Balanced Incomplete Block Designs and Binary Linear Codes are two combinatorial designs. Due to the vast application of codes in communication the field of coding theory progressed more rapidly than many other fields of combinatorial designs. On the other hand, Block Designs are applicable in statistics and designing experiments in different fields, such as biology, medicine, and agriculture. Finding the relationship between instances of these two designs can be useful in constructing instances of one from the other. Applying the properties of codes to corresponding instances of Balanced Incomplete Block Designs has been used previously to show the non-existence of some designs. In this research the relationship between (16,6,3)-designs and (25,12) codes was determined.

In this thesis we study the application of trace maps over Galois fields and Galois rings in the construction of non-binary linear and non-linear codes and mutually unbiased bases. Properties of the trace map over the Galois fields and Galois rings has been used very successfully in the construction of cocyclic Hadamard, complex Hadamard and Butson Hadamard matrices and consequently to construct linear codes over integers modulo prime and prime powers. These results provide motivation to extend this work to construct codes over integers modulo . The prime factorization of integers paved the way to focus our attention on the direct product of Galois rings and Galois fields of the same degree. We define a new map over the direct product of Galois rings and Galois fields by using the usual trace maps. We study the fundamental properties of the this map and notice that these are very similar to that of the trace map over Galois rings and Galois fields. As such this map called the trace-like map and is used to construct cocyclic Butson Hadamard matrices and consequently to construct linear codes over integers modulo . We notice that the codes construct in this way over the integers modulo 6 is simplex code of type . A further generalization of the trace-like map called the weighted-trace map is defined over the direct product of Galois rings and Galois fields of different degrees. We use the weighted-trace map to construct some non-linear codes and mutually unbiased bases of odd integer dimensions. Further more we study the distribution of over the Galois fields of degree 2 and use it to construct 2-dimensional, two-weight, self-orthogonal codes and constant weight codes over integers modulo prime.

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. Despite the large body of literature in quantum coding theory, many important questions, especially those centering on the issue of "good codes" are unresolved. In this dissertation the dominant underlying theme is that of constructing good quantum codes. It approaches this problem from three rather different but not exclusive strategies. Broadly, its contribution to the theory of quantum error correction is threefold. Firstly, it extends the framework of an important class of quantum codes - nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. In particular it provides many explicit constructions of stabilizer codes, most notably it simplifies the criteria by which quantum BCH codes can be constructed from classical codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. Prior to our work however, systematic methods to construct these codes were few and it was not clear how to fairly compare them with other classes of quantum codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work established a close link between subsystem codes and classical codes and it became clear that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels. This approach is based on a Calderbank- Shor-Steane construction that combines BCH and finite geometry LDPC codes.