Communication systems are described that use geometrically shaped constellations that have increased capacity compared to conventional constellations operating within a similar SNR band. In several embodiments, the geometrically shaped is optimized based upon a capacity measure such as parallel decoding capacity or joint capacity. In many embodiments, a capacity optimized geometrically shaped 
constellation can be used to replace a conventional 
constellation as part of a 
firmware upgrade to transmitters and receivers within a communication 
system. In a number of embodiments, the geometrically shaped 
constellation is optimized for an 
Additive White Gaussian Noise channel or a 
fading channel. In numerous embodiments, the communication uses adaptive rate encoding and the location of points within the geometrically shaped constellation changes as the 
code rate changes. One embodiment of the invention includes a 
transmitter configured to transmit signals to a 
receiver via a 
communication channel, wherein the 
transmitter, includes a coder configured to receive user bits and output encoded bits at an expanded output encoded 
bit rate, a mapper configured to map encoded bits to symbols in a symbol constellation, a modulator configured to generate a 
signal for transmission via the 
communication channel using symbols generated by the mapper. In addition, the 
receiver includes a demodulator configured to demodulate the received 
signal via the 
communication channel, a demapper configured to estimate likelihoods from the demodulated 
signal, a decoder that is configured to estimate decoded bits from the likelihoods generated by the demapper. Furthermore, the symbol constellation is a capacity optimized geometrically spaced symbol constellation that provides a given capacity at a reduced signal-to-
noise ratio compared to a signal constellation that maximizes dmin.