A CHARACTERIZATION OF CLASSES OF LINEAR TERNARY CYCLIC CODES, THEIR DESIGNS AND LATTICES

Abstract

Linear cyclic ternary codes defined over the Galois field GF(3) exhibit several ad vantages over their binary counterparts. They provide an extra option for each pulse resulting into a larger set of available codes at any given length. This the sis presents a comprehensive study of classes of linear cyclic ternary codes of length 25 ≤ n ≤ 50, their associated combinatorial designs, and lattice structures. While binary codes have been extensively studied, the properties and applications of longer ternary codes remain less explored. This research aimed at addressing this gap by providing an in-depth characterization of these codes and their related mathemati cal structures. Using computational methods implemented in Magma software, we generated and analyzed a diverse set of linear cyclic ternary codes over GF(3). The study examined key properties including minimum distance, weight distribution, and theoretical bounds for both the generated codes and their duals. We employed the Assmus-Mattson Theorem and Kramer-Mesner method to construct t-designs from these codes, revealing rich combinatorial structures. A significant contribution of this research is the exploration of the geometric properties of these codes through the construction and analysis of related lattices using Construction A. We investigated the relationships between code parameters and lattice properties, providing new in sights into the structure of ternary codes from a geometric perspective. Our findings extended the existing knowledge of ternary cyclic codes, particularly for lengths ex ceeding 25. We identified several new codes with favorable parameters, constructed previously unreported combinatorial designs, and characterized lattices with unique properties. The results demonstrated that ternary cyclic codes exhibit high struc tural regularity and often produce interesting designs and lattices with properties distinct from their binary counterparts. The research revealed strong interconnec tions between coding theory, combinatorial design theory, and lattice theory in the context of ternary codes. We provided a multifaceted characterization framework that integrates algebraic, combinatorial, and geometric perspectives, offering a holistic un derstanding of these codes. This study contributes to the theoretical advancement of non-binary codes and opens new avenues for their practical applications in error correction, cryptography, and communication systems. The comprehensive analysis and novel insights presented in this thesis lay a strong foundation for future research in ternary coding theory and its intersections with combinatorics and geometry.