Johnson Type Bounds on Constant Dimension Codes

TL;DR

This paper investigates the optimality of constant dimension codes in network coding, establishing bounds and showing Steiner structures as optimal codes that meet these bounds.

Contribution

It introduces Johnson type bounds on constant dimension codes and proves Steiner structures are optimal, achieving these bounds.

Findings

Steiner structures are optimal constant dimension codes.

Johnson type bound II improves upon the Wang-Xing-Safavi-Naini bound.

Known Steiner structures achieve the Johnson bounds.

Abstract

Very recently, an operator channel was defined by Koetter and Kschischang when they studied random network coding. They also introduced constant dimension codes and demonstrated that these codes can be employed to correct errors and/or erasures over the operator channel. Constant dimension codes are equivalent to the so-called linear authentication codes introduced by Wang, Xing and Safavi-Naini when constructing distributed authentication systems in 2003. In this paper, we study constant dimension codes. It is shown that Steiner structures are optimal constant dimension codes achieving the Wang-Xing-Safavi-Naini bound. Furthermore, we show that constant dimension codes achieve the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures. Then, we derive two Johnson type upper bounds, say I and II, on constant dimension codes. The Johnson type bound II slightly improves on the Wang-Xing-Safavi-Naini bound. Finally, we point out that a family of known Steiner structures is actually a family of optimal constant dimension codes achieving both the Johnson type bounds I and II.