Controlled Doping via High- Order Rootchecks in Graph Codes

Code construction for a data transmission channel with a limited number of degrees of freedom is a big challenge . For codes on graphs, a solution based on rootchecks has been proposed [1]. In this paper, we start by establishing a new pr oof for full diversity equivalence between the block-fading and the block-erasure channels. Then, we show how doping [2] can be made via high-order rootchecks. This controlled doping wil l help in increasing the diversity order of parity bits while boosting the coding gain of information bits in full-diversity root-LDP C codes. New ensembles of root-LDPC codes are designed such that 100% of parity bits achieve full diversity. I. I NTRODUCTION Consider a data transmission channel with a limited number of degrees of freedom. A transmitted codeword can be divided into B blocks where each block is observing a different channel state. For a given signal-to-noise ratio, it is not possi ble to guarantee a vanishing error probability at asymptotic co de length; Shannon capacity of such a non-ergodic channel is zero [3]. We restrict our study to the worst case B = 2 channel states. As an example, the block-fading channel with binary input and additive white Gaussian noise is defined by yn = α1 · xn + ηn, for n = 1 . . . N 2 , (1) yn = α2 · xn + ηn, for n = N