# Weighted Generating Functions and Configuration Results for Type II Lattices and Codes

@inproceedings{Kominers2009WeightedGF, title={Weighted Generating Functions and Configuration Results for Type II Lattices and Codes}, author={Scott Duke Kominers and Noam D. Elkies and Henry Cohn and John H. Conway and B. Gross and Abhinav Kumar and Gabriele Nebe and Ken Ono}, year={2009} }

We present an exposition of weighted theta functions, which are weighted generating functions for the norms and distribution of lattice vectors. We derive a decomposition theorem for the space of degree-d homogeneous polynomials in terms of spaces of harmonic polynomials and then prove that the weighted theta functions of Type II lattices are examples of modular forms. Our development of these results is structural, related to the infinite-dimensional representation theory of the Lie algebra…

## One Citation

Configurations of Extremal Type II Codes via Harmonic Weight Enumerators

- Mathematics, Computer ScienceJournal de Théorie des Nombres de Bordeaux
- 2019

It is shown that for n ∈ {8, 24, 32, 48, 56, 72, 96} every extremal Type II code of length n is generated by its codewords of minimal weight.

## References

SHOWING 1-10 OF 54 REFERENCES

Special Functions and the Theory of Group Representations

- Mathematics
- 1968

A standard scheme for a relation between special functions and group representation theory is the following: certain classes of special functions are interpreted as matrix elements of irreducible…

of Euclidean lattices , with some applications

- Mathematics
- 2009

By “Euclidean space” of dimension n we mean a real vector space of dimension n, equipped with a positive-definite inner product 〈·, ·〉. We usually call such a space “Rn” even when there is no…

Refined Configuration Results for Extremal Type II Lattices of Ranks 40 and 80

- Mathematics
- 2010

We show that, if L is an extremal Type II lattice of rank 40 or 80, then L is generated by its vectors of norm min(L) + 2. This sharpens earlier results of Ozeki, and the second author and Abel,…

Hahn Polynomials, Discrete Harmonics, and t-Designs

- Mathematics
- 1978

It is shown that certain Hahn polynomials and their q-analogues play in combinatorics a similar role as Gegenbauer polynomials in real Euclidean geometry. The concept of harmonic function on a fiber…

On the Classification of Type II Codes of Length 24

- Computer Science, MathematicsSIAM J. Discret. Math.
- 2010

A new, purely coding-theoretic proof of Koch's criterion on the tetrad systems of Type II codes of length 24 is given using the theory of harmonic weight enumerators and gives a new instance of the analogy between lattices and codes.

On Error-Correcting Codes and Invariant Linear Forms

- Computer Science, MathematicsSIAM J. Discret. Math.
- 1993

It is proved that the t-designs afforded by the codewords of any fixed weight exhibit extra regularity with respect to $( t + 2 )$-sets.

On the Classification and Enumeration of Self-Dual Codes

- Computer ScienceJ. Comb. Theory, Ser. A
- 1975

On even unimodular positive definite quadratic lattices of rank 32

- Mathematics
- 1986

Let Z be the ring of rational integers, and ~ the field of rational numbers. A finitely generated Z-module L in Q" with a positive definite metric is called a quadratic lattice. Since we treat only…