I find the below expression a lot in math, statistics and information theory m ∈ [1 : 2^nR]
expression is a text like the following:
a randomized encoder that generates a codeword X^n(m), m ∈ [1 : 2nR], according to a conditional pmf p(x^n|m)
Could some one explain it for me ?
The notation $(M,n)=(2^{nR},n)$ usually represents a binary code (not necessarily linear) with codewords of length $n$ and rate $R\in(0,1)$ which gives (up to a floor function) $2^{nR}$ codewords. Since we're interested in asymptotics we don't care about the floor function.
The codewords can be arranged so that they correspond to one of $2^{nR}$ messages with index $\{1,2,\ldots,2^{nR}\}$