Research On Combinatorial Approaches Of Encoding Two Classes Of Codes

Combinatorial design is a branch of combinatorics, it is a subject of analyzing and constructing the configuration of finite discrete objects of a given kind and size.Coding theory is the study of the properties of codes in information transmission.They all study the discrete objects and have close links. Coding is a transform of signal for a specific application and one important content is the research on encoding methods in order to create a code whose performance can be optimal. However, constructing a code of ”good” performance is not a simple matter. In this dissertation, we study two topics in coding theory from a combinatorial point-of-view: two-dimensional optical orthogonal codes having property of AM-OPPTS and equitable symbol weight codes.The dissertation is divided into two parts.In the first part, we focus on the encoding methods of AM-OPPTS(at most one-pulse per time slot) 2-D(m × n, k, ρ)-OOCs which have good auto-correlation and cross-correlation properties in fiber-optic code-division multiple-access. In chapter3, we explore the intimate relationship among AM-OPPTS 2-D(m × n, k, ρ)-OOCs,n-cyclic holey packings and n-cyclic holey mixed-di?erence packings(families). As a consequence, an upper bound that the optimal code can attain is derived. Based on the link, we are able to employ combinatorial methods, such as mixed-di?erence, recursive constructions, to construct optimal AM-OPPTS 2-D(m × n, k, ρ)-OOCs. As their applications, infinite families of such codes are produced.In the second part, we discuss a kind of important codes in power line communication, that is equitable symbol weight code which was proposed by Chee et al. in order to more precisely capture a code’s performance against permanent narrowband noise in power line communication. In chapter 4, we present the set-theoretic characterization of equitable symbol weight codes, that is generalized balanced tournament design. By the combinatorial characterization, recursive construction, starter-adder,di?erence matrix are established to construct infinite classes of HGBTDs. As an application of the established constructions, two series of new equitable symbol weight codes of optimal sizes meeting the Plotkin bound are constructed.