A Geometric View of the Service Rates of Codes Problem and its Application to the Service Rate of the First Order Reed-Muller Codes

Abstract

We investigate the problem of characterizing the service rate region of a coded storage system by introducing a novel geometric approach. The service rate is an important performance metric that measures the number of users that can be simultaneously served by the storage system. One of the most significant advantages of our introduced geometric approach over the existing approaches is that it allows one to derive bounds on the service rate of a code without explicitly knowing the list of all possible recovery sets. As an illustration of the power of our geometric approach, we derive upper bounds on the service rate of the first order Reed-Muller codes and the simplex codes. Then, we show how these upper bounds can be achieved. Moreover, utilizing the same geometric technique, we show that given the service rate region of a code, a lower bound on the minimum distance of the code can be obtained.

Service Rate Region: A New Aspect of Coded Distributed System Design

“…3) Geometric Approach: Finding the service rate region of a given storage scheme is an optimization problem. One natural way to look at this problem is through the geometric approach, introduced in [33], that provides a set of half-spaces whose intersection surrounds the service rate region of a given linear storage scheme. In other words, the geometric approach provides upper bounds (half-spaces) on the sum of each subset of arrival rates in any demand vector (λ 1 , • • • , λ k ) in the service rate region of a linear code in a more straightforward manner in comparison to other approaches.…”

“…We then give a brief description of the approach, and finally, using two examples, we explain how the service rate regions of the binary first order Reed-Muller codes and binary Simplex codes are obtained by the geometric technique. For a formal description and more details, see [33], where this approach is introduced.…”

“…The geometric approach equipped with Proposition 5 can be used to obtain the service rate region of Simplex and Reed-Muller codes. The service rate region of the binary [2 k − 1, k] Simplex code and binary [2 k−1 , k] first order Reed-Muller code are given in [33]. To illustrate the method, we here provide two examples: binary [7,3] Simplex and [8,4] first order Reed-Muller codes.…”

Efficient Storage Schemes for Desired Service Rate Regions

“…By construction R(T ) is a convex polytope and R(T )↓= R(T ), i.e., R(T ) is its own lower set. (See e.g., [11].) Note that in some cases, the values of the function T : 2 [k] → N can be modified without changing R(T ).…”

“…Also, it has been shown that the service rate problem can be viewed as a generalization of the batch codes problem. In [11], we characterized the service rate regions of the binary first order Reed-Muller codes and binary simplex codes using a novel geometric technique. Also, we showed that given the service rate region of a code, a lower bound on the minimum distance of the code can be derived.…”