201. On the Classification of MDS Codes.
- Author
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Kokkala, Janne I., Krotov, Denis S., and Ostergard, Patric R. J.
- Subjects
SINGLETON bounds ,CODING theory ,LINEAR codes ,SET theory ,HAMMING distance - Abstract
A q -ary code of length n , size M , and minimum distance d code. An (n,q^{k},n-k+1)_{q} code is called a maximum distance separable (MDS) code. In this paper, some MDS codes over small alphabets are classified. It is shown that every (k+d-1,q^{k},d)_{q} code with k\geq 3 , d \geq 3 , is equivalent to a linear code with the same parameters. This implies that the (6,5^4,3)5 code and the (n,7^{n-2},3)7 MDS codes for n\in \6,7,8\ are unique. The classification of one-error-correcting 8-ary MDS codes is also finished; there are 14, 8, 4, and 4 equivalence classes of (n,8^n-2,3)8 codes for $n=6,7,8$ , and 9, respectively. One of the equivalence classes of perfect (9,8^{7},3)_{8} codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction. [ABSTRACT FROM PUBLISHER]
- Published
- 2015
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