Accession Number : ADA554686

Title :   Subspace Arrangement Codes and Cryptosystems

Descriptive Note : Trident Scholar Project rept. no. 395


Personal Author(s) : Berg, James A

Full Text :

Report Date : 09 May 2011

Pagination or Media Count : 51

Abstract : Errors often occur in transferring electronic data, ranging from sensitive government information to everyday bar codes. Encoding information with an error-correcting code can alleviate the problem of corrupt or lost data. In order to not overburden computing systems, an efficient code must be used that will quickly encode and decode data while detecting and correcting a large number of errors. The goal of this project is to construct and develop efficient codes using recent advances in algebraic geometry, combinatorics, and commutative algebra. The mathematics of subspace arrangements and simplicial complexes lend themselves well for applications to coding theory. A subspace arrangement is a finite collection of subspaces in a vector space. A simplicial complex is an abstract generalization of a polygon or Euclidean solid. Fortunately, both simplicial complexes and subspaces arrangements can be described algebraically by a collection of polynomials, which can be used to construct a code. Then the combinatorial and geometric properties of subspace arrangements and simplicial complexes can be used to enumerate these efficient codes. Scripts and algorithms were written in the computer algebra systems Sage and Macaulay2 to compute properties of the codes. The data led to the main results of the project: formulas for the length, dimension, and minimum distance of polygon and skeletal simplicial complex evaluation codes. Scripts were written that aided in the construction of proofs for these formulas. The formulas give favorable lengths (short to minimize computation), dimensions (large to allow for more codewords), and minimum distances (large to allow more errors to be corrected and identified) of these polygon and skeletal simplicial complex evaluation codes. The last part of the project involved extensions to a cryptosystem based on these codes. A cryptosystem deals with enciphering a message, which is an algorithmic process designed to make a sent message unreadable


Subject Categories : Numerical Mathematics
      Theoretical Mathematics

Distribution Statement : APPROVED FOR PUBLIC RELEASE