Term Coding: An Entropic Framework for Extremal Combinatorics and the Guessing--Number Sandwich Theorem
S{\o}ren Riis

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 in variables, how large can its solution set be on an --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 such that the maximum code size satisfies (equivalently, ), and can be bounded or computed using entropy and polymatroid methods. We illustrate the framework with examples from extremal combinatorics…
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Taxonomy
Topicsgraph theory and CDMA systems · Cellular Automata and Applications · Limits and Structures in Graph Theory
