On the Parameters of Codes for Data Access

TL;DR

The paper tackles the problem of designing codes for coded distributed storage to realize a prescribed service rate region. It formalizes the system with a generator matrix $G \in \mathbb{F}_q^{k\times n}$, recovery sets, and a service rate region $\Lambda(G)$, and studies two main questions: (1) for fixed alphabet size $q$, what is the minimum number of servers $n$ (denoted $n_q(S)$) needed to include a given demand set $S$ in $\Lambda(G)$, and (2) for fixed $n$, what is the minimum field size $q$ (denoted $q_n(S)$) required to include $S$ in $\Lambda(G)$? The authors develop two analytical approaches, including a mixed-integer optimization framework to certify containment of $S$ in $\Lambda(G)$ and a projective-geometry-based bound derivation, leading to general upper and lower bounds and exact results in key cases. For sets of the form $S=\{X_i e_i\}$ they establish a field-size dependent lower bound $n_q(S) \ge \left\lceil \sum_{i=1}^k q^{1-i} X_i \right\rceil$ and, in the binary case, a constructive upper bound $n_2(S) \le \sum_{i=1}^k \left\lceil \frac{X_i}{2^{i-1}} \right\rceil$, with tightness under certain divisibility conditions and connections to simplex codes. These results guide practical design under computational and field-size constraints and are complemented by illustrative examples and nonexistence results for certain parameter regimes.

Abstract

This paper studies two crucial problems in the context of coded distributed storage systems directly related to their performance: 1) for a fixed alphabet size, determine the minimum number of servers the system must have for its service rate region to contain a prescribed set of points; 2) for a given number of servers, determine the minimum alphabet size for which the service rate region of the system contains a prescribed set of points. The paper establishes rigorous upper and lower bounds, as well as code constructions based on techniques from coding theory, optimization, and projective geometry.

Figures (2)

• Figure 1: The service rate region for Example \ref{['ex1']}.
• Figure 2: Set $S =\{(1,1),(2,0)\}$ is in $\Lambda(G_i)$ for $i \in \{1,2,3\}$ and $\Lambda(G_3) = \textnormal{conv}(S)$. Set $\textnormal{conv}(S)$ is shown in (a) and (b) with a different color. Observe that 1) any code whose service rate region contains the points in $S$ must contain the minimal region shown in (c), and 2) regions that are proper supersets of the minimal are achievable without enlarging the field size.

Theorems & Definitions (25)

• Definition 1
• Example 1
• Proposition 1
• Definition 2
• Example 2
• Example 3
• Remark 1
• Proposition 2
• Definition 3
• Example 4