Theoretical Bound and Compensation for Residual Error Probability of GRAND-CRC-Based Functional Safety Communication

In modern Industrial Internet of Things (IIoT) ecosystems, functional safety protocols are increasingly utilized to transmit safety protocol data unit (SPDU). Integrating the universal guessing random additive noise decoding (GRAND) algorithm with cyclic redundancy check (CRC)-coded SPDU can minimize SPDU retransmissions. However, it introduces residual error probability (REP) degradation that requires careful consideration. Using the IEC 61784-3 Standard and CRC assumptions, we derive a closed-form REP evaluation formula specific to SPDU length and maximum error correction capability. Our analysis reveals that, under the worst industrial conditions and for an SPDU length of 128 bits, the REP performance degrades by approximately 2:2 × 10<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> times when the CRC signature length is 24 bits and up to one bit is guessing decoded. This degradation becomes more pronounced with increased error correction capability. To address this, we propose to compensate the REP degradation by adopting longer CRC signature. This paper provides a theoretical framework for adopting the GRAND algorithm in functional safety communication, setting a foundation for enhancing reliability in IIoT applications..