Applications of Linear Programming to Coding Theory

Maximum-likelihood decoding is often the optimal decoding rule one can use, but it is very costly to implement in a general setting. Much effort has therefore been dedicated to find efficient decoding algorithms that either achieve or approximate the error-correcting performance of the maximum-likelihood decoder. This dissertation examines two approaches to this problem. In 2003 Feldman and his collaborators defined the linear programming decoder, which operates by solving a linear programming relaxation of the maximum-likelihood decoding problem. As with many modern decoding algorithms, is possible for the linear programming decoder to output vectors that do not correspond to codewords; such vectors are known as pseudocodewords. In this work, we completely classify the set of linear programming pseudocodewords for the family of cycle codes. For the case of the binary symmetric channel, another approximation of maximum-likelihood decoding was introduced by Omura in 1972. This decoder employs an iterative algorithm whose behavior closely mimics that of the simplex algorithm. We generalize Omura's decoder to operate on any binary-input memoryless channel, thus obtaining a soft-decision decoding algorithm. Further, we prove that the probability of the generalized algorithm returning the maximum-likelihood codeword approaches 1 as the number of iterations goes to infinity.

Ctenosaura similis (Black Spiny-Tailed Iguana), ,i>Gopherus Polyphemus (Gopher Tortoise) Concurrent Burrow Use

Ctenosaura similis is exotic to Florida (Meshaka et al. 2004. The Exotic Amphibians and Reptiles of Florida, Krieger Publ. Co., Malabar, Florida. 155 pp.), whereas Gopherus polyphemus is listed as a species of special concern by the state of Florida (Florida Wildlife Code Chap. 39 F.A.C.), and as a threatened species by the Florida Committee on Rare and Endangered Plants and Animals (FCREPA) (Moler 1992. Rare and Endangered Biota of Florida: Volume III, Reptiles and Amphibians. University Press of Florida, Gainesville, Florida. 291 pp.).