PERMUTATION GROUPS OF EXTENDED CYCLIC CODES OVER GALOIS RINGS
Author
BLACKFORD, JASON THOMAS
Year
1999
Advisor
RAY-CHAUDHURI, DIJEN K.
Abstract
Coding theory is a branch of mathematics that has many practical applications in the transmission of information, and is used heavily by computer scientists and engineers. A code over a ring $R$ is a set of vectors of $R^n$ , and an $R$-linear code is a submodule of $R^n$. It was recently discovered that certain binary codes with good parameters can be constructed from $Z_4$-linear codes. This discovery motivated the study of codes over integer residue rings and eventually codes over Galois rings. The permutation group of a code is the group of permutations on the coordinates of a vector that preserve the code. In particular, a cyclic code is a code that is left invariant under any cyclic shift of coordinates. It is good for a code to have a large permutation group, because it is useful for decoding and determining weight distributions. This dissertation focuses on the study of cyclic and extended cyclic codes over Galois rings, particularly those of characteristic 4. Many classical results about cyclic codes over finite fields are generalized. A slightly new approach is given that characterizes cyclic codes over Galois rings, one that involves multiple defining sets and Mattson-Solomon polynomials. Using this approach, all affine-invariant extended cyclic codes over the ring $GR(4, m)$ are classified. This dissertation suggests a method of determining their complete permutation groups, and creates some extended cyclic codes with permutation groups that are larger than those of codes appearing in previous literature. Then a conclusion is made on the asymptotic behavior of these codes.
BLACKFORD, JASON THOMAS
Year
1999
Advisor
RAY-CHAUDHURI, DIJEN K.
Abstract
Coding theory is a branch of mathematics that has many practical applications in the transmission of information, and is used heavily by computer scientists and engineers. A code over a ring $R$ is a set of vectors of $R^n$ , and an $R$-linear code is a submodule of $R^n$. It was recently discovered that certain binary codes with good parameters can be constructed from $Z_4$-linear codes. This discovery motivated the study of codes over integer residue rings and eventually codes over Galois rings. The permutation group of a code is the group of permutations on the coordinates of a vector that preserve the code. In particular, a cyclic code is a code that is left invariant under any cyclic shift of coordinates. It is good for a code to have a large permutation group, because it is useful for decoding and determining weight distributions. This dissertation focuses on the study of cyclic and extended cyclic codes over Galois rings, particularly those of characteristic 4. Many classical results about cyclic codes over finite fields are generalized. A slightly new approach is given that characterizes cyclic codes over Galois rings, one that involves multiple defining sets and Mattson-Solomon polynomials. Using this approach, all affine-invariant extended cyclic codes over the ring $GR(4, m)$ are classified. This dissertation suggests a method of determining their complete permutation groups, and creates some extended cyclic codes with permutation groups that are larger than those of codes appearing in previous literature. Then a conclusion is made on the asymptotic behavior of these codes.