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RLNC vs Reed-Solomon in Satellite Channels: Crossover Analysis and the Bridging Property

Angeline Szczepanski, Vipindev Adat Vasudevan

arXiv:2610.35649v1cs.NI

TL;DR

Satellite channels combine high round-trip times with burst-prone losses, and prior work had not rigorously characterized the RS–RLNC crossover at matched rates under physically grounded conditions. The paper derives a satellite channel model and analyzes and simulates the coding schemes at matched rates. RLNC becomes more efficient beyond the crossover and remains usable across RS failure regions, while burst structure can make RS impractical despite low average loss.

  • Problem

    Prior comparisons had not rigorously characterized the RS–RLNC crossover at matched rates under physically grounded satellite channel conditions, despite high RTTs limiting retransmission-based reliability.

    Source: §I · p. 1

  • Method

    The paper derives a physically grounded three-hop satellite loss model and compares RS and RLNC at matched code rates using analytical expressions and simulations.

    Source: §VII · p. 10

  • Results

    Above p*≈0.21, RLNC is decisively more efficient than RS(255, 204); under the derived channel, RS(255, 204) is impractical despite 1% average loss.

    Source: A. The Crossover and Bridging… · p. 9, §VII · p. 10

  • Takeaways & Limitations

    A single RLNC configuration spans the failure regions of the evaluated RS rates, and burst structure—not average loss alone—is critical for code selection in the derived satellite channel.

    Source: §VII · p. 10

    The satellite results use one MEO orbital configuration and one ground-station pair; LEO constellations require separate characterization.

    Source: C. Limitations · p. 10

Abstract

from arXiv · show

Satellite communication systems are increasingly relied upon for connectivity in remote areas and are expected to play a pivotal role in 6G networks, operating under high round-trip times and packet loss where forward error correction is essential. This work evaluates Reed-Solomon (RS) coding, Random Linear Network Coding (RLNC), and a combined RS+RLNC hybrid over a three-hop satellite channel whose loss statistics are derived from orbital geometry and link-budget calculations rather than assumed, comparing RS and RLNC at matched code rates of 0.800. The central finding is the \emph{bridging property} of RLNC: every RS code rate collapses above a crossover threshold $p^*$, while RLNC operates continuously through all failure regions with overhead growing as $1/(1-p)$. At rate~0.800, $p^* \approx 0.21$. Under the derived satcom channel with mean burst duration 166~minutes, RS$(255,204)$ is impractical despite an average loss rate of 1%, while RLNC requires 15.5~Mbit to recover all 1000 packets. An RS+RLNC hybrid reduces overhead by 19% relative to pure RLNC at BER$=10^{-3}$. To the authors' knowledge, this is the first analytical derivation of the rate-normalised crossover $p^*$ between RS and RLNC and the first experimental confirmation of the bridging property under a physically derived satellite channel model.

I. INTRODUCTION

Satellite links combine long round-trip times with time-varying, burst-prone losses, while prior comparisons did not rigorously establish the RS–RLNC crossover at matched rates. This work derives and evaluates that crossover and RLNC’s bridging property using a physically grounded three-hop satellite scenario.

I. INTRODUCTION

  • The channel challenge is structured burst loss from orbital-geometry-driven link outages, which memoryless models poorly capture; high RTT also limits ARQ and HARQ.These conditions motivate packet-level forward erasure correction without feedback.Source: §I · p. 1
  • Prior comparisons had not rigorously characterized the RS–RLNC crossover at matched code rates under physically grounded satellite conditions.Some comparisons used different rates or achievable rate rather than transmission overhead.Source: §I · pp. 1–2
  • At matched rate 0.800, the RS–RLNC comparison yields p*≈0.21; every RS code collapses above its finite p*, while one fixed RLNC scheme operates continuously through the evaluated RS failure regions.Source: §I · p. 2
  • With mean burst duration 166 minutes, RS(255, 204) is impractical despite 1% average loss, while RLNC requires 15.5 Mbit to recover all 1000 packets.The paper cautions that average loss alone is misleading for code selection on this satellite channel.Source: §I · p. 2
  • The concatenated RS+RLNC hybrid reduces overhead by approximately 19% versus pure RLNC at BER=10^-3, while providing bit-error and burst-erasure correction.On the pure-erasure satellite channel, the hybrid costs approximately 2× overhead.Source: §I · p. 2

