Related papers: Quaternary Legendre pairs II
This note reports on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type,…
This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of…
A finite sequence of numbers is perfect if it has zero periodic autocorrelation after a nontrivial cyclic shift. In this work, we study quaternionic perfect sequences having a one-to-one correspondence with the binary sequences arising in…
In this paper, we study non-linear differential equations associated with Legendre polynomials and their applications. From our study of non- linear differential equations, we derive some new and explicit identities for Legendre…
It is proved that a code $L(q)$ which is monomially equivalent to the Pless symmetry code $C(q)$ of length $2q+2$ contains the (0,1)-incidence matrix of a Hadamard 3-$(2q+2,q+1,(q-1)/2)$ design $D(q)$ associated with a Paley-Hadamard matrix…
Let $1\leq a<q$ be a pair of small integers such that $\gcd(a,q)=1$ and let $x>1$ be a large number. This note discusses the existence of a short sequence of primes $p\equiv a\bmod q$ between two squares $x^2$ and $(x+1)^2$.
We study the masses of the low-lying charmed mesons within the framework of heavy-hadron chiral perturbation theory. We work to third order in the chiral expansion, where meson loops contribute. In contrast to previous approaches, we use…
In this article, we consider a special class of Williamson type matrices which we call them near Williamson matrices. They are in fact four $n\times n$ $(-1, 1)$-matrices $A, B, C, D$ so that $A$ is circulant, $B,C,D$ are symmetric…
Double integrals that represent matrix elements of the power and logarithmic potentials in the Legendre polynoiomial basis on [-1,1] are found in a closed form. Several proofs are given, which involve different special functions and…
Self-dual codes have been studied actively because they are connected with mathematical structures including block designs and lattices and have practical applications in quantum error-correcting codes and secret sharing schemes.…
A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and 14 is given, and a new parametrization scheme for obtaining new examples of affine parametric families of complex Hadamard matrices is provided. On the…
Let $n_k(s)$ be the maximal length $n$ such that a quaternary additive $[n,k,n-s]_4$-code exists. We solve a natural asymptotic problem by determining the lim sup $\lambda_k$ of $n_k(s)/s,$ and the smallest value of $s$ such that…
Two arrays form a periodic complementary pair if the sum of their periodic autocorrelations is a delta function. Finding such pairs, however, is challenging for large arrays whose entries are constrained to a small alphabet. One such…
There is a one-to-one correspondence between $\ell$-quasi-cyclic codes over a finite field $\mathbb F_q$ and linear codes over a ring $R = \mathbb F_q[Y]/(Y^m-1)$. Using this correspondence, we prove that every $\ell$-quasi-cyclic self-dual…
Golay complementary sequences have been put a high value on the applications in orthogonal frequency-division multiplexing (OFDM) systems since its good peak-to-mean envelope power ratio(PMEPR) properties. However, with the increase of the…
Legendre published the first attempted proof of the law of Quadratic Reciprocity. But, in its final form (1797), it had a gap. Some 125 years later Herman Teege published the first rigorous proof of the unproven hypothesis which formed that…
The normal sequences NS(n) and near-normal sequences NN(n) play an important role in the construction of orthogonal designs and Hadamard matrices. They can be identified with certain base sequences (A;B;C;D), where A and B have length n+1…
We systematically explore the new method of construction (known as the propus construction) of symmetric Hadamard matrices for small orders, $4v$. In particular we give the first examples of symmetric Hadamard matrices of order $156=4\cdot…
The construction of self-dual codes over small fields such that their minimum distances are as large as possible is a long-standing challenging problem in the coding theory. In 2009, a family of binary self-dual cyclic codes with lengths…
We review a lattice construction arising from quaternion algebras over number fields and use it to obtain some known extremal and densest lattices in dimensions 8 and 16. The benefit of using quaternion algebras over number fields is that…