Universal Communication over Modulo-additive Channels with an Individual Noise Sequence

Which communication rates can be attained over an unknown channel where the relation between the input and output can be arbitrary? A channel where the output is any arbitrary (possibly stochastic) function of the input that may vary arbitrarily in time with no a-priori model? In this paper we provide an operational definition of a “capacity” (the maximal possible rate) for such an arbitrary infinite vector channel, which is similar in spirit to the finite-state compressibility of a sequence defined by Lempel and Ziv. This capacity is the highest rate achieved by a designer that knows the particular relation that indeed exists between input and output for all times, yet is constrained to use a fixed finite-length block communication scheme (i.e., use the same scheme over each block). In the case where the relation between input and output is constrained to be “modulo additive” that is the channel generates the output sequence by adding (modulo the channel alphabet) an arbitrary individual sequence to the input sequence, this capacity is upper bounded by 1 minus the finite state compressibilty of the noise sequence, multiplied by the logarithm of the alphabet size. We present a communication scheme with feedback that attains this rate universally without prior knowledge of the noise sequence.