TY - JOUR

T1 - Network-Coding Solutions for Minimal Combination Networks and Their Sub-Networks

AU - Cai, Han

AU - Chrisnata, Johan

AU - Etzion, Tuvi

AU - Schwartz, Moshe

AU - Wachter-Zeh, Antonia

N1 - Funding Information:
Manuscript received September 13, 2019; revised February 20, 2020; accepted May 9, 2020. Date of publication May 20, 2020; date of current version October 21, 2020. The work of Tuvi Etzion was supported in part by the Bernard Elkin Chair in Computer Science. The work of Moshe Schwartz and Antonia Wachter-Zeh was supported in part by a German Israeli Project Cooperation (DIP) Grant under Grant PE2398/1-1 and Grant KR3517/9-1. This article was presented at the 2019 International Symposium on Information Theory. (Corresponding author: Moshe Schwartz.) Han Cai and Moshe Schwartz are with the School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva 8410501, Israel (e-mail: hancai@aliyun.com; schwartz@ee.bgu.ac.il).
Publisher Copyright:
© 1963-2012 IEEE.

PY - 2020/11/1

Y1 - 2020/11/1

N2 - Minimal multicast networks are fascinating and efficient combinatorial objects, where the removal of a single link makes it impossible for all receivers to obtain all messages. We study the structure of such networks, and prove some constraints on their possible solutions. We then focus on the combination network, which is one of the simplest and most insightful network in network-coding theory. Of particular interest are minimal combination networks. We study the gap in alphabet size between vector-linear and scalar-linear network-coding solutions for such minimal combination networks and some of their sub-networks. For minimal multicast networks with two source messages we find the maximum possible gap. We define and study sub-networks of the combination network, which we call Kneser networks, and prove that they attain the upper bound on the gap with equality. We also prove that the study of this gap may be limited to the study of sub-networks of minimal combination networks, by using graph homomorphisms connected with the q-analog of Kneser graphs. Additionally, we prove a gap for minimal multicast networks with three or more source messages by studying Kneser networks. Finally, an upper bound on the gap for full minimal combination networks shows nearly no gap, or none in some cases. This is obtained using an MDS-like bound for subspaces over a finite field.

AB - Minimal multicast networks are fascinating and efficient combinatorial objects, where the removal of a single link makes it impossible for all receivers to obtain all messages. We study the structure of such networks, and prove some constraints on their possible solutions. We then focus on the combination network, which is one of the simplest and most insightful network in network-coding theory. Of particular interest are minimal combination networks. We study the gap in alphabet size between vector-linear and scalar-linear network-coding solutions for such minimal combination networks and some of their sub-networks. For minimal multicast networks with two source messages we find the maximum possible gap. We define and study sub-networks of the combination network, which we call Kneser networks, and prove that they attain the upper bound on the gap with equality. We also prove that the study of this gap may be limited to the study of sub-networks of minimal combination networks, by using graph homomorphisms connected with the q-analog of Kneser graphs. Additionally, we prove a gap for minimal multicast networks with three or more source messages by studying Kneser networks. Finally, an upper bound on the gap for full minimal combination networks shows nearly no gap, or none in some cases. This is obtained using an MDS-like bound for subspaces over a finite field.

KW - Linear network coding

KW - combination network

KW - graph coloring

KW - minimal networks

KW - q-Kneser graphs

UR - http://www.scopus.com/inward/record.url?scp=85094634142&partnerID=8YFLogxK

U2 - 10.1109/TIT.2020.2995845

DO - 10.1109/TIT.2020.2995845

M3 - Article

AN - SCOPUS:85094634142

SN - 0018-9448

VL - 66

SP - 6786

EP - 6798

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 11

M1 - 9097246

ER -