In the present paper we give a derivation of Elias' Omega code from physics principles by combining a constrained variational formulation of prefix coding with a renormalization flow on codeword distributions. Starting from a Lagrangian that minimizes average codelength under the Kraft-McMillan constraint, we show that the implied distribution is a fixed point of a coarse-graining map, yielding the canonical iterated log-sum length, asymptotically up to an additive constant. This establishes completeness and asymptotic optimality, and connects universal integer coding with coarse-grained entropy, uncertainty-type bounds, and entropy relations familiar from statistical physics.

翻译：本文通过将前缀编码的约束变分表述与码字分布上的重整化流相结合，从物理学原理出发推导出Elias Omega编码。从在Kraft-McMillan约束下最小化平均码长的拉格朗日量出发，我们证明隐含分布是粗粒度映射的不动点，渐进地（至多相差一个加性常数）给出典型的迭代对数求和长度。这确立了完备性与渐进最优性，并将通用整数编码与粗粒度熵、不确定性类型界限以及统计物理学中熟悉的熵关系联系起来。