A. Satellite Scenario

  • The comparison uses a three-hop relay path between Bangalore and Sydney, with two satellites in circular 10,000 km semi-major-axis orbits over a 24-hour simulation.The end-to-end link is available only when both ground-station elevation angles exceed 5°.Source: A. Satellite Scenario · p. 2

B. Link Budget Model

The link-budget model derives received signal quality from the time-varying satellite geometry and link parameters across the three hops. Regenerative repeaters combine the hops, with the weakest hop dominating.

B. Link Budget Model

  • At each time step, hop ranges are determined from satellite positions, and free-space path loss is calculated for each hop.The antenna-gain model uses aperture efficiency η=0.55.Source: B. Link Budget Model · p. 2
  • Received Eb/N0 is calculated from transmit power and gain, receiver G/T, Boltzmann’s constant, and the 10^6 bps bit rate.The system assumes regenerative repeaters at each satellite.Source: B. Link Budget Model · p. 2
  • The three hops combine in the linear domain, with the weakest hop dominating; the link margin uses a required Eb/N0 of 1 dB.Key link-budget parameters are summarized in Table I.Source: B. Link Budget Model · p. 2

C. Packet Loss Probability Model

The model maps time-varying link margin to packet loss and represents orbital outages as correlated bursts rather than independent losses. A fitted Gilbert-Elliott channel summarizes this burst structure, which average loss alone does not capture for block codes.

C. Packet Loss Probability Model

  • The link margin Λ(k) is mapped to per-step packet loss probability with a sigmoid, while elevation below 5° sets loss to 1.The mapping parameters are chosen to reproduce qualitative empirical BER/margin curves rather than fitted to measured data.Source: C. Packet Loss Probability Model · pp. 2–3

D. Gilbert-Elliott Channel Model

  • Orbital outages create extended periods with loss probability 1, while open-link windows with low loss last tens of minutes, motivating a Gilbert-Elliott model.The cited model is a two-state Markov channel.Source: D. Gilbert-Elliott Channel Model · p. 3
  • The Gilbert-Elliott model distinguishes a good open-link state from a bad outage or severe-degradation state; its transition parameters determine state probabilities and mean burst and good-period durations.The marginal packet-loss probability combines the state probabilities with their corresponding losses.Source: D. Gilbert-Elliott Channel Model · p. 3
  • Average loss alone is insufficient for block-code performance: rare long bursts are more damaging than frequent short bursts at the same average loss.The GE parameters are estimated by maximum likelihood from the simulated link-availability sequence.Source: D. Gilbert-Elliott Channel Model · p. 3

III. FORWARD ERROR CORRECTION SCHEMES

The paper contrasts fixed-rate, symbol-level Reed-Solomon blocks with packet-level RLNC, which accumulates coded packets across rounds. Both are specified at matched rate 0.800, while RLNC's probability of receiving an innovative packet remains high for the stated generation size.

A. Reed-Solomon Coding

  • RS(255, 204) and RLNC both use rate 0.800, but RS has a fixed block rate while RLNC can absorb loss through additional rounds.The evaluated RS configurations use N = 255; RLNC uses Kgen = 16 and Ncoded = 20.Source: A. Reed-Solomon Coding · p. 3, B. Random Linear Network Coding · p. 4
  • RS codes correct symbol-level errors and decode on an erasure channel only when at most ρ = N − K symbols are lost.The evaluated configurations have rates 0.875, 0.800, and 0.749, respectively.Source: A. Reed-Solomon Coding · p. 3

B. Random Linear Network Coding

  • RLNC groups packets into generations and transmits random linear combinations, decoding once the received coefficient matrix reaches rank Kgen.Gaussian elimination over GF(2^8) recovers the original packets at full rank.Source: B. Random Linear Network Coding · pp. 3–4
  • For Kgen = 16 over GF(2^8), each received packet is innovative with probability above 1 − 2^-8 ≈ 0.996 at every rank.The paper treats linear-dependence failures as negligible in practice.Source: B. Random Linear Network Coding · p. 4
  • Block RLNC sends 20 coded packets per round without waiting for feedback; a lost round delays decoding rather than failing the generation.This block approach is chosen because satellite links have large RTTs, and its per-round rate matches RS(255, 204).Source: B. Random Linear Network Coding · p. 4

C. RS+RLNC Hybrid Coding

The hybrid applies RS within packets and RLNC across packet erasures, combining symbol-error and burst-erasure protection. For RS alone, decoding requires no more than ρ erasures in a block, and overhead diverges as loss rises and decoding probability falls.

