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In this paper, we investigate the error floors of non-binary low-density parity-check (LDPC) codes transmitted over the memoryless binary-input output-symmetric (MBIOS) channels. We provide a necessary and sufficient condition for successful decoding of zigzag cycle codes over the MBIOS channel by the BP decoder. We consider an expurgated ensemble of nonbinary LDPC codes by using this condition, and hence exhibit lower error floors. to analyze and to lower the decoding error rate in the error floor by using the above condition. Finally, we show lower bounds of the error floors for the expurgated LDPC code ensembles over the MBIOS channel.

SUMMARYWe propose a digital image watermarking method satisfying information hiding criteria (IHC) for robustness against JPEG compression, cropping, scaling, and rotation. When a stego-image is cropped, the marking positions of watermarks are unclear. To detect the position in a cropped stego-image, a marker or synchronization code is embedded with the watermarks in a lattice pattern. Attacks by JPEG compression, scaling, and rotation cause errors in extracted watermarks. Against such errors, the same watermarks are repeatedly embedded in several areas. The number of errors in the extracted watermarks can be reduced by using a weighted majority voting (WMV) algorithm. To correct residual errors in output of the WMV algorithm, we use a high-performance error-correcting code: a low-density parity-check (LDPC) code constructed by progressive edgegrowth (PEG). In computer simulations using the IHC ver. 4 the proposed method could a bit error rate of 0, the average PSNR was 41.136 dB, and the computational time for synchronization recovery was less than 10 seconds. The proposed method can thus provide high image quality and fast synchronization recovery.

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In this paper, we analyze (2, k)-regular non-binary low-density parity-check codes over the binary erasure channels. We propose a method to improve the error floors by optimizing labels in zigzag cycles in the Tanner graph. We analyze the error floors for codes designed by the proposed optimization method and show that the error floors are decreasing in the size of Galois field.