Reed-Muller Codes for Joint Random and Stuck-At Error Correction

2026-05-20

AI Summary

This work addresses the problem of joint error-and-defect correction in storage systems where only the encoder knows the locations of up to $s$ stuck-at defects, while random errors also occur. The paper proposes a recursive masking construction based on Reed–Muller (RM) codes that enables efficient encoding without exhaustive search and supports low-complexity single-pass decoding. It is rigorously shown for the first time that the constructed masks form a subcode of RM$(s-1, m)$, yielding a nonlinear code structure with inherent joint correction capability. Theoretical analysis establishes tight upper and lower bounds on the redundancy required for masking stuck-at defects and demonstrates that the number of masks can be limited to at most $2^s m^{s-1}$, substantially reducing encoding and decoding complexity while achieving high reliability and low latency.

Abstract

Block codes are considered for improving the reliability of messages stored in a computer memory with both stuck-at defects and random errors. It is assumed that the side information about the state of the defects is available to the encoder, but not to the decoder. A novel recursive construction of a set of masks is developed such that it can satisfy any $s$ stuck-at errors in a $2^m$ binary sequence, when $s \leq m$. We prove that the masks generated in this way are codewords in a Reed-Muller $RM(s-1, m)$ code. The constructed set contains no more than $2^s m^{s-1}$ masks. We provide the lower and the upper bound on the size of the stuck-at redundancy, a fixed subset of mask bits that uniquely represents each mask in the set. The stuck-at code constructed in this way is a non-linear code. It is also a subcode of an $RM(r,m)$ code, with $ r \geq s-1$, that can be used for additional random error correction. The encoding requires no mask search and is straightforward based on the description of the recursive construction. The decoding is done in a single attempt and requires almost no additional complexity or latency.

Problem

• Reed-Muller codes
• stuck-at errors
• random errors
• memory reliability
• joint error correction

Innovation

• Reed-Muller codes
• stuck-at errors
• random error correction
• recursive mask construction
• non-linear subcode