Term Coding: An Entropic Framework for Extremal Combinatorics and the Guessing--Number Sandwich Theorem

TL;DR

This paper introduces a new entropic framework called Term Coding that transforms extremal combinatorics and guessing problems into quantitative questions, connecting them to graph entropy and polymatroid methods.

Contribution

It establishes a guessing-number sandwich theorem linking term coding to graph guessing numbers, providing a canonical structure and entropy-based bounds for solution set sizes.

Findings

The maximum code size relates to the guessing number α as log_n S_n(Γ)=α+o(1).

The framework applies to extremal combinatorics and network coding examples.

Explicit bounds and computations are achieved using entropy and polymatroid techniques.

Abstract

Term Coding asks: given a finite system of term identities $\Gamma$ in $v$ variables, how large can its solution set be on an $n$--element alphabet, when we are free to choose the interpretations of the function symbols? This turns familiar existence problems for quasigroups, designs, and related objects into quantitative extremal questions. We prove a guessing-number sandwich theorem that connects term coding to graph guessing numbers (graph entropy). After explicit normalisation and diversification reductions, every instance yields a canonical directed dependency structure with guessing number $\alpha$ such that the maximum code size satisfies $\log_n \Sn(\Gamma)=\alpha+o(1)$ (equivalently, $\Sn(\Gamma)=n^{\alpha+o(1)}$), and $\alpha$ can be bounded or computed using entropy and polymatroid methods. We illustrate the framework with examples from extremal combinatorics (Steiner-type identities, self-orthogonal Latin squares) and from information-flow / network-coding style constraints (including a five-cycle instance with fractional exponent and small storage/relay maps).