On m-MDS codes over finite fields

Abstract

Let ^⊥ be the dual code of the q-ary linear code =[n, k, d]_q and S()=n+1−k−d=m(≥0) be the Singleton defect of . Several sufficient and necessary conditions for that =[n, k, d]_q satisfies S()=S(^⊥)=m are obtained. Furthermore, all q-ary Hamming m-MDS codes are determined.

Keywords:

• MDS code
• m-MDS code
• parity-check matrix
• generator matrix
• Hamming code
• the minimum distance
• Singleton defect

2000 AMS Subject Classifications:

• 11T23
• 11T71
• 94B05

Acknowledgements

The first author is supported by the National Science Foundation of China (10990011), the Ph.D. Programmes Foundation of Ministry of Education of China (20095134120001), and Sichuan Provincial Advance Research Programme for Excellent Youth Leaders of Disciplines in Science of China (2011JQ0037). The author would like to thank reviewers for some helpful suggestions.