C. RS+RLNC Hybrid Coding

  • RS+RLNC uses inner RS to correct symbol errors within received packets and outer RLNC to correct packet erasures across the channel.The concatenated scheme provides simultaneous bit-error and burst-erasure protection, unlike either pure scheme alone.Source: C. RS+RLNC Hybrid Coding · p. 4
  • On a pure erasure channel, inner RS adds overhead without correction benefit, so pure RLNC is optimal; the hybrid is advantageous when residual bit errors are present.The latter condition is characterized in Experiment 6.Source: C. RS+RLNC Hybrid Coding · p. 4

A. RS Decoding Success Probability

  • An RS block decodes on the packet-erasure channel if and only if at most ρ = N − K packets are erased.For large N and ρ, the binomial sum is computed in logspace using the Stirling approximation to avoid numerical overflow.Source: A. RS Decoding Success Probability · p. 4
  • As loss increases, RS decoding probability falls rapidly and transmission overhead diverges; larger ρ delays, but does not prevent, collapse.The divergence rate depends strongly on ρ.Source: A. RS Decoding Success Probability · p. 4

B. RLNC Decoding Success Probability

RLNC accumulates degrees of freedom across rounds, so a lost round delays decoding rather than failing the generation. The analysis defines a rate-matched RS/RLNC crossover and finds that every RS rate eventually becomes impractical, while RLNC continues through those failure regions.

B. RLNC Decoding Success Probability

P RLNC

C. The Crossover Loss Rate p∗and the Bridging Property

  • At matched rate 0.800, RLNC becomes more efficient than RS(255, 204) above p* ≈ 0.21, and every RS rate has a finite crossover beyond which RS overhead collapses.The figure marks loss ranges where RS codes become impractical; the best available RS-rate envelope collapses by p ≈ 0.27.Source: Figure 1 · p. 5, C. The Crossover Loss Rate p∗and the… · p. 5
  • The crossover p* is solved numerically and depends on RS parameters and RLNC rate; in the rate-matched comparison, higher ρ shifts p* upward.The analytical expressions predict simulated transmission cost within 1% for RS and within 5% for RLNC at low-to-moderate loss across four RS rates.Source: C. The Crossover Loss Rate p∗and the… · p. 5

D. Burst Channel Penalty: RS vs. RLNC

Under the Gilbert–Elliott model, burst length—not only average packet loss—constrains RS because a burst exceeding its erasure capacity guarantees block failure. RLNC instead accumulates innovative packets across rounds, so a burst delays decoding rather than necessarily causing failure.

D. Burst Channel Penalty: RS vs. RLNC

  • For RS, a burst longer than erasure capacity ρ guarantees decoding failure; under the GE model, the burst-based failure bound is P(RS block fails | burst) ≥ P(LB > ρ) = (1 − r)^ρ.This bound is independent of per-packet loss probability, so marginal loss can understate RS overhead.Source: D. Burst Channel Penalty: RS vs. RLNC · p. 5
  • RLNC can tolerate a lost round as delay: decoding fails only if total innovative packets across rounds fall below Kgen = 16.RLNC continues accumulating innovative packets in later rounds, unlike RS’s hard burst-length limit.Source: D. Burst Channel Penalty: RS vs. RLNC · p. 5
  • Burst-channel RS overhead can far exceed its i.i.d. counterpart at the same average loss, while RLNC overhead remains approximately its i.i.d. overhead evaluated at the within-window loss rate.The burst structure can push RS above its crossover threshold even when average loss is low.Source: D. Burst Channel Penalty: RS vs. RLNC · p. 5

V. SIMULATION SETUP AND RESULTS

The simulations compare RS and RLNC at matched rate 0.800, then test their fixed-loss behavior and crossover. RS tracks theory before collapsing, whereas RLNC remains bounded across the loss sweep; the matched-rate experiments locate the crossover and test the bridging property.

