On Coset Leader Graphs of Structured Linear Codes

Abstract

We suggest a new approach to obtain bounds on locally correctable and some locally testable binary linear codes, by arguing that these codes (or their subcodes) have coset leader graphs with high discrete Ricci curvature.The bounds we obtain for locally correctable codes are worse than the best known bounds obtained using quantum information theory, but are better than those obtained using other methods, such as the "usual" information theory. (We remark that our methods are completely elementary.)The bounds we obtain for a family of locally testable codes improve the best known bounds.

Lin-Lu-Yau curvature and diameter of amply regular graphs

“…Discrete Bonnet-Myers theorem has recently found important applications in coding theory: it provides a completely elementary way to derive bounds on locally correctable and some locally testable binary linear codes [14].…”

Visible Rank and Codes with Locality

“…There is an exponentially large gap between upper and lower bounds on the trade-off between code dimension and code length for -LCCs. The best known code constructions have dimension only ((log ) −1 ) (achieved by generalized Reed-Muller codes or certain lifted codes [GKS13]), whereas the best known upper bound on the dimension of -LCCs is much larger and equals ( ( −2)/( −1) ) [KT00, Woo12,IS20] . Narrowing this huge gap has remained open for over two decades.…”