Confusion-Erasure Bounds of Error-Bounded Decoders under QAM

I. INTRODUCTION

• BLER conflates detectable block erasures with silent block confusions, a distinction important to safety-critical services but not rigorously analyzed in the finite-blocklength regime.Prior analysis was restricted to PSK, whose uniform symbol energies place codewords on one hypersphere.Source: §I · p. 1
• Square QAM requires bounds that account for multiple symbol distances: one uses RMS distance, while the other averages pairwise confusion over the per-shell distribution.The latter avoids the loose worst-case nearest-neighbour approximation.Source: §I · p. 1
• As SNR or redundancy increases, confusion bounds fall many orders of magnitude below ε, leaving detectable erasures as the dominant residual failure.The paper identifies the code’s inherent error detection as the mechanism for flagging these erasures.Source: §I · p. 1

II. ERROR-BOUNDED DECODER

• An ε-bounded decoder rejects list outputs, assigning correct decoding, erasure, or confusion according to whether the received word falls within the transmitted codeword’s region, outside all regions, or within another’s.The decision radius is constrained by R ≤ Dmin/2 so decision regions do not overlap.Source: §II · pp. 1–2
• The block erasure rate is Pers = ε − Pcon, so upper and lower bounds on confusion correspond respectively to lower and upper bounds on erasure.Source: §II · p. 2

III. BOUNDS ANALYSIS

• Pairwise confusion depends on the Euclidean distance between codewords, so bounding that distance is the next step in the analysis.Source: §III · p. 2

A. Bounds of Minimum Euclidean Distance

• At fixed Hamming distance, squared Euclidean distance varies with symbol magnitudes and phases, so it is characterized through its expectation rather than a single value.Source: A. Bounds of Minimum Euclidean Distance · p. 2
• The per-symbol squared-distance coefficient is Δ = 2 Ē_s (1 − η̄), giving the ensemble relation D² = Δd for Hamming distance d.The relation holds for a fixed constellation and symbol-usage distribution; it is not a per-pair equality.Source: A. Bounds of Minimum Euclidean Distance · p. 3
• Under uniform symbol use, Ē_s is the average constellation symbol energy and η̄ = −1/(M−1); individual pair distances still fluctuate around their conditional mean.Source: A. Bounds of Minimum Euclidean Distance · p. 3
• The RMS distance at Hamming distance d is the representative distance used in the bounds, justified by Lemma 3.Source: A. Bounds of Minimum Euclidean Distance · p. 3
• The minimum Hamming distance is bounded below by the non-overlap feasibility floor and above by the Hamming bound.The feasibility condition uses the maximum squared symbol distance to ensure distinct decision regions do not overlap.Source: A. Bounds of Minimum Euclidean Distance · p. 3

B. Bound of Block Confusion Rate

• For fixed ε and σ, pairwise confusion decreases with Euclidean distance and is convex for distances D ≥ 2R(ε).Source: B. Bound of Block Confusion Rate · p. 3
• The expected pairwise confusion within a Hamming-distance shell is at least its value at the RMS distance, supporting a lower bound on block confusion.The RMS substitution follows from the stated monotonicity and convexity of pairwise confusion.Source: B. Bound of Block Confusion Rate · p. 3
• Within a shell, squared distance is a sum of d independent per-symbol squared distances, and averaging pairwise confusion over that distribution gives the ensemble union-bound term.Source: B. Bound of Block Confusion Rate · p. 3
• The upper bound uses the expected distance spectrum, extends the sum down to the conservative minimum-distance floor, and thereby overestimates block confusion.Source: B. Bound of Block Confusion Rate · pp. 3–4

IV. SENSITIVITY OF CONFUSION RATE BOUNDS

• The section analyzes sensitivity of the confusion-rate bounds to energy and blocklength.The supplied passage introduces these two variables as the section’s focus.Source: §IV · p. 4

A. Lower Bound versus Energy

• For fixed (M, k, n, σ, ε), the lower bound P LB_con decreases monotonically and convexly in average symbol energy.The proof is deferred to Appendix A.Source: A. Lower Bound versus Energy · p. 4

