On the Error Rate of Binary BCH Codes under Error-and-erasure Decoding

Determining the exact decoding error probability of linear block codes is an interesting problem. For binary BCH codes, McEliece derived methods to estimate the error probability of a simple bounded distance decoding (BDD) for BCH codes. However, BDD falls short in many applications. In this work, we consider error-and-erasure decoding and its improved variants. We derive closed-form expressions for their error probabilities and validate them through simulations. Then, we illustrate their use in assessing concatenated coding schemes.

💡 Research Summary

This paper addresses the long‑standing problem of obtaining exact decoding error probabilities for binary BCH codes when soft‑decision information is exploited through error‑and‑erasure (EaE) decoding. While McEliece’s classic work provides analytical tools for simple bounded‑distance decoding (BDD), BDD alone is insufficient for many modern applications that demand higher coding gains. The authors therefore focus on a two‑step EaE decoder (EaED) that augments BDD with a single reliability class of erasures, and they derive closed‑form expressions for its decoding transition probability (DTP) P(R = r | U = u, E = e), where U, E, and R denote the numbers of input errors, input erasures, and residual errors after decoding, respectively.

The analysis begins with a concise review of BCH code parameters, weight enumerators, and the three channel models considered: the binary symmetric channel (BSC), the EaE channel, and the binary‑input AWGN channel, the latter being reduced to an EaE channel via a threshold T. For BDD, the authors re‑derive the DTP (Theorem 1) by counting, for each possible pair (a,b) of flipped correct bits and corrected erroneous bits, the number of received patterns that lead to a particular residual weight r. This yields exact expressions for the probabilities of success, failure, and miscorrection as functions of the code’s weight distribution.

The core contribution is the derivation of the DTP for EaED (Theorem 2). EaED generates two random filling patterns for the erasures, runs BDD on each filled word, and selects the output with the smaller Hamming distance to the original erasure pattern (or randomly if tied). The authors decompose the overall transition probability into a sum over the possible number e₁ of ones in the first filling pattern, weighting each term by the binomial probability of e₁ and the corresponding BDD DTP conditioned on the effective error count u + e₁. Closed‑form expressions are provided for the regimes where 2u + e < d_min (guaranteed success) and for the intermediate regimes where e₁ lies within