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This thesis is dedicated to the constructions of modular lattices with algebraic methods. The goal is to develop new methods as well as constructing new lattices. There are three methods considered: construction from number fields, construction from totally definite quaternion algebras over number fields and construction from linear codes via generalized Construction A. The construction of Arakelov-modular lattices, which result in modular lattices, was first introduced in [6] for ideal lattices from cyclotomic fields. We generalize this construction to other number fields and also to totally definite quaternion algebras over number fields. We give the characterization of Arakelov-modular lattices over the maximal real subfield of a cyclotomic field with prime power degree and totally real Galois fields with odd degrees. Characterizations of Arakelov-modular lattices of trace type, which are special cases of Arakelov- modular lattices, are given for quadratic fields and maximal real subfields of cyclotomic fields with non-prime power degrees. Furthermore, we give the classification of Arakelov-modular lattices of level l for l a prime over totally definite quaternion algebras with base field the field of rationals. Construction A is a well studied method to obtain lattices from codes via quotient of different rings, such as rings of integers, in which case mostly cyclotomic number fields have been considered. In this thesis, we will study Construction A over all totally real and CM fields. Using Construction A, the intersection between a lattice constructed from a linear complementary dual (LCD) code and its dual lattice is investigated. This is an attempt to find an equivalent definition to LCD codes for lattices. Several new constructions of existing extremal lattices as well as a new extremal lattice are obtained from the above mentioned methods. The mathematical concepts used in this thesis mainly involve algebraic number theory, class field theory, non commutative algebra and coding theory. Doctor of Philosophy (SPMS) |