B. Lower Bound versus Blocklength

• The lower bound is piecewise decreasing in blocklength within each dUB_H,min interval and globally decreasing for sufficiently large n.The distance limit dUB_H,min is nondecreasing in n; the QAM proof carries over the PSK monotonicity argument because per-shell distance and noise dimension are constant in n.Source: B. Lower Bound versus Blocklength · p. 4

C. Upper Bound versus Energy

• For fixed (M, k, n, σ, ε), the upper bound is piecewise continuous in average symbol energy, with jumps at energy thresholds, and decreases convexly within each continuity interval.The lower-bound Hamming-distance floor decreases with energy, as the relevant distance ratio decreases when average symbol energy rises.Source: C. Upper Bound versus Energy · p. 4

D. Upper Bound versus Blocklength

• For fixed (M, k, Es, σ, ε), the upper bound P UB_con decreases in n for sufficiently large n as redundancy exponentially suppresses shell populations.The decision radius grows with n and raises pairwise confusion, but the resulting increase in dLB_H,min removes nearest shells; at sufficiently large n, redundancy suppression dominates.Source: D. Upper Bound versus Blocklength · p. 4

V. NUMERICAL EXPERIMENT RESULTS

• The simulations use uniformly selected, unshaped square 16-QAM and 64-QAM symbols, with ε = 0.05 and σ² = 0.5.The passage states that nonuniform codebook use would rescale Δ through η̄ without changing the analysis or qualitative behavior.Source: §V · p. 4
• As blocklength grows, the lower bound decreases within each dUB_H,min interval and globally for large n, while the upper bound also decreases for large n.At fixed k, adding redundancy makes the codebook sparser within an interval; the upper-bound decline accompanies the rising dLB_H,min floor.Source: §V · p. 4
• As SNR rises, the lower bound decreases convexly, while the upper bound decreases within intervals and jumps when a newly feasible nearest shell enters the sum.The lower bound decreases monotonically; higher SNR stretches the constellation and widens the feasible window.Source: §V · p. 4
• With k = 16 and Es/N0 = 8 dB, both Hamming-distance limits increase with blocklength; the minimum feasible distance rises because R(ε) grows with n.The figure plots confusion-rate bounds versus blocklength for 16-QAM and 64-QAM.Source: §V · p. 4, Figure 2 · p. 5
• At high SNR, confusion is negligible relative to ε, leaving residual failures almost entirely as detectable erasures that can be handled by HARQ and network coding.The 64-QAM bounds decay more slowly and stay above the 16-QAM bounds; higher-order constellations require more transmit energy for confusion to become negligible.Source: §V · p. 5

• As SNR or redundancy increases, confusion falls many orders below the target, leaving erasures dominant and detectable without a separate CRC.Source: §VI · p. 5
• The bounds decrease with average symbol energy and blocklength, at a rate governed by constellation order.Source: §VI · p. 5
• Fig. 3 plots confusion-rate bounds against Es/N0 for (32, 16) codebooks.Source: Figure 3 · p. 5
• Future work would incorporate actual LDPC or polar code distance spectra and extend the framework to probabilistically shaped and non-square constellations.Source: §VI · p. 5

APPENDIX A PROOF OF THEOREM 3

• For fixed (M, k, n), the proof sets α = Δ/Es = 2(1 − η̄) before deriving the lower-bound properties.Source: Appendix A · p. 5
• For x ≥ 2R(ε), g′(x) ≤ 0 and g′′(x) ≥ 0 yield a lower confusion bound that decreases with average symbol energy.Source: Appendix A · p. 6
• The proof shows that the lower confusion bound is convex in average symbol energy because its second-derivative terms are nonnegative.Source: Appendix A · p. 6

APPENDIX B PROOF OF LEMMA 5

• The minimum-Hamming-distance term is an integer-valued staircase in average symbol energy, with jumps when the ceiling argument is an integer.Source: Appendix B · p. 6
• The upper confusion bound decreases with average symbol energy within intervals where dLB_H,min is constant, since its derivative integrand is negative.Source: Appendix B · p. 6
• Conditioned on shell d, the Euclidean distance scales with the square root of average symbol energy times Wd, which depends only on constellation shape and the per-shell distance distribution.Source: Appendix B · p. 6
• The upper confusion bound is convex in average symbol energy within intervals where dLB_H,min is constant.Source: Appendix B · p. 6