A note on MDS codes, n-Arcs and complete designs

Authors

Vladimir Tonchev, Michigan Technological UniversityFollow

Document Type

Article

Publication Date

5-2003

Abstract

A generalized incidence matrix of a design over GF(q) is any matrix obtained from the (0, 1)-incidence matrix by replacing ones with nonzero elements from GF(q). The dimension d q of a design D over GF(q) is defined as the minimum value of the q-rank of a generalized incidence matrix of D. It is proved that the dimension d_q of the complete design on n points having as blocks all w-subsets, is greater that or equal to n − w + 1, and the equality d_q = n − w + 1 holds if and only if there exists an [n, n − w + 1, w] MDS code over GF(q), or equivalently, an n-arc in PG(w − 2, q).

Publisher's Statement

© Kluwer Academic Publishers 2003. Publisher’s version of record: https://doi.org/10.1023/A:1024121012598

Publication Title

Designs, Codes and Cryptography

Recommended Citation

Tonchev, V. (2003). A note on MDS codes, n-Arcs and complete designs. Designs, Codes and Cryptography, 29(1-3), 247-250. http://doi.org/10.1023/A:1024121012598
Retrieved from: https://digitalcommons.mtu.edu/math-fp/123