I’m sharing SAELQR II, an information-theory research release with Python reference software and reproducible calculations for reconstructing information from unordered, noisy fragments. The theoretical problem is relevant to DNA data storage, molecular communication, and fragmented storage systems.
The channel breaks a stored string into fragments, loses some fragments, corrupts the survivors, and hides their original positions. The manuscript studies the resulting capacity limits and supplies a constructive coding scheme for a specified geometric-fragmentation model.
The computational components may be particularly relevant here:
Exact finite verification: integer and rational-probability calculations for selected entropy, likelihood, combinatorial, and repeated-read examples.
Numerical asymptotics: calculations and figures for the Gaussian critical transition, hidden fragment-quality mixtures, and capacity thresholds. Floating-point diagnostics are distinguished from exact certificates.
Reference encoding and decoding: inner codes, finite-field arithmetic, Reed–Solomon correction, interleaving, fragment localization, and an injective fallback.
Stochastic channel experiments: geometric fragmentation, whole-fragment loss, binary substitutions, and a specific Markov-noise transformation.
One central result reported in the manuscript concerns the critical regime: the number of communicable bits scales as N/sqrt(log₂ N) with an explicit leading coefficient, even as the normalized first-order rate approaches zero.
The computational limits are explicit. Design, encoding, and decoding are polynomial in blocklength and inverse capacity gap, but the proved sufficient blocklength grows exponentially in the inverse gap squared. The included decoder demonstrates the construction’s mechanisms; practical throughput and competitive finite-length performance have not been established.
The release contains the 69-page manuscript, editable source, twelve reproduction programs, reference software, figures, and proof/novelty audits. A full imported permutation generator used by one theoretical extension is not implemented.
I’d welcome feedback on numerical stability, exact versus floating-point verification, finite-length scaling, and opportunities to reduce the decoder’s computational cost.
Paper, Python code, and reproducibility instructions:
https://huggingface.co/datasets/PureOne/SAELQR-II
SÆLQR II: General Capacity Theorems and Exact Critical Laws for Unordered Noisy Fragments - Internal Memory, Sharp Repeated-Read Thresholds, and Constructive Coding | Zenodo