34 research outputs found
Eisenstein polynomials associated to binary codes
The Eisenstein polynomial is the weighted sum of all classes of
Type II codes of fixed length.
In this note, we investigate the ring of the Eisenstein
polynomials in genus
The invariants of the Clifford groups
The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m (m not
3) is a subgroup of index 2 in a certain ``Clifford group'' C_m (an
extraspecial group of order 2^(1+2m) extended by an orthogonal group). This
group and its complex analogue CC_m have arisen in recent years in connection
with the construction of orthogonal spreads, Kerdock sets, packings in
Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs.
In this paper we give a simpler proof of Runge's 1996 result that the space
of invariants for C_m of degree 2k is spanned by the complete weight
enumerators of the codes obtained by tensoring binary self-dual codes of length
2k with the field GF(2^m); these are a basis if m >= k-1. We also give new
constructions for L_m and C_m: let M be the Z[sqrt(2)]-lattice with Gram matrix
[2, sqrt(2); sqrt(2), 2]. Then L_m is the rational part of the mth tensor power
of M, and C_m is the automorphism group of this tensor power. Also, if C is a
binary self-dual code not generated by vectors of weight 2, then C_m is
precisely the automorphism group of the complete weight enumerator of the
tensor product of C and GF(2^m). There are analogues of all these results for
the complex group CC_m, with ``doubly-even self-dual code'' instead of
``self-dual code''.Comment: Latex, 24 pages. Many small improvement
On the image of code polynomials under theta map
The theta map sends code polynomials into the ring of Siegel modular forms of
even weights. Explicit description of the image is known for and the
surjectivity of the theta map follows. Instead it is known that this map is not
surjective for . In this paper we discuss the possibility of an
embedding between the associated projective varieties. We prove that this is
not possible for and consequently we get the non surjectivity of the
graded rings for the remaining case
Kneser-Hecke-operators in coding theory
The Kneser-Hecke-operator is a linear operator defined on the complex vector
space spanned by the equivalence classes of a family of self-dual codes of
fixed length. It maps a linear self-dual code over a finite field to the
formal sum of the equivalence classes of those self-dual codes that intersect
in a codimension 1 subspace. The eigenspaces of this self-adjoint linear
operator may be described in terms of a coding-theory analogue of the Siegel
-operator
OBSERVATION ON THE WEIGHT ENUMERATORS FROM CLASSICAL INVARIANT THEORY
The purpose of this paper is to collect computations related to the weight enumerators and to present some relationships among invariant rings.The latter is done by combining two maps,the Brou\'e-Enguehard map and Igusa's \rho homomorphism. For the completeness of the story,some formulae are given which are not necessarily used in the present manuscript.Sections 1 and 2 contain no new result
On the integral ring spanned by genus two weight enumerators
It is known that the weight enumerator of a self-dual doubly-even code in genus g = 1 can be uniquely written as an isobaric polynomial in certain homogeneous polynomials with integral coeffcients. We settle the case where g = 2 and prove the non-existence of such polynomials under some conditions
Equivariant theory for codes and lattices I
In this paper, we present a generalization of Hayden's theorem [7, Theorem
4.2] for -codes over finite Frobenius rings. A lattice theoretical form of
this generalization is also given. Moreover, Astumi's MacWilliams identity [1,
Theorem 1] is generalized in several ways for different weight enumerators of
-codes over finite Frobenius rings. Furthermore, we provide the Jacobi
analogue of Astumi's MacWilliams identity for -codes over finite Frobenius
rings. Finally, we study the relation between -codes and its corresponding
-lattices.Comment: 22 page