Recursive Decoding and Its Performance for Low-Rate Reed–Muller Codes

Abstract

Abstract-Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length and fixed order . An algorithm is designed that has complexity of order log and corrects most error patterns of weight up to (1 2 ) given that exceeds 1 2 . This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity.To evaluate decoding capability, we develop a probabilistic technique that disintegrates decoding into a sequence of recursive steps. Although dependent, subsequent outputs can be tightly evaluated under the assumption that all preceding decodings are correct. In turn, this allows us to employ second-order analysis and find the error weights for which the decoding error probability vanishes on the entire sequence of decoding steps as the code length grows.

Error exponents for two soft decision decoding algorithms of Reed-Muller codes

“…form the vectors y 0 and y 00 of length n=2: The following recursive algorithm m r is introduced in [7] and is discussed there in more detail. First, we form the vector…”

“…Recursive estimates [7] show that m r has complexity j m r j 2n min(r; m r) + n(m r + 1) 3nm=2: Our goal now is to estimate the error rate of the algorithm, which will be done in the following sections.…”

“…Let q (3) properly tracks the actual posterior probability q v i ; similarly to original identities (2). However, the other estimate y u i of (4) does not track the corresponding quantity q u i (see [7] for more details), which has a different value,…”

“…Recursive algorithms also achieve the highest error-correcting levels among all non-exponential algorithms known for RM codes. In particular, for long RM codes of any xed code rate, recursive decoding of [7] corrects most error patterns up to the decoding radius (d ln d) =2, which is twice the threshold of majority decoding.…”

“…Therefore, we establish some partial ranking of different decoding steps. This layout mostly follows [7] and [8]. Then in Section IV, we address the new AWGN setting.…”