Upper Bound on the Bit Error Probability of Systematic Binary Linear Codes via Their Weight Spectra

Algorithm 1

A list decoding algorithm.

(1)	We denote
	in which
	and
	Make hard decisions on the information part of the received vector , resulting in a vector of length k. Then, the channel becomes a memoryless binary symmetric channel (BSC) with cross probability
(2)	List all sequences of length k within the Hamming sphere with center at of radius , where is a positive integer. The resulting list is denoted as .
(3)	Encode each sequence in by the encoding algorithm of the systematic code, resulting in a list of code words, denoted as .
(4)	Find the codeword that is closest to . Output the information part of as the decoding result.