Grassmannian codes from multiple families of mutually unbiased bases

We explore the underlying algebraic structure of Mutually Unbiased Bases (MUBs), and their application to code design. Columns in MUBs have inner products with absolute values less or equal to 1/√N. MUBs provide a systematic way of generating optimal codebooks for various coding and precoding applications. A maximal set of MUBs (MaxMUBs) in N = 2^m dimensions, with m Z, can produce codebooks of QPSK lines with good distance properties and alphabets which limit processing complexity. We expand the construction by identifying that in N = 2^m dimensions there exists N^(m-1)/2 families of MUB, each with N matrices. Inner products of columns of these matrices are less or equal to 1/√2. As an example, we construct Grassmannian line codes from the columns of these matrices. Then decoding or encoding these codebooks can be performed without multiplications, and with a number of additions that scales linearly with the number of codewords, irrespectively of the dimension.