3590 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 9, SEPTEMBER 2013 Improved Low-Density Parity Check Accumulate (LDPCA) Codes Chao Yu and Gaurav Sharma, Fellow, IEEE Abstract—We present improved constructions for Low-Density Parity-Check Accumulate (LDPCA) codes, which are rate- adaptive codes commonly used for distributed source coding (DSC) applications. Our proposed constructions mirror the traditional LDPCA approach; higher rate codes are obtained by splitting the check nodes in the decoding graph of lower rate codes, beginning with a lowest rate mother code. In a departure from the uniform splitting strategy adopted by prior LDPCA codes, however, the proposed constructions introduce non-uniform splitting of the check nodes at higher rates. Codes are designed by a global minimization of the average rate gap between the code operating rates and the corresponding theoretical lower bounds evaluated by density-evolution. In the process of formulating the design framework, the paper also contributes a formal definition of LDPCA codes. Performance improvements provided by the proposed non- uniform splitting strategy over the conventional uniform splitting approach used in prior work are substantiated via density evolution based analysis and DSC codec simulations. Optimized designs for our proposed constructions yield codes with a lower average rate gap than conventional designs and alleviate the trade-off between the performance at different rates inherent in conventional designs. A software implementation is provided for the codec developed. Index Terms—LDPCA codes, LDPC codes, distributed source coding, code design. I. I NTRODUCTION E MERGING applications of battery-powered mobile de- vices and sensor networks have motivated the develop- ment of DSC techniques that exploit inter-dependency be- tween sensor data at different nodes to reduce communication requirements, thereby improving energy-efficiency and oper- ating times [1]–[4]. Although information theoretic results for DSC appeared nearly 40 years ago [5], [6], practical code constructions that achieve close to promised performance have only been devel- oped in the past decade. Most constructions, and the discussion in this paper, restrict attention to the binary-input memoryless side-informed coding scenario: a block x =[x 1 ,x 2 ,...x L ] T of L independent bits available at one terminal, the encoder, Manuscript received November 20, 2012; revised April 7, June 16, and August 1, 2013. The editor coordinating the review of this paper and approving it for publication was D. Declercq. This work was supported in part by the National Science Foundation under grant number ECS-0428157. C. Yu is with the Department of Electrical and Computer Engineer- ing, University of Rochester, Rochester, NY 14627-0126 USA (e-mail: chyu@ece.rochester.edu). G. Sharma is with the Department of Electrical and Computer Engineering, the Department of Biostatistics and Computational Biology, and the Depart- ment of Oncology, University of Rochester, NY 14627-0126 USA (e-mail: gaurav.sharma@rochester.edu). Digital Object Identifier 10.1109/TCOMM.2013.13.120892 needs to be communicated to a second terminal, the decoder, that has a priori information consisting of a corresponding block of side information y =[y 1 ,y 2 ,...y L ] T , where 1 the elements of Y are independent and for each 1 ≤ i ≤ L, Y i is (potentially) correlated with X i and independent of X ∼i def = [X 1 ,X 2 ,...,X i-1 ,X i+1 ,X i+2 ,...X L ] T . The en- coder generates a vector of j bits ˜ p = [p 1 ,p 2 ,...p j ] T , which is (noiselessly) sent to the decoder and using which the decoder must recover x. The objective is to minimize the rate r =(j/L) required per encoded bit by exploiting, in the decoding process, the side information y that is available at the decoder but not at the encoder. Practical side-informed coding methods leverage channel coding techniques: y is interpreted as the noisy output of a virtual channel with input x and error correction decoding is used to recover x at the decoder. DSC constructions have been developed based on trellis codes [7], Turbo codes [8], and Low-Density Parity-Check (LDPC) codes [9]. Information theoretic results for side-informed coding imply that the conditional entropy per symbol H (X|Y)/L is the minimum required rate (on average). In the memoryless setting where the pairs of random variables {(X i ,Y i )} L i=1 are drawn independently from the same joint distribution p XY (x, y), the minimum required rate becomes the conditional entropy H (X |Y ). To simplify practical implementations and handle the vary- ing correlation (between X and Y ) encountered in DSC ap- plications, rate-adaptive DSC techniques have been developed based on punctured Turbo codes [8], [10], [11], or using an LDPC-Accumulate (LDPCA) construction [12] that builds on LDPC codes. LDPCA codes, in particular, offer superior performance for side-informed source coding and have been adopted for several DSC applications such as distributed video coding [13]–[16] and image authentication [17]. Previously reported LDPCA codes follow the framework introduced in [12], [18]. Rate adaptivity is obtained by begin- ning with an LDPC code at the lowest rate from which higher rate codes are obtained by uniformly splitting a fraction of the check nodes in the decoding graph to define additional check nodes, which then define the additional bits to be communicated from the encoder to the decoder for the higher operating rate. Within this framework, optimized designs were developed in [19]. An examination of the performance of these previously reported codes (See results in Section IV), reveals 1 We adopt the standard notational convention where upper case letters represent the random variables corresponding to their lower case counterparts, both being bold when these are vectors. Elements of a vector are represented by corresponding non-bold subscripted variables. The notation H for denoting parity check matrices (with various superscripts) is the exception to the notational convention. 0090-6778/13$31.00 c  2013 IEEE