New lower bound on the Shannon capacity of C7 from circular graphs

Abstract

We give an independent set of size 367 in the fifth strong product power of C 7 , where C 7 is the cycle on 7 vertices. This leads to an improved lower bound on the Shannon capacity of C 7 : Θ(C 7 ) ≥ 367 1/5 > 3.2578. The independent set is found by computer, using the fact that the set {t · (1, 7, 7 2 , 7 3 , 7 4 ) | t ∈ Z 382 } ⊆ Z 5 382 is independent in the fifth strong product power of the circular graph C 108,382 . Here the circular graph C k,n is the graph with vertex set Z n , the cyclic group of order n, in which two distinct vertices are adjacent if and only if their distance (mod n) is strictly less than k.

Variable-Length Coding for Zero-Error Channel Capacity

“…1: The zero-error capacity of the graph C 7 , corresponding to the noisy-typewriter channel with 7 letters, is unknown. Some recent lower and upper bounds are stated in [26].…”

“…1. Its zero-error capacity is still unknown, despite several attempts to build zero-error codes on these channels, see [31], [32], and [26]. In Fig.…”

The zero-error capacity of binary channels with 2-memories

“…A breakthrough in this field was given by Lovász [12] who proved that Θ(G) = √ 5. Despite many results on Shannon capacity of cycles with 7 vertices C 7 [14,15,17], the capacity problem for C 7 is still open. For disjoint union of two graphs, Alon [2] showed that there exist graphs G and H such that Θ(G + H) > Θ(G) + Θ(H), which disproved a conjectured given by Shannon [16].…”

On zero-error codes produced by greedy algorithms

“…Recently, Polak and Schrijver (2019) and Mathew and Östergård (2017) made some progress in research on channels represented by strong powers of cycles. Moreover, Boche and Deppe (2020) proved that the zero-error capacity is uncomputable in the Banach-Mazur and Borel-Turing senses.…”