Communication system and methods using very large multiple-in multiple-out (MIMO) antenna systems with extremely large class of fast unitary transformations

A method, comprising:

• generating, via a processor associated with a first communication device, a plurality of symbols based on an incoming data;
• applying, via the processor, a permutation to each symbol from the plurality of symbols, to produce a plurality of permuted symbols;
• transforming, via the processor, each permuted symbol from the plurality of permuted symbols based on at least one primitive transformation matrix, to produce a plurality of transformed symbols;
• applying, to each transformed symbol from the plurality of transformed symbols and via the processor, a precode matrix selected from a codebook of transformation matrices, the codebook of transformation matrices accessible to the first communication device and to a second communication device different from the first communication device, to produce a plurality of precoded symbols;
• and
• sending, via the processor and to the second communication device, a signal representing a plurality of signals, each signal from the plurality of signals representing a precoded symbol from the plurality of precoded symbols, each signal from the plurality of signals transmitted using a unique antenna from a plurality of antennas of the first communication device.

A non-transitory, processor-readable medium storing instructions that, when executed by a processor that is associated with a first communication device, cause the processor to:

• generate a plurality of symbols based on an incoming data;
• apply a permutation to each symbol from the plurality of symbols, to produce a plurality of permuted symbols;
• transform each permuted symbol from the plurality of permuted symbols based on at least one primitive transformation matrix, to produce a plurality of transformed symbols;
• apply, to each transformed symbol from the plurality of transformed symbols, a precode matrix selected from a codebook of transformation matrices, the codebook of transformation matrices accessible to the first communication device and to a second communication device different from the first communication device, to produce a plurality of precoded symbols;
• and
• send a signal to cause transmission, to the second communication device, of a plurality of signals, each signal from the plurality of signals representing a precoded symbol from the plurality of precoded symbols, each signal from the plurality of signals transmitted using a unique antenna from a plurality of antennas of the first communication device.