Related papers: Quaternary Legendre pairs II
We introduce quaternary Legendre pairs of length $\ell$. In contrast to binary Legendre pairs they can exist for even $\ell$ as well. First we show that they are pertinent to the construction of quaternary Hadamard matrices of order…
We give the first general constructions of even length quaternary Legendre pairs: there is a quaternary Legendre pair of length $(q-1)/2$ for every prime power $q$ congruent to $1$ modulo $4$, and there is a quaternary Legendre pair of…
By assuming a type of balance for length $\ell=87$ and non-trivial subgroups of multiplier groups of Legendre pairs (LPs) for length $\ell=85$, we find LPs of these lengths. We then study the power spectral density (PSD) values of…
There are 20 odd integers v less than 200 for which the existence of Legendre pairs of length v is undecided. The smallest among them is v=77. We have constructed Legendre pairs of lengths 91, 93 and 123 reducing the number of undecided…
We prove a proposition that connects constant-PAF sequences and the corresponding Legendre pairs with integer PSD values. We show how to determine explicitly the complete spectrum of the $(\ell/3)$-rd value of the discrete Fourier transform…
We provide a search method for Legendre pairs of composite length based on generating binary matrices with fixed row and column sums from compressed, complementary integer vectors. This approach yielded the first construction of a Legendre…
If $N=2^k > 8$ then there exist exactly $[(k-1)/2]$ pairwise nonequivalent $Z_4$-linear Hadamard $(N,2N,N/2)$-codes and $[(k+1)/2]$ pairwise nonequivalent $Z_4$-linear extended perfect $(N,2^N/2N,4)$-codes. A recurrent construction of…
Recently, simplicial complexes are used in constructions of several infinite families of minimal and optimal linear codes by Hyun {\em et al.} Building upon their research, in this paper more linear codes over the ring $\mathbb{Z}_4$ are…
We present a new method for constructing affine families of complex Hadamard matrices in every even dimension. This method has an intersection with the Di\c{t}\u{a} construction and it generalizes the Sz\"oll\H{o}si's method. We reproduce…
In this article, we construct infinite families of quaternary (that is, over the ring $\mathbb{Z}_4$) $\mathcal{C}_{D}$-codes, where the defining set $D$ is derived utilizing a two-generator simplicial complex, and determine their Lee…
Motivated by two Legendre-type formulas for overpartitions, we derive a variety of their companions as Legendre theorems for overpartition pairs. This leads to equalities of subclasses of overpartitions and overpartition pairs.
We show that 138 odd values of n less than 10000 for which one knows how to construct a Hadamard matrix of order 4n have been overlooked in the recent handbook of combinatorial designs. There are four additional odd n, namely 191, 5767,…
In 2013, Nebe and Villar gave a series of ternary self-dual codes of length $2(p+1)$ for a prime $p$ congruent to $5$ modulo $8$. As a consequence, the third ternary extremal self-dual code of length $60$ was found. We show that the ternary…
In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to device new quantum algorithms. We show how the properties of a weighing matrix can be used to construct…
A classification of Hadamard matrices of order $2p+2$ with an automorphism of order $p$ is given for $p=29$ and $31$. The ternary self-dual codes spanned by the newly found Hadamard matrices of order $60$ with an automorphism of order $29$…
An extensive table of pairs of functions linked by the Legendre transformation is presented. Many special functions and formulas that are used in the sciences are included in the pairs. Formulations are provided for finding the Legendre…
We have extended the Paley constructions for Hadamard matrices and obtained some series of Hadamard matrices. Especially Paley construction-II is applicable for odd prime power q is congruent to 1(mod 4) however our method is applicable for…
Our splitter theorem for internally 4-connected binary matroids studies pairs of the form (M,N), where N and M are internally 4-connected binary matroids, M has a proper N-minor, and if M' is an internally 4-connected matroid such that M…
Periodic Golay pairs are a generalization of ordinary Golay pairs. They can be used to construct Hadamard matrices. A positive integer $v$ is a (periodic) Golay number if there exists a (periodic) Golay pair of length $v$. Taking into the…
We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the $7$-modular and $11$-modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a…