The Basics of Bases

The maximum length of MDS codes for a given field and dimension, and their structure in the optimal case, remain unknown. However, at least asymptotically, the Reed-Solomon codes (RS codes) are optimal in the family of MDS codes. These RS codes are fundamental in technological applications ranging from computer drives to CD and DVD players to all manner of digital imaging, such as the amazing pictures transmitted by Voyager II. Here, Bruen and Bruen explain how many bases are contained in a spanning set and obtain the fundamental lower bound for the number of bases in a spanning set in a vector space. Moreover, using a little algebraic geometry they sketch an embedding result for finite fields.