Department of Mathematics Colloquium

University of Idaho

Spring 2013

Thursday,  March 19, 3:30-4:20 pm, room TLC 047

Refreshments in Brink 305 at 3:00 pm

Linear Codes from Commutative
Algebraic Perspectives


Stefan Tohaneanu

Department of Mathematics

Western University


Given a finite set of points in a projective or Euclidean space, the "Exact Fitting Problem" determines the maximum number of these points contained in some hyperplane. Besides the applications in Optimization Theory or Computer Graphic Design and Pattern Recognition, this problem is strongly related to important questions in Coding Theory. If one considers any generating matrix of a linear code and thinks of the columns of this matrix as the homogeneous coordinates of points in a projective space (assuming that there are no proportional nor zero columns), solving the Exact Fitting Problem for these points also determines the minimum distance of the code. I will present a Commutative Algebraic approach to solve this problem and I will also show how some information contained in the graded minimal free resolution of the ideal of the set of points constructed from the columns of the generating matrix, gives lower bounds for the minimum distance of the code. The talk will address to a general audience, with a minimal background in Linear Algebra, and some knowledge in Abstract Algebra (e.g., rings, modules, ideals, the ring of polynomials).