Distance Bounds for an Ensemble of LDPC Convolutional Codes

Abstract

An ensemble of <emphasis xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex>$(J,K)$</tex> </formula></emphasis> -regular low-density parity- check (LDPC) convolutional codes is introduced and existence-type lower bounds on the minimum distance <emphasis xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex>$d _ {\rm L}$</tex></formula></emphasis> of code segments of finite length <emphasis xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex>$L$</tex> </formula></emphasis> and on the free distance <emphasis xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"> <tex>$d _{\rm free}$</tex></formula></emphasis> are derived. For sufficiently large constraint lengths <emphasis xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex>$\nu$</tex> </formula></emphasis> , the distances are shown to grow linearly with <emphasis xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex>$\nu$</tex></formula></emphasis> and the ratio <emphasis xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex>$d_ {\rm L}/\nu$</tex></formula></emphasis> approaches the ratio <emphasis xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex>$d _{ {\rm free}}/\nu$</tex> </formula></emphasis> for large <emphasis xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex>$L$</tex> </formula></emphasis> . Moreover, the ratio of free distance to constraint length is several times larger than the ratio of minimum distance to block length for Gallager's ensemble of (J,K)-regular LDPC block codes.