IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 3, MARCH 2007 1215 Invariance Properties of Binary Linear Codes Over a Memoryless Channel With Discrete Input Ali Abedi, Member, IEEE, and Amir K. Khandani, Member, IEEE Abstract—This work studies certain properties of the probability density function (pdf) of the bit log-likelihood ratio (LLR) for binary linear block codes over a memoryless channel with discrete input and discrete or contin- uous output. We prove that under a set of mild conditions, the pdf of the bit LLR of a specific bit position is independent of the transmitted codeword. It is also shown that the pdf of a given bit LLR when the corresponding bit takes the values of zero and one are symmetric with respect to each other (reflection of one another with respect to the vertical axis). For the case of channels with binary input, a sufficient condition for two bit positions to have the same pdf is presented. Index Terms—Bit decoding, block codes, geometrically uniform, log-like- lihood ratio (LLR), probability density function (pdf), regular channel, symmetric channel. I. INTRODUCTION The use of linear binary codes to label the points of signal constella- tions has been the subject of many investigations. The distance invari- ance property in such schemes guarantees that the (frame) error proba- bility under maximum-likelihood (ML) decoding is independent of the transmitted codeword. Another class of decoding algorithms, known as bit decoding, directly compute the probability of the individual bits. Recently, bit decoding algorithms have received increasing attention as they deliver bit reliability information which has been used in many ap- plications including turbo decoding. Asymptotic performance analysis of bit decoding has received attention in [2]–[4] where an ensemble of coset codes is used to handle the complications due to the lack of dis- tance invariance. It would be of interest to study such distance invari- ance properties for finite-block lengths which is the motivation behind current work. In [5], it is shown that for a binary-input, output-sym- metric channel (as defined in [5]), the conditional probability of error is independent of the transmitted codeword. In multilevel coding (MLC), each bit in the constellation label is pro- tected by a different binary code. Accordingly, in multistage decoding (MSD), these component codes are successively decoded. It is known that the combination of MLC and MSD can achieve capacity. With this viewpoint, the transmission of multiple label bits is separated into the transmission over binary-input component channels [6]. Reference [4] argues that the application of bit decoding in MLC/MSD is com- plicated as the analysis of the all-zero codeword alone does not nec- essarily suffice. Refernece [4] also shows that for a Gray labeled am- plitude shift keying (ASK) constellation, the binary-input component channels do not posses such invariance property. This motives the au- thors to use low-density parity-check (LDPC) coset codes to overcome Manuscript received May 9, 2003; revised October 22, 2006. This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) and by Communications and Information Technology Ontario (CITO). The material in this correspondence was presented in part at the Conference on Information Sciences and Systems, Princeton, NJ, March 2002. This work is a continuation of earlier work in which the case of AWGN channel with binary phase shift keying (BPSK) modulation is considered. A. Abedi is with the Department of Electrical and Computer Engineering, University of Maine, Orono ME 04469 USA (e-mail: abedi@eece.maine.edu). A. K. Khandani is with the Coding and Signal Transmission Laboratory, De- partment of Electrical and Computer Engineering, University of Waterloo, Wa- terloo, ON N2L 3G1, Canada (e-mail: khandani@cst.uwaterloo.ca). Communicated by R. Koetter, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2006.890719 this problem. We will later show that for Natural labeling 1 and ASK modulation, these binary-input component channels indeed satisfy the derived necessary conditions, and consequently, an all-zero codeword can be used to analyze their performance. This correspondence is organized as follows. In Section II, the model used to analyze the problem is presented. All notations and assumptions are given in this section. Some theorems are proved on bit decoding al- gorithms in Section III. Throughout the correspondence, higher indices represent the elements of the sets (for example, different codewords), vectors are shown in bold face, and lower indices represent subsequent components of a sequence (for example, sequence of bits within a code- word). II. MODELING Assume a binary linear code with codewords of length is given. Notation is used to refer to the th codeword and its components. We partition the code into a subcode and its coset according to the value of the th bit position of its codewords, i.e., (1) We denote bit-wise binary addition of two codewords on the code book as . Note that the subcode is closed under binary addition. Each codeword will be partitioned into blocks of bits, assuming , to be transmitted over a channel with a discrete input al- phabet set composed of elements. Notation , , , is used for these blocks, which will be called -blocks hereafter. For example, codeword is composed of . We assume that there exists a one-to —one correspondence between the possible -blocks and the input symbols of the channel. The set of -blocks referred as forms a group under binary addition. The channel has discrete inputs and discrete or continuous output . In some cases, to clarify the underlying dependencies, we use the notation to explicitly show that the channel output is a probabilistic function of the channel input. The channel is described by the transition probability , or simply by . For channels with a discrete output and stands for the probability mass function (pmf). For channels with con- tinuous output where is the set of real numbers, is the size of vector and denotes the probability density function (pdf). We consider two classes of channels: i) channels with a geometrical rep- resentation, and ii) channels without a geometrical representation. For channels without a geometrical representation, “channel input” always refers to the corresponding binary label ( -block). For channels with a geometrical representation, depending on the context, the “channel input” may refer to the corresponding binary label or to the actual signal point. In all cases, the channel is assumed to be memoryless. Consider the situation of sending a codeword through the channel. Each -block, , , will be trans- mitted and a symbol , , will be received at the channel output. The log-likelihood- ratio (LLR) of the th bit position (assuming is transmitted) is given by (2) where is the value of the th bit in and stands for natural logarithm. Assuming (3) 1 Note that the channel capacity is not affected by the method of labeling. 0018-9448/$25.00 © 2007 IEEE