Structure-Preserving Error-Correcting Codes for Polynomial Frames

Modern FFT/NTT analytics, coded computation, and privacy-preserving ML interface routinely move polynomial frames across NICs, storage, and accelerators. However, even rare silent data corruption (SDC) can flip a few ring coefficients and cascade through downstream arithmetic. Conventional defenses are ill-matched to current low-latency pipelines: detect-and-retransmit adds RTTs, while byte-stream ECC ignores the algebraic structure and forces format conversions. To that end, we propose a structure-preserving reliability layer that operates in the encoded data's original polynomial ring, adds a small amount of systematic redundancy, and corrects symbol errors/flagged erasures without round-trip or format changes. We construct two complementary schemes: one for odd length $N_{odd}$ via a Hensel-lifted BCH ideal with an idempotent encoder, and one for power-of-two length $N_{2^m}$ via a repeated-root negacyclic code with derivative-style decoding. In particular, to stay robust against clustered errors, a ring automorphism provides in-place interleaving to disperse bursts. Implementation wise, on four frame sizes $N\!=\!1024, 2048, 4096, 8192$, we meet a per-frame failure target of $10^{-9}$ at symbol error rates $10^{-6}\text{--}10^{-5}$ with $t\!=\!8\text{--}9$, incurring only $0.20\%\text{--}1.56\%$ overhead and tolerating $\sim\!32\text{--}72$\,B unknown-error bursts (roughly doubled when flagged as erasures) after interleaving. By aligning error correction with ring semantics, we take a practical step toward deployable robustness for polynomial-frame computations from an algebraic coding perspective.