1 General Properties of Lattices.- 2 Geometric Inequalities.- 3 Perfection and Eutaxy.- 4 Root Lattices.- 5 Lattices Related to Root Lattices.- 6 Low-Dimensional Perfect Lattices.- 7 The Voronoi Algorithm.- 8 Hermitian Lattices.- 9 The Configurations of Minimal Vectors.- 10 Extremal Properties of Families of Lattices.- 11 Group Actions.- 12 Cross-Sections.- 13 Extensions of the Voronoi Algorithm.- 14 Numerical Data.- 15 Appendix 1: Semi-Simple Algebras and Quaternions.- 16 Appendix 2: Strongly Perfect Lattices.- References.- List of Symbols.

Perfect Lattices in Euclidean Spaces

We generalize the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. The resulting non commutative rings are no longer left and right Euclidean. Codes that are principal ideals in quotient rings of skew polynomial rings by a two sided ideals are studied. As an application, skew constacyclic self-dual codes over GR(4, 2) are constructed. Euclidean self-dual codes give self-dual Z(4)-codes. Hermitian self-dual codes yield 3-modular lattices and quasi-cyclic self-dual Z(4)-codes.

https://hal.archives-ouvertes.fr/hal-00359833/document

Skew constacyclic codes over Galois rings

Quasi-twisted codes with constacyclic constituent codes

Two class association schemes consist of either strongly regular
graphs (SRG) or doubly regular tournaments (DRT). We construct self-dual codes from the adjacency matrices of these schemes. This generalizes the construction of Pless ternary Symmetry codes, Karlin binary Double Circulant codes, Calderbank and Sloane quaternary double circulant codes, and Gaborit Quadratic Double Circulant codes (QDC). As new examples SRG's give 4 (resp. 5) new Type I (resp. Type II) [72, 36, 12] codes. We construct a [200, 100, 12] Type II code invariant under the Higman-Sims group, a [200, 100, 16] Type II code invariant under the Hall-Janko group, and more generally self-dual binary
codes attached to rank three groups.

Double circulant codes from two class association schemes

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79---95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(32,2), GR(33,2) and GR(34,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(52,2), GR(53,2) and GR(72,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(112,2) we have MDS self-dual codes of lengths up to 12.

/pdf/construction-of-mds-self-dual-codes-over-galois-rings-1ng0wp6fwd.pdf

Construction of MDS self-dual codes over Galois rings

Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual &#x1d53d;4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.

Ann Marie Natividad

Papers

Eisenstein lattices, Galois rings and quaternary codes

On quintic quasi-cyclic codes