Codes inside shells of low-dimensional integer lattices.

<p>Let <em>S(n,t,k)</em> be the maximum size of a code containing only vectors of the <em>k</em>th shell of the <em>n</em>-dimensional integer lattice such that the inner product between distinct vectors does not exceed <em>t</em>. The file A.pdf contains examples of codes leading to lower bounds for <em>S(n,t,k)</em>, one example for the each lower bound. By <em>(n,k,t,N)</em> we denote a code containing <em>N</em> vectors of the <em>k</em>th shell of the <em>n</em>-dimensional integer lattice such that the inner product between any two different vectors is at most <em>t</em>. The following format to describe example codes is used:</p> <p>&mdash; If a code is constructed using a signed permutation group, then a list containing generators of the corresponding group <em>G</em> is given first followed by a list containing used block system and <em>G</em>-orbits forming the code. Each orbit is represented by a list containing orbit representative and an orbit size (in that order). Finally, a code may also contain explicitly added vectors. In this case corresponding list contains two items: a number of explicitly added vectors and an another list containing the vectors.</p> <p>&mdash; If a permutation group is used, then the format stays the same except the list containing block system is not present.</p> <p>&mdash; Some codes are constructed by scaling spherical codes obtained by <em>Y_n</em> constructions described in [1]. If one part of an example code is constructed by using a signed permutation group, then <em>Y_n</em>-constructed part also utilizes the same block system to describe components of codevectors.</p> <p>&mdash; Notation <em>W(n,w)</em> is used to denote a weighing matrix of order <em>n</em> and weight <em>w</em> used to map vectors of a <em>(n,k,t,N)</em>-code into <em>kw</em>th shell of <em>n</em>-dimensional integer lattice to produce a <em>(n,kw,tw,N)</em>-code.</p> <p>The file B.pdf lists some examples of large spherical codes containing vectors from multiple shells. To obtain spherical codes scale given vectors appropiately. If only parameters of a code are given without the construction, then this code is one of the examples given in A.pdf.</p> <p>[1] Ericsson, T.; Zinoviev, V.: Codes on Euclidean Spheres. Elsevier, Amsterdam, 2001.</p>