For a positive integer n we evaluate the convolution sum Sigma(al+bm=n) sigma(l)sigma(3)(m) for (a,b)=(1,7),(7,1),(1,8),(8,1),(1,9),(9,1). We then use these evaluations together with knownevaluations of other convolution sums to determine the numbers of representations of n by the forms x(1)(2) + x(2)(2) + x(3)(2) + x(4)(2) + 2(x(5)(2) + x(6)(2) + x(7)(2) + x(8)(2) + x(9)(2) + x(10)(2) + x(11)(2) + x(12)(2)), x(1)(2) + x(2)(2) + x(3)(2) + x(4)(2) + 2(x(5)(2) + x(6)(2) + x(7)(2) + x(8)(2) + x(9)(2) + x(10)(2) + x(11)(2) + x(12)(2)), x(1)(2) + x(1)x(2) + x(2)(2) + x(3)(2) + x(3)x(4) + x(4)(2) + 3(x(5)(2) + x(5)x(6) + x(6)(2) + x(7)(2) + x(7)x(8) + x(8)(2) + x(9)(2) + x(9)x(10) + x(10)(2) + x(11)(2) + x(11)x(12) + x(12)(2)), x(1)(2) + x(1)x(2) + x(2)(2) + x(3)(2) + x(3)x(4) + x(4)(2) + x(5)(2) + x(5)x(6) + x(6)(2) + x(7)(2) + x(7)x(8) + x(8)(2) + 3(x(9)(2) + x(9)x(10) + x(10)(2) + x(11)(2) + x(11)x(12) + x(2)(12)). We use a modular form approach.

For a positive integer n we evaluate the convolution sum ∑ _al+bm=nσ (l)σ _ 3 (m) for (a,b) = (1,7) , (7, 1), (1, 8), (8, 1), (1, 9) and (9, 1). We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of n by the forms x_1^2 + x_2^2 + x_3^2 + x_4^2 + 2(x_5^2 + x_6^2 + x_7^2 + x_8^2 + x_9^2 + x_10^2 + x_11^2 + x_12^2), x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2 + x_8^2 + 2(x_9^2 + x_10^2 + x_11^2 + x_12^2), x_1^2 + x_1x_2 + x_2^2 +x_3^2 + x_3x_4 + x_4^2 + 3(x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2 + x_9^2 + x_9x_10 + x_10^2 + x_11^2 + x_11x_12 + x_12^2) , x_1^2 + x_1x_2 + x_2^2 +x_3^2 + x_3x_4 + x_4^2 + x_5^2 + x_5x_6 + x_6^2 + x_7^2 + x_7x_8 + x_8^2 + 3(x_9^2 + x_9x_10 + x_10^2 + x_11^2 + x_11x_12 + x_12^2) . We use a modular form approach.

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.

We consider the problem of finding maximal sets of shift-inequivalent decimations of Sidelnikov-Lempel-Cohn-Eastman (SLCE) sequences (as well as the equivalent problem of determining the multiplier groups of the almost difference sets associated with these sequences). We derive a numerical necessary condition for a residue to be a multiplier of an SLCE almost difference set. Using our necessary condition, we show that if $p$ is an odd prime and $S$ is an SLCE almost difference set over $\mathbb{F}_p,$ then the multiplier group of $S$ is trivial. Consequently, for each odd prime $p,$ we obtain a family of $\phi(p-1)$ shift-inequivalent balanced periodic sequences (where $\phi$ is the Euler-Totient function) each having period $p-1$ and nearly perfect autocorrelation.

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 determine the convolution sum Sigma(al+bm=n) sigma(l)sigma(m) for (a, b) = (1, 48), (3, 16), (1, 54), (2, 27) for all positive integers n. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of n by the octonary quadratic forms k(x(1)(2) +x(1)x(2) + x(2)(2) + x(3)(2)+ x(3)x(4) + x(4)(2)) + l(x(5)(2) + x(5)x(6) + x(6)(2) + x(7)(2) + x(7)x(8) + x(8)(2)) for (k, l) = (1, 16), (1, 18), (2, 9). A modular form approach is used.

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 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.

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.

Binary sequences with good autocorrelation properties and large linear complexity are useful in stream cipher cryptography. The Sidelnikov-Lempel-Cohn-Eastman (SLCE) sequences have nearly optimal autocorrelation. However, the problem of determining the linear complexity of the SLCE sequences is still open. It is well known that one can gain insight into the linear complexity of a sequence if one can say something about the divisors of the gcd of a certain pair of polynomials associated with the sequence. Helleseth and Yang (IEEE Trans. Inf. Theory 49(6), 1548–1552 2002), Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) and Meidl and Winterhof (Des. Codes Crypt. 8, 159–178 2006) were able to obtain some results of this type for the SLCE sequences. Kyureghyan and Pott (Des. Codes Crypt. 29, 149–164 2003) mention that it would be nice to obtain more such results. We derive new divisibility results for the SLCE sequences in this paper. Our approach is to exploit the fact that character values associated with the SLCE sequences can be expressed in terms of a certain type of Jacobi sum. By making use of known evaluations of Gauss and Jacobi sums in the “pure” and “small index” cases, we are able to obtain new insight into the linear complexity of the SLCE sequences.

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]

We determine the convolution sums Sigma(l+27m= n) sigma(l)sigma(m) and Sigma(l+32m= n) sigma(l)sigma(m) for all positive integers n. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of n by the octonary quadratic forms x(1)(2) + x(1)x(2) + x(2)(2) + x(3)(3) + x(3)x(4) + x(4)(2) + 9(x(5)(2) + x(5)x(6) + x(6)(2) + x(7)(2) + x(7)x(8) + x(8)(2)) and x(1)(2) + x(2)(2) + x(3)(2) + x(4)(2) + 8(x(5)(2) + x(6)(2) + x(7)(2) + x(8)(2)). A modular form approach is used.

We present an explicit evaluation of the double Gauss sum G(a,b,c;S;p^n):=∑_x,y=0^p^n-1 e^2π i S(ax^2+bxy+cy^2)/p^n, where a, b, c are integers such that (a,b,c)=1, p is a prime, n is a positive integer, and S is an integer coprime to p.

We determine the convolution sums [Formula: see text] and [Formula: see text] for all positive integers [Formula: see text]. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of [Formula: see text] by the octonary quadratic forms [Formula: see text] and [Formula: see text]. A modular form approach is used.

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.

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 express products of Lambert series as power series. We use these Lambert series-to-power series identities to obtain new Liouville identities with two functions. Many of the known Liouville identities follow from our new identities. We explore the relationships between Lambert series, new Liouville identities, and convolution sums. We then obtain recursive formulae for various convolution sums.