Byte weight enumerators and modular forms of genus r

2015 ◽  
Vol 14 (06) ◽  
pp. 1550080
Author(s):  
Anuradha Sharma ◽  
Amit K. Sharma
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Zeta Functions ◽  
Siegel Modular Forms ◽  
Type I ◽  
Type Ii ◽  
Jacobi Forms ◽  
Weight Enumerators ◽  
Type Ii Codes ◽  
Real Quaternions

For a positive integer m, let R be either the ring ℤ2m of integers modulo 2m or the quaternionic ring Σ2m = ℤ2m + αℤ2m + βℤ2m + γℤ2m with α = 1 + î, β = 1 + ĵ and [Formula: see text], where [Formula: see text] are elements of the ring ℍ of real quaternions satisfying [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we obtain Jacobi forms (or Siegel modular forms) of genus r from byte weight enumerators (or symmetrized byte weight enumerators) in genus r of Type I and Type II codes over R. Furthermore, we derive a functional equation for partial Epstein zeta functions, which are summands of classical Epstein zeta functions associated with quadratic forms.

scholarly journals EISENSTEIN POLYNOMIALS ASSOCIATED TO BINARY CODES

2009 ◽  
Vol 05 (04) ◽  
pp. 635-640 ◽  
Author(s):  
MANABU OURA
Keyword(s):  
Binary Codes ◽  
Weighted Sum ◽  
Type Ii ◽  
Weight Enumerators ◽  
Fixed Length ◽  
Type Ii Codes ◽  
Genus 2

The Eisenstein polynomial is the weighted sum of the weight enumerators of all classes of Type II codes of fixed length. In this note, we investigate the ring generated by Eisenstein polynomials in genus 2.

TOWARDS THE SIEGEL RING IN GENUS FOUR

2008 ◽  
Vol 04 (04) ◽  
pp. 563-586 ◽  
Author(s):  
MANABU OURA ◽  
CRIS POOR ◽  
DAVID S. YUEN
Keyword(s):  
Linear Combination ◽  
Irreducible Component ◽  
Modular Forms ◽  
Quotient Ring ◽  
Siegel Modular Forms ◽  
Weight Enumerators ◽  
New Techniques ◽  
First Order ◽  
The Difference ◽  
Infinite Set

Runge gave the ring of genus three Siegel modular forms as a quotient ring, R3/〈J(3)〉 where R3 is the genus three ring of code polynomials and J(3) is the difference of the weight enumerators for the e8 ⊕ e8 and [Formula: see text] codes. Freitag and Oura gave a degree 24 relation, [Formula: see text], of the corresponding ideal in genus four; where [Formula: see text] is also a linear combination of weight enumerators. We take another step towards the ring of Siegel modular forms in genus four. We explain new techniques for computing with Siegel modular forms and actually compute six new relations, classifying all relations through degree 32. We show that the local codimension of any irreducible component defined by these known relations is at least 3 and that the true ideal of relations in genus four is not a complete intersection. Also, we explain how to generate an infinite set of relations by symmetrizing first order theta identities and give one example in degree 32. We give the generating function of R5 and use it to reprove results of Nebe [25] and Salvati Manni [37].

ON A CORRESPONDENCE BETWEEN JACOBI CUSP FORMS AND ELLIPTIC CUSP FORMS

2013 ◽  
Vol 09 (04) ◽  
pp. 917-937 ◽  
Author(s):  
B. RAMAKRISHNAN ◽  
KARAM DEO SHANKHADHAR
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Cusp Forms ◽  
Jacobi Forms ◽  
Binary Quadratic Forms ◽  
Congruence Subgroups ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Forms Of Higher Degree ◽  
Appl Math

In this paper, we prove a generalization of a correspondence between holomorphic Jacobi cusp forms of higher degree (matrix index) and elliptic cusp forms obtained by K. Bringmann [Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms, Math. Z.253 (2006) 735–752], for forms of higher levels (for congruence subgroups). To achieve this, we make use of the method adopted by M. Manickam and the first author in Sec. 3 of [On Shimura, Shintani and Eichler–Zagier correspondences, Trans. Amer. Math. Soc.352 (2000) 2601–2617], who obtained similar correspondence in the degree one case. We also derive a similar correspondence in the case of skew-holomorphic Jacobi forms (matrix index and for congruence subgroups). Such results in the degree one case (for the full group) were obtained by N.-P. Skoruppa [Developments in the theory of Jacobi forms, in Automorphic Functions and Their Applications, Khabarovsk, 1988 (Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990), pp. 168–185; Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms, J. Reine Angew. Math.411 (1990) 66–95] and by M. Manickam [Newforms of half-integral weight and some problems on modular forms, Ph.D. thesis, University of Madras (1989)].

Efficient Singularity-Free Workspace Approximations Using Sum-of-Squares Programming

2020 ◽  
Vol 12 (6) ◽  
Author(s):  
Wankun Sirichotiyakul ◽  
Volkan Patoglu ◽  
Aykut C. Satici
Keyword(s):  
Quadratic Forms ◽  
Optimization Technique ◽  
Robotic Manipulators ◽  
Additional Constraint ◽  
Sum Of Squares ◽  
Type I ◽  
Polynomial Functions ◽  
Type Ii ◽  
Trigonometric Functions ◽  
Outer Approximations

Abstract In this paper, we provide a general framework to determine inner and outer approximations to the singularity-free workspace of fully actuated robotic manipulators, subject to Type-I and Type-II singularities. This framework utilizes the sum-of-squares optimization technique, which is numerically implemented by semidefinite programming. In order to apply the sum-of-squares optimization technique, we convert the trigonometric functions in the kinematics of the manipulator to polynomial functions with an additional constraint. We define two quadratic forms, describing two ellipsoids, whose volumes are optimized to yield inner and outer approximations of the singularity-free workspace.

scholarly journals Modular forms of degree n and representation by quadratic forms

1979 ◽  
Vol 74 ◽  
pp. 95-122 ◽  
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Circle Method ◽  
Positive Definite ◽  
Siegel Modular Forms ◽  
Partial Result ◽  
Definite Integral ◽  
Integral Matrices

Let A(m), B(n) be positive definite integral matrices and suppose that B is represented by A over each p-adic integers ring Zp. Using the circle method or theory of modular forms in case of n = 1, B, if sufficiently large, is represented by A provided that m ≥ 5. The approach via the theory of modular forms has been extended by [7] to Siegel modular forms to obtain a partial result in the particular case when n = 2, m ≥ 7.

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