#### Log-domain decoding of quantum LDPC codes over binary finite fields

##### Kao-Yueh Kuo, Ching-Yi Lai

A stabilizer code over GF($q=2^l$) corresponds to a classical additive code over GF($q^2 = 2^{2l}$) that is self-orthogonal with respect to a binary symplectic inner product. We study the decoding of quantum low-density parity-check (LDPC) codes over finite fields GF($q=2^l$) for $l\ge 1$ by the sum-product algorithm (SPA), also known as belief propagation (BP). Previously, all the BP decoding of quantum codes are studied in linear domain. In this paper, we propose a BP decoding algorithm for quantum codes over GF($2^l$) in log domain by passing scalar messages derived from log-likelihood ratios (LLRs) of the channel statistics. Techniques such as message normalization or offset for improving BP performance can be naturally applied in this algorithm. Moreover, the implementation cost of this LLR-BP is relatively small compared to the linear-domain BP, which is of the same case as in the classical coding research. Several computer simulations are provided to demonstrate these advantages.

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