Simultaneous Rational Function Codes: Improved Analysis Beyond Half the Minimum Distance with Multiplicities and Poles

Abstract: In this paper, we extend the work of Abbondati et al. (2024) on decoding simultaneous rational function codes by addressing two important scenarios: multiplicities and poles (zeros of denominators). First, we generalize previous results to rational codes with multiplicities by considering evaluations with multi-precision. Then, using the hybrid model from Guerrini et al. (2023), we extend our approach to vectors of rational functions that may present poles. Our contributions include: a rigorous analysis of the decoding algorithm's failure probability that generalizes and improves several previous results, an extension to a hybrid model handling situations where not all errors can be assumed random, and a new improved analysis in the more general context handling poles within multiplicities. The theoretical results provide a comprehensive probabilistic analysis of reconstruction failure in these more complex scenarios, advancing the state of the art in error correction for rational function codes.

• The paper presents new decoding algorithms for interleaved rational function codes that extend beyond half the minimum distance using multiplicities and poles.
• It develops tight, exponentially decaying failure probability bounds under both random and hybrid error models by leveraging interleaving structure.
• The analysis broadens the applicability of rational function codes to distributed computing and fault-tolerant systems by addressing evaluation multiplicities and poles consistently.

Robust Decoding Beyond Unique Decoding Radius for Simultaneous Rational Function Codes with Multiplicities and Poles

Overview

The paper "Simultaneous Rational Function Codes: Improved Analysis Beyond Half the Minimum Distance with Multiplicities and Poles" (2508.05284) presents an extensive theoretical analysis and new algorithmic results for decoding interleaved rational function codes in the presence of both evaluation multiplicities and poles. The authors generalize and strengthen previous bounds on decoding failure probability when reconstructing vectors of rational functions under adversarial and random error models, significantly extending the unique decoding regime.

Simultaneous Rational Function Codes with Multiplicities and Poles

The focus is on simultaneous reconstruction of vectors of rational functions over finite fields, with modular reductions at multiple evaluation points, each potentially with a specified multiplicity (Hermite interpolation). The core coding-theoretic object is the Simultaneous Rational Function (SRF) code, where codewords consist of evaluations (with multiplicities) of $\ell$ rational functions sharing a common denominator. Unlike standard Reed-Solomon-based codes, these codes must account for the possibility that some evaluation points are poles of the denominator, resulting in undefined modular reductions at those points.

Building on previous work (notably [abbondati2024decoding], [guerrini2023simultaneous]), the current study systematically incorporates multiplicities (enabling Hermite-like superpositions) and the presence of poles into the analysis of decoding algorithms. The main aim is to understand, and rigorously bound, the probability of decoding failure when correcting errors beyond half the (weighted) minimum distance (“list-decoding” regimes), both for error models where errors are uniformly random and those with both random and fixed error subsets (hybrid or semi-adversarial errors).

Decoding Algorithms and Key Equations

The decoding procedure generalizes the classic error-and-erasure paradigm for Reed-Solomon and rational function codes. The algorithm solves for vectors $(\varphi, \psi_1, ..., \psi_\ell)$ of polynomials satisfying modular key equations derived from the received word, the code structure, and the CRT. The solution vector is selected to have minimal degree, and is post-processed to extract the reconstructed rational functions.

A central insight is that, when interleaving ($\ell > 1$), the shared algebraic structure of the code can be leveraged to decode reliably even beyond the traditional unique decoding radius. The analysis depends on the structure of the error locator polynomials and the effective (weighted/multiplicity) Hamming distance in this more general setting.

Main Theoretical Results

1. Failure Probability Bounds:

For both random and hybrid error models, the authors provide exponentially decaying upper bounds (as a function of the excess error weight over the unique decoding radius) on the probability of decoding failure. For a fixed code and error parameters, if $\ell$ is the interleaving multiplicity and $t$ the maximal error weight considered, then the probability of decoding failure is bounded by a term of order $q^{-(\ell+1)(t-t^*)}$ for $t^*$ below the unique decoding radius, improving substantially over previous linear bounds (i.e., bounds of type $O(e/q)$).

2. Extension to Hybrid and Semi-Adversarial Models:

The analysis extends to error models where a subset of errors is fixed (adversarial) and the remainder is random, matching the so-called hybrid error model of [guerrini2023simultaneous] and the semi-adversarial model introduced in [brakensiek2025unique]. The result shows that even when correction exceeds the unique decoding guarantee, the failure probability remains exponentially small in the size of the random error set, and the correction capacity is essentially characterized as a function of the interleaving level and the distribution of random versus adversarial error support.

3. Decoding with Poles and Multiplicities:

A major technical contribution is the extension of the above analysis to the most general setting—arbitrary multiplicities and the presence of poles (zeros of the denominator), which requires careful adaptation of the modular reduction and error analysis framework. Using a multi-precision (valued) encoding, the authors prove that their bounds hold in this setting, and demonstrate that the minimal distance and decoding radii are preserved, modulo mild technical constraints on the code parameters.

4. Minimal Distance Analysis:

The paper provides new lower (and, under combinatorial constraints, tight) bounds on the minimal distance of SRF codes with both multiplicities and poles, showing that in many natural cases the “unique decoding” radius is indeed half the (weighted) distance.

The key numerical insight is that, with high interleaving ($\ell$ large), the codes can be decoded up to almost the full minimum distance with exponentially small probability of failure, extending the classic capacity of RS and Hermite codes.

Strong and Contradictory Claims

• The new failure probability bounds are exponentially tight in the interleaving parameter $\ell$, and strictly improve all previously published bounds for these codes in the presence of multiplicity and poles.
• The analysis shows that, for sufficiently large $\ell$, the zone between the unique decoding radius and the minimum distance can be almost entirely covered, except with negligible failure probability.
• The decoding algorithm does not require “multiplicity balancing” as in [guerrini2023simultaneous], and is thus insensitive to the distribution of error multiplicities—removing a key limitation of previous work.

Practical and Theoretical Implications

The results have significant implications for algebraic coding in computational and distributed settings, especially for fault-tolerant algorithms using rational function evaluation/interpolation, modular CRT-based methods, and LCC applications (e.g., verifiable distributed linear algebra, coded computation).

• Robustness in Distributed Computation: The ability to correct substantially more errors (especially random or hybrid sets) enables more efficient and robust distributed computation over networks susceptible to noise or adversarial behavior.
• Extension to General Code Structures: The techniques enable “list-decoding”-like reliability in a broad class of algebraic codes beyond classical Reed-Solomon, including codes with Hermite evaluation multiplicities and poles, broadening their applicability in practice.
• Future Algorithms: The improved bounds open the way to practical implementations of robust decoding for rational Hermite codes, and suggest general strategies for using interleaving to defeat random and partially adversarial errors in related algebraic code families.

On the theoretical side, the paper strengthens the fundamental connection between interleaved code structure, error model randomness, and the probabilistics of syndrome decoding. The analysis techniques could likely be extended to other non-linear algebraic codes and advanced modular reduction schemes.

Conclusion

This paper systematically advances the analysis of rational function codes, extending robust interleaved decoding beyond unique decoding for codes with both multiplicity and poles. The exponentially decreasing failure probability bounds as a function of the number of interleaved codewords and error weight, even in hybrid adversarial-random models, represent a new high-water mark for such codes. The results remove major limitations of previous analyses (notably dependence on multiplicity balancing and the restriction to pole-free settings) and directly connect algebraic code design to practical distributed computation scenarios. Future work will likely focus on fast implementations and further generalizations of these robust decoding strategies.