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**1 - 1**of**1**### DISCRETE MATHEMATICS On sublattices of the hexagonal lattice

, 1994

"... How many sublattices of index N are there in the planar hexagonal lattice? Which of them are the best from the point of view of packing density, signal-to-noise ratio, or energy? We answer the first question completely and give partial answers to the other questions. 1. Questions Let A denote the fa ..."

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How many sublattices of index N are there in the planar hexagonal lattice? Which of them are the best from the point of view of packing density, signal-to-noise ratio, or energy? We answer the first question completely and give partial answers to the other questions. 1. Questions Let A denote the familiar hexagonal lattice shown in Fig. 1. We shall investigate four questions arising from digital communications, especially cellular radio: (Q1) How many sublattices does A have of index N? (Q2) Which sublattice F of index N has the greatest minimal norm/ ~ (defined in (1))? (Q3) Which sublattice has the highest signal-to-noise ratio S (defined in (2))? (Q4) Which sublattice has a fundamental region of minimal energy M (defined in (4))? Applications of these results will be discussed in a separate paper [2]. 2. Comments We assume throughout that A is the hexagonal lattice defined in Fig. 1, having Gram matrix ( _ I/2- ~/2) and determinant . The index of a sublattice M in a lattice L is the order of the quotient group L/M. (Q1) Two sublattices of A are called equivalent if they differ only by a rotation and possibly a reflection that sends A to itself. We wish to find the number of inequivalent sublattices of a given index. Surprisingly, the solution does not seem to be given in the * Corresponding author.