Topics in Integral Matrices and Abelian Group Codes

• Main Author: Rushanan, Joseph John
• Format: Others
• Language: en
• Published: 1986
• Online Access: https://thesis.library.caltech.edu/1572/1/Rushanan_rj_1986.pdf; Rushanan, Joseph John (1986) Topics in Integral Matrices and Abelian Group Codes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/CZA5-PX53. https://resolver.caltech.edu/CaltechETD:etd-05022003-113743 <https://resolver.caltech.edu/CaltechETD:etd-05022003-113743>

<p>This thesis consists of two independent chapters.</p> <p>The first chapter concerns the Smith Normal Form (SNF) over the integers ℤ of integral matrices. We consider the SNF of a matrix <i>A</i> to be the ratio of two ℤ-modules -- a finitely generated abelian group; this is called the <i>Smith group</i> of <i>A</i>. The Smith group provides a unified setting to present both new and old results. The new results concern the relationship between the eigenvalues of an integral matrix and its SNF. In particular, the multiplicities of integer eigenvalues are shown to relate to the multiplicities in the type of the Smith group. Bounds are also given for the exponent of the Smith group. In some cases, these are best possible. The old results discussed are the interlacing of the SNF in the case of augmented matrices and the symmetries of the SNF for certain combinatorial matrices. The latter results are extended to rectangular matrices. Numerous examples are given throughout, along with many conjectures based on computation.</p> <p>The second chapter generalizes the work of Pless, et al. on duadic codes and Q-codes. We take abelian group codes to be ideals in the group ring <b>F</b>[<i>G</i>], where <i>G</i> is a finite abelian group of odd order <i>n</i> and <b>F</b> is a finite field with characteristic relatively prime to <i>n</i>. We define <i>generalized Q-codes</i> from a pair of idempotents of <b>F</b>[<i>G</i>] and an automorphism of <i>G</i> which together obey two simple equations. These codes are (<i>n</i>, (<i>n</i>+1)/2) and (<i>n</i>, (<i>n</i>-1)/2) linear codes. We show that all of the properties of duadic and Q-codes generalize. In particular, we extend the results on the relationship of these codes to projective planes with regular automorphism group <i>G</i>. When <b>F</b> has characteristic 2, we give simple numerical conditions on <i>G</i> and <b>F</b> which determine when generalized Q-codes exist. We also give some techniques for constructing these codes.</p>