Coding theory, Communication theory and Computer algebra
Cyclic Orbit Codes (with A.-L. Trautman, M. Braun and J. Rosenthal) IEEE Transactions on Information Theory, 59(11):7386-7404, November 2013.
ABSTRACT: A constant dimension code consists of a set of k-dimensional subspaces of a finite field. Orbit codes are constant dimension codes which are defined as orbits of a subgroup of the general linear group, acting on the set of all subspaces of a finite field. If the acting group is cyclic, the corresponding orbit codes are called cyclic orbit codes. In this paper, we show how orbit codes can be seen as an analog of linear codes in the block coding case. We investigate how the structure of cyclic orbit codes can be utilized to compute the minimum distance and cardinality of a given code and propose different decoding procedures for a particular subclass of cyclic orbit codes.
Kötter Interpolation in Skew Polynomial Rings (with S. Liu and F.R. Kschischang ) Designs, Codes and Cryptography, January 2013.
ABSTRACT: Skew polynomials are a noncommutative generalization of ordinary polynomials that, in recent years, have found applications in coding theory and cryptography. Viewed as functions, skew polynomials have a well-defined evaluation map; however, little is known about skew-polynomial interpolation. In this work, we apply Kötter’s interpolation framework to free modules over skew polynomial rings. As a special case, we introduce a simple interpolation algorithm akin to Newton interpolation for ordinary polynomials.
An Algebraic Approach for Decoding Spread Codes (with E. Gorla and J. Rosenthal ) Advances in Mathematics of Communications, 6(4):443-466, November 2012
ABSTRACT: In this paper we study spread codes: a family of constant-dimension codes for random linear network coding. In other words, the codewords are full-rank matrices of size k×n with entries in a finite field. Spread codes are a family of optimal codes with maximal minimum distance. We give a minimum-distance decoding algorithm with polynomial complexity in n and k over an extension field. Our algorithm is more efficient than the previous ones in the literature, when the dimension k of the codewords is small with respect to n. The decoding algorithm takes advantage of the algebraic structure of the code, and it uses original results on minors of a matrix and on the factorization of polynomials over finite fields.
MATH 8560 - theory of Error-Correcting Codes
Topics include code constructions such as Hammig, cyclic, BCH, Reed-Solomon, Goppa, algebraic geometry, finite geometry, low-density parity check, convolutional and polynomial codes; code parameters and bounds; and decoding algorithms. Students are expected to have completed a graduate level course in matrix analysis before enrolling in this course.