Improved tangential sphere bound on the ML decoding error probability of linear binary block codes in AWGN and block fading channels

Abstract

Recently, the added-hyperplane (AHP) bound was proposed on the foundation of the tangential sphere bound (TSB) of Poltyrev. AHP utilises a Bonferroni-type inequality (known as the Hunter bound) together with the Gallager first bounding technique (GFBT) and is tighter than TSB; however, it suffers from a performance-degrading overhead. Another inequality from the Hunter-bound family is applied to the GFBT and a novel technique has been proposed to waive the need for global geometrical properties of the code, removing the aforementioned overhead. Also, a star-structured graph is proposed as the corresponding spanning tree for the Hunter bound. The improved tangential sphere bound (ITSB) is tighter than TSB and AHP and does not impose any overhead or extra optimisation. ITSB is thus the tightest upper bound on the performance of linear binary block codes over AWGN channel. ITSB is then applied to different block (slow) fading channels as well as low-density parity-check codes.