A coding theorem on generalized \(R\)-norm entropy (Q2761621)

The generalized \(R\)-norm entropy of a distribution \(P= (p_1,\dots, p_N)\), \(p_i> 0\), \(\sum p_i= 1\), is defined by NEWLINE\[NEWLINEH_{{R\over 2-\beta}}(P)= {R\over R+\beta-2} \Biggl[1- (\sum p^{{R\over 2-\beta}}_i)^{{2-\beta\over R}}\Biggr],\text{ where }0< \beta< 2,\;R+\beta\neq 2;NEWLINE\]NEWLINE if \(\beta= 1\) and \(R\to 1\) this approaches the Shannon entropy. Applications of \(R\)-norm entropy in coding theory are studied; a theorem on bounds of generalized mean codeword length in terms of the \(R\)-norm entropy is proven.