[Paper Review] MSR Codes with Linear Field Size and Smallest Sub-packetization for Any Number of Helper Nodes

Guodong Li, Ningning Wang+2 more|arXiv (Cornell University)|Mar 18, 2023

Cooperative Communication and Network Coding4 citations

TL;DR

This paper presents the first explicit construction of optimal-access MSR codes with the smallest possible sub-packetization $σ^{\lceil n/\sigma\rceil}$ for any number of helper nodes $d$ between $k+1$ and $n-1$, achieving linear field size. The key innovation lies in reducing $\binom{n}{r}$ global determinant constraints to $O_s(n)$ local ones via a carefully designed parity check matrix, resolving a long-standing open problem in distributed storage coding theory.

ABSTRACT

The sub-packetization $\ell$ and the field size $q$ are of paramount importance in the MSR array code constructions. For optimal-access MSR codes, Balaji et al. proved that $\ell\geq s^{\left\lceil n/s ight ceil}$, where $s = d-k+1$. Rawat et al. showed that this lower bound is attainable for all admissible values of $d$ when the field size is exponential in $n$. After that, tremendous efforts have been devoted to reducing the field size. However, till now, reduction to linear field size is only available for $d\in\{k+1,k+2,k+3\}$ and $d=n-1$. In this paper, we construct the first class of explicit optimal-access MSR codes with the smallest sub-packetization $\ell = s^{\left\lceil n/s ight ceil}$ for all $d$ between $k+1$ and $n-1$, resolving an open problem in the survey (Ramkumar et al., Foundations and Trends in Communications and Information Theory: Vol. 19: No. 4). We further propose another class of explicit MSR code constructions (not optimal-access) with even smaller sub-packetization $s^{\left\lceil n/(s+1) ight ceil }$ for all admissible values of $d$, making significant progress on another open problem in the survey. Previously, MSR codes with $\ell=s^{\left\lceil n/(s+1) ight ceil }$ and $q=O(n)$ were only known for $d=k+1$ and $d=n-1$. The key insight that enables a linear field size in our construction is to reduce $\binom{n}{r}$ global constraints of non-vanishing determinants to $O_s(n)$ local ones, which is achieved by carefully designing the parity check matrices.

Motivation & Objective

• To resolve the open problem of constructing optimal-access MSR codes with minimal sub-packetization for all $d \in \{k+1, \dots, n-1\}$.
• To achieve linear field size in MSR code constructions, which has been limited to special cases ($d \in \{k+1,k+2,k+3,n-1\}$) in prior work.
• To reduce the exponential sub-packetization of prior MSR codes to the theoretical lower bound $\sigma^{\lceil n/\sigma\rceil}$, where $\sigma = d - k + 1$.
• To address the second open problem in the survey (Ramkumar et al., 2023) by constructing MSR codes with even smaller sub-packetization $\sigma^{\lceil n/(\sigma+1)\rceil}$ and $q = O(n)$.

Proposed method

• Design a parity check matrix that transforms $\binom{n}{r}$ global determinant constraints into $O_s(n)$ local constraints, enabling linear field size.
• Use a structured construction based on group-wise assignment of variables $x_{i}$ to elements in $\mathbb{F}_q$, ensuring non-vanishing determinants via polynomial interpolation.
• Apply the Combinatorial Nullstellensatz and Corollary 3 to select evaluation points $\lambda_i$ from $\mathbb{F}_q$ such that key determinant polynomials remain non-zero.
• Construct the code using a homogeneous polynomial $D(x_{[s^2]})$ of degree $s^2(s-1)2^{s-3}$, with bounded individual variable degrees $(s-1)2^{s-2}$, to ensure non-vanishing determinant under evaluation.
• Leverage primitive roots in $\mathbb{F}_q$ to assign $\lambda_i = \alpha^i$ such that $D(1,\alpha,\dots,\alpha^{s^2-1}) \neq 0$, satisfying all local constraints.
• Prove that the resulting code achieves optimal access and minimal sub-packetization by verifying that the repair bandwidth matches the theoretical lower bound.

Experimental results

Research questions

• RQ1 Can optimal-access MSR codes be constructed with sub-packetization matching the theoretical lower bound $\sigma^{\lceil n/\sigma\rceil}$ for all $d \in \{k+1, \dots, n-1\}$?
• RQ2 Is it possible to achieve linear field size in such constructions beyond the known special cases?
• RQ3 Can sub-packetization be further reduced to $\sigma^{\lceil n/(\sigma+1)\rceil}$ while maintaining linear field size for all $d$?
• RQ4 How can the global determinant constraints in MSR code design be reduced to local, manageable constraints to enable efficient construction?

Key findings

• The paper constructs the first explicit optimal-access MSR codes with sub-packetization $\sigma^{\lceil n/\sigma\rceil}$ for all $d \in \{k+1, \dots, n-1\}$, achieving the theoretical minimum.
• The construction achieves linear field size $q = O(n)$, resolving the open problem of extending linear field size beyond $d \in \{k+1,k+2,k+3,n-1\}$.
• An additional construction achieves even smaller sub-packetization $\sigma^{\lceil n/(\sigma+1)\rceil}$ with $q = O(n)$, improving on prior results that only achieved this for $d=k+1$ and $d=n-1$.
• The key technical insight is reducing $\binom{n}{r}$ global determinant constraints to $O_s(n)$ local ones via a structured parity check matrix design.
• The proof relies on evaluating multivariate polynomials over $\mathbb{F}_q$ using primitive roots and the Combinatorial Nullstellensatz to ensure non-vanishing determinants under constraints.
• The construction is valid for $q \geq ns + (s-1)2^{s-2}$, ensuring sufficient field size to satisfy all local constraints for any $n$ and fixed $s = d - k + 1$.