A. Simulation Implementation

  • The simulation uses packet-level erasures, a 10^9-bit transmission cap, and matched rate 0.800 for RLNC and RS(255, 204); it evaluates fixed-loss, orbital satcom, and Gilbert–Elliott burst modes.The RS simulation sends full N = 255-packet blocks, while RLNC uses Kgen = 16 and Ncoded = 20.Source: A. Simulation Implementation · pp. 5–6

B. Experiments 1 and 2: Fixed Packet Loss Rate Results

  • RS(255, 204) remains practical to p≈0.20, where expected losses per block equal ρ = 51, then its block-failure probability rises sharply; simulation closely tracks theory before the cap.The 950-, 990-, and 1000-packet recovery thresholds are nearly coincident because block-level failure dominates.Source: B. Experiments 1 and 2: Fixed Packet… · p. 6, Figure 2 · p. 6
  • RLNC overhead remains bounded and smoothly growing across p∈[0, 0.6], unlike RS collapse; finite-generation start-up overhead puts simulation slightly above the theoretical 1.25/(1−p) curve.The receiver cannot start a new round immediately at generation boundaries.Source: B. Experiments 1 and 2: Fixed Packet… · p. 6, Figure 3 · p. 6

C. Experiment 3: Crossover Points and the Bridging Property

  • Above p∗≈0.21, RLNC becomes more efficient than RS(255, 204) at matched rate 0.800, while every evaluated RS configuration eventually collapses and RLNC remains below 8 Mbit across the range.Increasing ρ shifts the RS collapse threshold rightward; a single RLNC configuration spans all three RS failure regions.Source: Figure 4 · p. 6, C. Experiment 3: Crossover Points and… · pp. 6–7
  • Multiple RLNC rates can be selected within an RS rate gap, each producing a distinct crossover point and continuously filling the bridging region.The operating points shown cross 50% received-block percentage at distinct loss probabilities.Source: Figure 5 · p. 7
  • At p=0.1, theory and simulation are compared across four RS rates and rate-matched RLNC; RS agrees within 1%, while RLNC agrees within 5% for rates 0.502–0.800 but diverges by 38% at rate 0.937.At rate 0.937, the small generation size amplifies the finite-generation effect.Source: C. Experiment 3: Crossover Points and… · p. 7, Table IV · p. 7

D. Multiple RLNC Operating Points Within the Bridging Region

The satcom and burst-duration experiments show how correlated outages penalize RS beyond what average loss suggests. RLNC stays practical under the derived satellite channel, while increasing burst duration drives RS failure sharply upward and RLNC overhead upward more gradually.

D. Multiple RLNC Operating Points Within the Bridging Region

E. Experiment 4: Satcom Time-Varying Channel

  • The satcom GE fit gives mean burst duration 166.5 minutes and open-link loss ϵG=0.0100; outage windows average about 166 steps, more than three times RS(255, 204)’s ρ=51.Any burst longer than 51 steps guarantees RS block failure, so essentially every outage window causes complete RS decoding failure.Source: E. Experiment 4: Satcom Time-Varying… · p. 7
  • At open-link loss ϵG=0.010, RLNC requires approximately 15.5 Mbit to recover all 1000 packets, while RS(255, 204) exceeds the 500 Mbit cap despite i.i.d. theory predicting approximately 1 Mbit.The burst structure pushes RS into its collapse regime around p≈0.20 even though open-link loss is only 1%.Source: Figure 7 · p. 7, E. Experiment 4: Satcom Time-Varying… · pp. 7–8

F. Experiment 5: Burst Duration Sweep

  • At matched rate 0.800, RS failure rises to near 100% well before the satcom mean burst duration, while RLNC overhead grows from approximately 1.3 Mbit to 5.0 Mbit as bursts lengthen from 1 to 50 steps.Within-state losses compound burst losses in the GE model; RLNC continues accumulating across rounds,.Source: F. Experiment 5: Burst Duration Sweep · p. 8

G. Experiment 6: RS+RLNC Hybrid

The hybrid results show that inner RS coding can help modestly when bit errors accompany erasures, but costs more on pure-erasure satellite channels. The paper’s broader result is that RLNC bridges RS failure regions, with a rate-0.800 crossover near p∗≈0.21.

