to obtain a different bound for the weights of a linear code. In contrast to the Roos bound, the role of the codes A and B in our bound is symmetric. We use the bound to prove the actual minimum distance for a class of dual BCH codes of length q²−1 over Fq. We give cyclic codes [63,38,16] and [65,40,16] over F₈ that are better than the known [63,38,15] and [65,40,15] codes.     

On self-orthogonal group ring codes

We obtain structural results about group ring codes over F[G], where F is a finite field of characteristic p > 0 and the Sylow p-subgroup of the Abelian group G is cyclic. As a special case, we characterize cyclic codes over finite fields in the case the length of the code is divisible by the characteristic of the field. By the same approach we study cyclic codes of length m over the ring R = F _q [u], u ^r  = 0 with r  > 0, gcd(m, q) = 1. Finally, we give a construction of quasi-cyclic codes over finite fields.      

Mappings preserving two hyperbolic distances

Suppose that X is the set of points of a hyperbolic geometry of finite or infinite dimension , and that is a fixed real number and N>1 a fixed integer. Let be a mapping such that for every if h(x, y)=, then h(f,(x),f,(y)) , and if h(x,y) = N, then h(f,(x),f,(y)) , where h,(p,q) designates the hyperbolic distance of p,q . Then f is an isometry of X. Note that there is no regularity assumption on f, like continuity or even differentiability. Moreover, we present an example showing that the assumption that one fixed distance > 0 is preserved does not characterize hyperbolic isometries. Received 17 February 2000.     

Taking pth Roots Modulo Polynomials over Finite Fields

In this contribution we show how to find y(x) in the polynomial equation y(x) ^p t(x) mod f(x), where t(x), y(x) and f(x) are polynomials over the field GF(p ^m). The solution of such equations are thought for in many cases, e.g., for p = 2 it is a step in the so-called Patterson Algorithm for decoding binary Goppa codes.     

Inverse spectral problem for a generalized Sturm-Liouville equation with complex-valued coefficients

The present paper is the first to prove that one of the columns of the monodromy matrix and two of the three coefficients (piecewise analytic on the interval [0, 1]) of the equation (f(x)y′)′+(r(x)−λ ² q(x))y = 0 uniquely determine the third coefficient on this interval provided that the values of the functions f(x) and q(x) lie in the lower (or upper) open complex halfplane and on the positive part of the real axis. This unknown coefficient can be reconstructed by finding the unique zero minimum of a specially constructed functional depending on the solutions of the corresponding Cauchy problem and the given elements of the monodromy matrix.     

Limit-point / limit-circle classification of second-order differential operators arising in PT quantum mechanics

We consider a second-order differential equation −y^″ + q(x)y(x) = λy(x) with complex-valued potential q and eigenvalue parameter λ ∈ ℂ. In PT quantum mechanics the potential q is given by q(x) = −(ix)^N+2 on a contour Γ ⊂ ℂ. Via a parametrization we obtain two differential equations on [0, ∞) and (−∞, 0]. We give a limit-point/limit-circle classification of this problem via WKB-analysis. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tensor product q-Bernstein polynomials

An affine de Casteljau type algorithm to compute q-Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bézier patch and a tensor product q-Bernstein Bézier patch. Finally, q-Bernstein polynomials B _n,m (f;x,y) for a function f(x,y), (x,y)∈[0,1]×[0,1] are defined and fundamental properties are discussed. AMS subject classification (2000)  65D17     

A comparative study of functional equations characterising sine and cosine

A comparative study of the functional equationsf(x+y)f(x–y)=f ²(x)–f ²(y),f(y){f(x+y)+f(x–y)}=f(x)f(2y) andf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} which characterise the sine function has been carried out. The zeros of the functionf satisfying any one of the above equations play a vital role in the investigations. The relation of the equationf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} with D'Alembert's equation,f(x+y)+f(x–y)=2f(x)f(y) and the sine-cosine equationg(x–y)=g(x)g(y) +f(x)f(y) has also been investigated.     

All q m -ary Cyclic Codes with Cyclic q-ary Image Are Known

We give a simpler presentation of recent work byLeonard, Séguin and Tang-Soh-Gunawan. Our methodsimply as a new result that even in the repeated-root casethere are no more q ^m -arycyclic codes with cyclic q-ary image than thosegiven by Séguin.     

Decomposer and associative functional equations

In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:

f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)