# Applications of Coding Theory to the Construction of Modular Lattices

@article{Bachoc1997ApplicationsOC, title={Applications of Coding Theory to the Construction of Modular Lattices}, author={Christine Bachoc}, journal={J. Comb. Theory, Ser. A}, year={1997}, volume={78}, pages={92-119} }

We study self-dual codes over certain finite rings which are quotients of quadratic imaginary fields or of totally definite quaternion fields over Q. A natural weight taking two different nonzero values is defined over these rings; using invariant theory, we give a basis for the space of invariants to which belongs the three variables weight enumerator of a self-dual code. A general bound for the weight of such codes is derived. We construct a number of extremal self-dual codes, which are the…

## 172 Citations

EISENSTEIN LATTICES, GALOIS RINGS AND QUATERNARY CODES

- Computer Science
- 2006

Self-dual codes over the Galois ring GR(4,2) are investigated, of special interest are quadratic double circulant codes and Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.

A generalisation of Turyn's construction of self-dual codes (Research into Vertex Operator Algebras, Finite Groups and Combinatorics)

- Computer Science, Mathematics
- 2011

The present note interprets this construction as a sum of tensor products of codes and uses it to construct certain new extremal (or at least very good) self-dual codes (for example an extremal doubly-even binary code of length 80).

HERMITIAN LATTICES OVER IMAGINARY QUADRATIC FIELDS

- Mathematics

We introduce a family of bi-dimensional theta functions which give uniformly explicit formulae for the theta series of hermitian lattices over imaginary quadratic fields constructed from codes over…

Skew constacyclic codes over Galois rings

- Mathematics, Computer ScienceAdv. Math. Commun.
- 2008

It is shown that skew constacyclic self-dual codes over GR(4, 2) are constructed using Galois rings instead of finite fields as coefficients, and the resulting non commutative rings are no longer left and right Euclidean.

AND HERMITIAN LATTICES OVER IMAGINARY QUADRATIC FIELDS

- Mathematics

We introduce a family of bi-dimensional theta functions which give uniformly explicit formulae for the theta series of hermitian lattices over imaginary quadratic fields constructed from codes over…

Linear Codes over Finite Chain Rings

- Mathematics, Computer ScienceElectron. J. Comb.
- 2000

It is proved that there exists a one-to-one correspondence between so-called fat linear codes over chain rings and multisets of points in projective Hjelmslev geometries, in the sense that semilinearly isomorphic codes correspond to equivalent multiset and vice versa.

Quadratic residue codes over a non-chain ring extension of F 2

- Computer Science

This work defines Euclidean and Hermitian self-dual families of codes as extended quadratic residue codes over the ring F2+vF2 in terms of their idempotent generators and shows that these codes share the properties analogous to that of quadratics residue code over finite fields.

On the Equivalence of Codes over Finite Rings

- Mathematics, Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 2004

It is shown that for every finite ring R which is a direct sum of local and homogeneous semilocal subrings, if every Hamming-weight preserving R-linear transformation of a codeC1 onto a code C2 is a monomial transformation then R is a Frobenius ring.

A new shortening method and Hermitian self-dual codes over F2+vF2

- Computer ScienceDiscret. Math.
- 2020

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