An upper bound on the covering radius of the logarithmic lattice for cyclotomic number fields

Abstract

Recently, in coding theory and cryptography, it has been important the diversified use of lattices. One use of them is to cover a space. Each lattice has a covering radius, a number corresponding to the radius of a ball whose translations by all the points of the lattice cover efficiently the space generated by a basis of it. A way to obtain lattices algebraically is from subgroups of the multiplicative group of units of a number field via the logarithm embedding. This includes the logarithm lattice. In this work, it is presented an upper bound on the covering radius of the logarithm lattice obtained from the units of general cyclotomic number fields via the logarithm embedding, which generalizes an upper bound present in a previous work for cyclotomic number fields of prime-power indices.