A note on the Shannon capacity of run-length-limited codes (Corresp.)

Abstract

It is proven that 100-percent efficient fixed-rate codes for run-length-limited (RLL) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(d,k)</tex> and RLL charge-constrained <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(d, k; c)</tex> channels are possible in only two eases, namely <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(d,k; c)=(0,1;1)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1,3;3)</tex> . Specifically, the binary Shannon capacity of RLL <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(d, k)</tex> constrained systems is shown to be irrational for all values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(d, k),0 \leq d &lt; k</tex> . For RLL charge-constrained systems with parameters <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(d, k;c)</tex> , the binary capacity is irrational for all values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(d, k; c),0 \leq d &lt; k,2c \geq k + 1</tex> , except <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(0,1; 1)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1,3;3)</tex> , which both have binary capacity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</tex> .