We explore the theta functions of nineteen positive-definite integral non-diagonal quaternary quadratic forms of discriminant 784 with levels 28 or 56. We express these theta functions in terms of Eisenstein series and cusp forms, which we then use to give explicit formulas for the representation number of a positive integer n by their corresponding non-diagonal quaternary quadratic forms. We also find the theta functions of the genera to which those non-diagonal quaternary quadratic forms belong. Finally, we express the theta function of each non-diagonal quadratic form in terms of the theta functions of certain diagonal quaternary quadratic forms and a cusp form.

In this letter, a code-domain nonorthogonal multiple access (NOMA) technique based on an algebraic design is studied. We propose an improved low-density spreading (LDS) sequence design based on projective geometry. In terms of its bit error rate (BER) performance, our proposed improved LDS code set outperforms the existing LDS designs over the frequency-nonselective Rayleigh fading and additive white Gaussian noise (AWGN) channels. We demonstrated that achieving the best BER depends on the minimum distance.

The purpose of this paper is to present some examples of positive-definite integral nondiagonal quaternary quadratic forms whose representation numbers can be determined explicitly using the theory of modular forms. Very few such examples appear in the literature. The seven forms presented were selected because they each belong to a genus containing exactly two form classes for which the single genus mate is a diagonal form whose representation number has been determined recently.

We find all the eta quotients in the spaces M-1(Gamma(0)(12), (d/.)) (d = -3, -4) of modular forms and determine their Fourier coefficients, where (d/.) is the Legendre-Jacobi-Kronecker symbol.

We consider the problem of determining the cross-correlation values of the sequences in the families comprised of constant multiples of M-ary Sidelnikov sequences over 𝔽_q, where q is a power of an odd prime p. We show that the cross-correlation values of pairs of sequences from such a family can be expressed in terms of certain Jacobi sums. This insight facilitates the computation of the cross-correlation values of these sequence pairs so long as ϕ(M)^ϕ(M)≤ q. We are also able to use our Jacobi sum expression to deduce explicit formulae for the cross-correlation distribution of a family of this type in the special case that there exists an integer x such that p^x ≡ -1 M.

We determine explicit formulas for the number of representations of a positive integer n by quaternary quadratic forms with coefficients 1, 2, 5 or 10. We use a modular forms approach.

We evaluate the convolution sum W-a,W-b(n) := Sigma(al+bm=n) sigma(l)sigma(m) for (a,b) = (1, 28), (4, 7), (2, 7) for all positive integers n. We use a modular form approach. We also re-evaluate the known sums W-1,(14)(n) and W-1(,7)(n) with our method. We then use these evaluations to determine the number of representations of n by the octonary quadratic form x(1)(2) + x(2)(2) + x(3)(2) +x(4)(2) + 7(x(5)(2) + x(6)(2) + x(7)(2) + x(8)(2)). Finally we express the modular forms Delta(4)(,7)(z), Delta(4)(,14,1)(z) and Delta(4)(,14,2)(z) (given in [10, 14]) as linear combinations of eta quotients.

We express all the newforms of weight $2$ and levels $30$, $33$, $35$, $38$, $40$, $42$, $44$, $45$ as linear combinations of eta quotients and Eisenstein series, and list their corresponding strong Weil curves. Let $p$ denote a prime and $E (\zz_p)$ denote the the group of algebraic points of an elliptic curve $E$ over $\zz_p$. We give a generating function for the order of $E (\zz_p)$ for certain strong Weil curves in terms of eta quotients and Eisenstein series. We then use our generating functions to deduce congruence relations for the order of $E (\zz_p)$ for those strong Weil curves.

Using modular forms we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.

We find bases for the spaces M-2 (Gamma(0)(24), (d/center dot)) (d = 1, 8, 1 2, 2 4) of modular forms. We determine the Fourier coefficients of all 3 5 theta products phi[alpha(1), alpha(2), alpha(3), alpha(4)] (z) in these spaces. We then deduce formulas for the number of representations of a positive integer n by diagonal quaternary quadratic forms with coefficients 1, 2, 3 or 6 in a uniform manner, of which 1 4 are Ramanujan's universal quaternary quadratic forms. We also find all the eta quotients in the Eisenstein spaces E-2 (Gamma(0)(24), (d/center dot)) (d = 1, 8, 1 2, 2 4) and give their Fourier coefficients.

We find bases for the spaces $M_2\Big(\Gamma_0(24),\Big(\frac{d}{\cdot}\Big)\Big)$ ($d=1,8,12, 24$) of modular forms. We determine the Fourier coefficients of all $35$ theta products $\varphi[a_1,a_2,a_3,a_4](z)$ in these spaces. We then deduce formulas for the number of representations of a positive integer $n$ by diagonal quaternary quadratic forms with coefficients $1$, $2$, $3$ or $6$ in a uniform manner, of which $14$ are Ramanujan's universal quaternary quadratic forms. We also find all the eta quotients in the Eisenstein spaces $E_2\Big(\Gamma_0(24),\Big(\frac{d}{\cdot}\Big)\Big)$ ($d=1,8,12,24$) and give their Fourier coefficients.

Let [Formula: see text] denote a complex variable with [Formula: see text]. For a positive integer [Formula: see text] let [Formula: see text] If [Formula: see text] we define [Formula: see text] for each nonnegative integer [Formula: see text]. In this paper, we determine results of the type [Formula: see text]

Let $k\geq 2$ be an integer and $j$ an integer satisfying $1\leq j \leq 4k-5$. We define a family $\{ C_{j,k}(z) \}_{1\leq j \leq 4k-5} $ of eta quotients, and prove that this family constitute a basis for the space $S_{2k} (\Gamma_0 (12))$ of cusp forms of weight $2k$ and level $12$. We then use this basis together with certain properties of modular forms at their cusps to prove an extension of the Ramanujan-Mordell formula.

An explicit formula is given for the representation number of each of the 75 reduced, positive-definite, integral, primitive, quaternary quadratic forms [Formula: see text], which belong to a genus with discriminant [Formula: see text] containing one and only one form class.

We determine explicit formulae for the number of representations of a positive integer n by the quaternary quadratic forms a_1x_1^2+a_2x_2^2+a_3x_3^2+a_4x_4^2, where a_1, a_2, a_3, a_4 in {1,2,7,14}. We use a modular form approach.

For all natural numbers n, we discuss the evaluation of the convolution sum, (l,m) ∈ℕ_0^2 α l+β m=n∑σ(l)σ(m), where αβ=14,22,26. We generalize the extraction of the convolution sum using Eisenstein forms of weight 4 for all pairs of positive integers (α,β). We also determine formulae for the number of representations of a positive integer by the octonary quadratic forms a (x_1^2 + x_2^2 + x_3^2 + x_4^2)+ b (x_5^2 + x_6^2 + x_7^2 + x_8^2), where (a,b)= (1,1), (1,3), (2,3), (1,9). These numbers of representations of a positive integer are applications of the evaluation of certain convolution sums by J. G. Huard et al., A. Alaca et al. and D. Ye.

We determine the coefficients of the Fourier series of a class of eta quotients of weight 2. For example, we show that [Formula: see text] where [Formula: see text] and [Formula: see text] are Jacobi–Kronecker symbols. We prove our results using the theory of modular forms.

Using modular forms we determine the number of representations of a positive integer by certain diagonal octonary quadratic forms with coefficients 1, 3 or 9.