On Ingleton-Violating Finite Groups

Wei Mao, Matthew Thill, Babak Hassibi

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


Given n discrete random variables, its entropy vector is the 2n-1-dimensional vector obtained from the joint entropies of all non-empty subsets of the random variables. It is well known that there is a close relation between such an entropy vector and a certain group-characterizable vector obtained from a finite group and n of its subgroups; indeed, roughly speaking, knowing the region of all such group-characterizable vectors is equivalent to knowing the region of all entropy vectors. This correspondence may be useful for characterizing the space of entropic vectors and for designing network codes. If one restricts attention to abelian groups then not all entropy vectors can be obtained. This is an explanation for the fact shown by Dougherty et al. that linear network codes cannot achieve capacity in general network coding problems (since linear network codes come from abelian groups). All abelian group-characterizable vectors, and by fiat all entropy vectors generated by linear network codes, satisfy a linear inequality called the Ingleton inequality. General entropy vectors, however, do not necessarily have this property. It is, therefore, of interest to identify groups that violate the Ingleton inequality. In this paper, we study the problem of finding nonabelian finite groups that yield characterizable vectors, which violate the Ingleton inequality. Using a refined computer search, we find the symmetric group S5 to be the smallest group that violates the Ingleton inequality. Careful study of the structure of this group, and its subgroups, reveals that it belongs to the Ingleton-violating family PGL(2,q) with a prime power q ≥q 5 , i.e., the projective group of 2× 2 nonsingular matrices with entries in Fq. We further interpret this family of groups, and their subgroups, using the theory of group actions and identify the subgroups as certain stabilizers. We also extend the construction to more general groups such as PGL(n,q) and GL(n,q). The families of groups identified here are therefore good candidates for constructing network codes more powerful than linear network codes, and we discuss some considerations for constructing such group network codes.
Original languageEnglish (US)
Pages (from-to)183-200
Number of pages18
JournalIEEE Transactions on Information Theory
Issue number1
StatePublished - Jan 2017
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2021-04-06
Acknowledgements: This work was supported in part by the National Science Foundation under Grant CNS-0932428, Grant CCF-1018927, Grant CCF-1423663, and Grant CCF- 1409204, in part by Qualcomm Inc., in part by the NASA's Jet Propulsion Laboratory through the President and Director's Fund, in part by King Abdulaziz University, and in part by the King Abdullah University of Science and Technology.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

  • Library and Information Sciences
  • Information Systems
  • Computer Science Applications


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