| Author(s): |
Qureshi, Claudio
;
Costa, Sueli I. R.
;
IEEE
Total Authors: 3
|
| Document type: | Journal article |
| Source: | 2016 INFORMATION THEORY AND APPLICATIONS WORKSHOP (ITA); v. N/A, p. 4-pg., 2016-01-01. |
| Abstract | |
Perfect codes in a given metric are the ones which attain the sphere packing bound. In the Hamming metric perfect linear codes over finite fields are known only to exist for a restricted number of parameters. In the Lee metric, perfect linear codes in Z(n) are classified for n = 2 and the long standing Golomb-Welch conjecture that there are no perfect codes for n >= 3, except for packing radius 1, still remains open. We approach here perfect codes over finite rings in the maximum metric (also known as L-infinity or Chebyshev metric). This metric has been recently considered in coding schemes for flash memory. A perfect linear code in Z(q)(n) in the maximum metric corresponds to an integer lattice cube tiling. We present a complete classification of two-dimensional perfect codes, constructions of perfect codes from codes of smaller dimensions and generator matrices for n-dimensional perfect linear q-ary codes in the maximum metric. From these results a description or all perfect codes in the Euclidean metric for n = 2, 3 is derived. (AU) | |
| FAPESP's process: | 13/25977-7 - Security and reliability of Information: theory and practice |
| Grantee: | Marcelo Firer |
| Support Opportunities: | Research Projects - Thematic Grants |