Finite Dimensional Lattice Codes With Self Error-Detection and Retry Decoding

Lattice codes with the optimal decoding coefficient are capacity-achieving when dimension <inline-formula> <tex-math notation="LaTeX">$N \rightarrow \infty $ </tex-math></inline-formula>. In communications systems, finite dimensional lattice codes are needed, where the optimal decoding coefficient may still fail decoding even when <inline-formula> <tex-math notation="LaTeX">$R\lt C$ </tex-math></inline-formula>. This paper presents a new retry decoding scheme for finite dimensional lattice-based transmissions. When decoding errors are detected, the receiver is allowed to adjust the value of decoding coefficients and retry decoding, instead of requesting a retransmission immediately which causes high latency. This scheme is considered for both point-to-point single user transmission and compute-forward (CF) relaying with power unconstrained relays, by which a lower word error rate (WER) is achieved than conventional one-shot decoding with optimal coefficients. A lattice/lattice code construction, called CRC-embedded lattice/lattice code, is presented to provide physical layer error detection to enable retry decoding. For CF relaying, a lattice code design is given so that the decoder is able to detect errors from CF linear combinations without requiring individual users’ messages. The numerical results show gains of up to 1.31 dB and 1.08 dB at error probability <inline-formula> <tex-math notation="LaTeX">$10^{-5}$ </tex-math></inline-formula> for a 2-user CF relay using 128- and 256-dimensional lattice codes with optimized CRC length and 2 decoding trials in total.