On the Weights of General MDS Codes
Abstract
The weight spectra of MDS codes of length $ n $ and dimension $ k $ over the arbitrary alphabets are studied. For all $ q $-ary MDS codes of dimension $ k $ containing the zero codeword, it is shown that all $ k $ weights from $ n $ to $ n-k+1 $ are realized. The remaining case $ n=q+k-1 $ is also determined. Additionally, we prove that all binary MDS codes are equivalent to linear MDS codes. The proofs are combinatorial, and self contained.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2019
- DOI:
- 10.48550/arXiv.1910.05634
- arXiv:
- arXiv:1910.05634
- Bibcode:
- 2019arXiv191005634A
- Keywords:
-
- Mathematics - Combinatorics;
- 94A45;
- 94B25
- E-Print:
- IEEE Trans. Inform. Theory 66 (2020), no. 9, 5414-5418