Abstract
Optimal rank-metric codes in Ferrers diagrams are considered. Such codes consist of matrices having zeros at certain fixed positions and can be used to construct good codes in the projective space. First, we consider rank-metric anticodes and prove a code-anticode bound for Ferrers diagram rank-metric codes. The size of optimal linear anticodes is given. Four techniques and constructions of Ferrers diagram rank-metric codes are presented, each providing optimal codes for different diagrams and parameters for which no optimal solution was known before. The first construction uses maximum distance separable codes on the diagonals of the matrices, the second one takes a subcode of a maximum rank distance code, and the last two combine codes in small diagrams to a code in a larger diagram. The constructions are analyzed and compared, and unsolved diagrams are identified.
| Original language | English |
|---|---|
| Article number | 7394177 |
| Pages (from-to) | 1616-1630 |
| Number of pages | 15 |
| Journal | IEEE Transactions on Information Theory |
| Volume | 62 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - Apr 2016 |
Funding
T. Etzion was supported in part by the Israeli Science Foundation, Jerusalem, Israel, under Grant 10/12. E. Gorla and A. Ravagnani were supported by the Swiss National Science Foundation under Grant 200021-150207. A. Wachter-Zeh was supported in part by a Minerva Post-Doctoral Fellowship and in part by the European Union's Horizon 2020 research and innovation programme under the Sklodowska-Curie grant agreement No. 655109.
Keywords
- anticodes
- Ferrers diagrams
- Gabidulin codes
- rank-metric codes
- subspace codes