Quantum error-correcting codes and 4-dimensional arithmetic hyperbolic manifolds

Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new homological quantum error correcting codes. They are LDPC codes with linear rate and distance $n^\epsilon$. Their rate is evaluated via Euler characteristic arguments and their distance using $\mathbb{Z}_2$-systolic geometry. This construction answers a queston of Z\'emor, who asked whether homological codes with such parameters could exist at all.Comment: 21 page

Lattice polytopes in coding theory

In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also prove a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations.Comment: 11 pages, 3 figure

Incidence structures from the blown-up plane and LDPC codes

In this article, new regular incidence structures are presented. They arise from sets of conics in the affine plane blown-up at its rational points. The LDPC codes given by these incidence matrices are studied. These sparse incidence matrices turn out to be redundant, which means that their number of rows exceeds their rank. Such a feature is absent from random LDPC codes and is in general interesting for the efficiency of iterative decoding. The performance of some codes under iterative decoding is tested. Some of them turn out to perform better than regular Gallager codes having similar rate and row weight.Comment: 31 pages, 10 figure

Higher Hamming weights for locally recoverable codes on algebraic curves

We study the locally recoverable codes on algebraic curves. In the first part of this article, we provide a bound of generalized Hamming weight of these codes. Whereas in the second part, we propose a new family of algebraic geometric LRC codes, that are LRC codes from Norm-Trace curve. Finally, using some properties of Hermitian codes, we improve the bounds of distance proposed in [1] for some Hermitian LRC codes. [1] A. Barg, I. Tamo, and S. Vlladut. Locally recoverable codes on algebraic curves. arXiv preprint arXiv:1501.04904, 2015