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Optimal leading coefficient problem for AWGN spherical codes

For every pair of integers n,kn,k satisfying 3≤k≤n−13\leq k\leq n-1, the optimal leading coefficient in the high-SNR maximum-likelihood error-probability expansion for SNR-dependent (n+k,n)(n+k,n) AWGN spherical codes is conjectured to satisfy Kn+k,n∗=4k(k−1)K_{n+k,n}^{\ast}=4k(k-1), where Kn+k,n∗K_{n+k,n}^{\ast} denotes the SNR-wise optimal coefficient.

References

Primary source

arXiv

Additional references

coding-theoryinformation-theoryoptimization
Posted

Progress summary

Refreshed
Claimed progress

A new preprint reports that allowing the code to change with signal strength improves the best high-SNR error coefficient in some cases, but does not settle every dimension.

The problem asks whether the code with the largest minimum distance also minimizes the leading high-SNR error coefficient for AWGN spherical codes. The latest preprint reports equality in the small-code regime and an improvement from signal-strength-dependent perturbations for larger codes.

August 2026 development

A preprint, Beyond Minimum Distance: The Optimal Leading Coefficient in the High-SNR Error-Probability Expansion for AWGN Spherical Codes, claims equality of fixed and SNR-wise coefficients for 2≤M≤n+12\le M\le n+1, and a strict improvement at M=n+2M=n+2. It explicitly leaves the regime 3≤k≤n−13\le k\le n-1 conjectural.

Current status (as of August 2026): The preprint claims progress for 2≤M≤n+12\le M\le n+1 and M=n+2M=n+2, while the regime 3≤k≤n−13\le k\le n-1 remains conjectural and the claims are unverified.

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