G. Experiment 6: RS+RLNC Hybrid

  • The hybrid uses an outer RLNC code at rate 0.800 with three inner RS configurations; on a pure erasure channel, inner RS adds overhead without correction benefit.At BER=10−3, the hybrids perform similarly across the full fixed-loss range because their inner RS codes have similar correction capabilities.Source: G. Experiment 6: RS+RLNC Hybrid · p. 8, Figure 10 · p. 9
  • At BER=10−3 and p=0.6, the hybrids require 6.28–6.53 Mbit versus 7.78 Mbit for pure RLNC, a 16–19% reduction; at BER=0, pure RLNC requires 2.24 Mbit versus 4.41–4.53 Mbit for hybrids.At BER=10−3, two inner RS rates fall slightly below pure RLNC at p=0.3, while the third is slightly above.Source: G. Experiment 6: RS+RLNC Hybrid · p. 8, Figure 9 · p. 8, Figure 10 · p. 9
  • On the pure-erasure satcom channel, pure RLNC requires 15.5 Mbit, while each hybrid requires approximately 32 Mbit because inner RS doubles bits per transmission.Pure RLNC is most efficient for pure burst-erasure channels; the hybrid is preferred only when bit errors and burst erasures both occur.Source: G. Experiment 6: RS+RLNC Hybrid · p. 8, Figure 11 · p. 9

A. The Crossover and Bridging Property: Summary

  • At matched rate 0.800, RS and RLNC perform similarly below p∗≈0.21, then RLNC becomes decisively more efficient; all evaluated RS rates eventually collapse while one RLNC configuration spans their failure regions.The evaluated RS-code envelope collapses by p≈0.27, and the results are validated on fixed-loss and physically derived satcom channels.Source: A. The Crossover and Bridging… · p. 9

B. Practical Implications

RLNC remains practical across loss conditions where fixed-rate RS becomes impractical, and burst duration is more informative than average loss for satellite code selection. The paper also identifies limits in its channel models and proposes adaptive switching and other extensions for future work.

B. Practical Implications

  • With mean burst duration reaching 166 minutes, RLNC maintains practical overhead where a fixed block code would need ρ > 166 to match it.The authors identify estimated burst duration, rather than average loss rate, as the relevant design metric.Source: B. Practical Implications · p. 9
  • One RLNC configuration covers operating conditions that would otherwise require selecting among multiple fixed-rate alternatives.The paper describes RLNC as degrading gracefully beyond the point where matching fixed-rate alternatives would no longer be practical.Source: B. Practical Implications · p. 9
  • The RS+RLNC hybrid provides bit-error and burst-erasure protection at approximately 2× the overhead of pure RLNC on the satcom channel.The authors recommend it when significant residual bit errors coexist with burst erasures.Source: B. Practical Implications · p. 9
  • Increasing Kgen at a fixed code rate shifts p* slightly lower, suggesting RS below p* and RLNC above it, with estimated burst duration as the switching signal.The authors present this adaptive strategy as a future direction, not as a tested scheme.Source: B. Practical Implications · pp. 9–10

C. Limitations

  • The fixed-p sweep assumes independent erasures, the Gilbert-Elliott model simplifies burst behavior, and satcom results cover one MEO orbit and ground-station pair.More accurate finite-state Markov models require measurement data unavailable in this work; LEO constellations need separate characterization.Source: C. Limitations · p. 10

VII. CONCLUSION AND FUTURE WORK

  • The paper derives a physically grounded loss model and closed-form performance expressions, validating them against MATLAB simulation across six experiments.Future work includes larger generation sizes, adaptive switching, sliding-window RLNC, and validation against real LEO packet traces using more accurate channel models.Source: §VII · p. 10
  • Above p*≈0.21, RLNC is decisively more efficient than RS(255, 204), while both perform similarly below that threshold.The reported advantage above p* is attributed to burst resilience, not a rate advantage; a single RLNC configuration spans the RS rate family.Source: §VII · p. 10
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