[Paper Review] High-Rate Regenerating Codes Through Layering

Birenjith Sasidharan, P. Vijay Kumar|NOT FOUND REPOSITORY (Indian Institute of Science Bangalore)|Jan 25, 2013

Advanced Data Storage Technologies11 references4 citations

TL;DR

This paper proposes explicit constructions of high-rate exact-repair regenerating codes with a layered structure, achieving performance beyond the space-sharing line between MSR and MBR points. By introducing canonical and non-canonical layered codes, the authors demonstrate the existence of exact-repair codes at interior points on the storage-repair-bandwidth tradeoff, particularly for parameters like (n, k, d) = (n, n−1, n−1), with low field size and no helper-node computation.

ABSTRACT

In this paper, we provide explicit constructions for a class of exact-repair regenerating codes that possess a layered structure. These regenerating codes correspond to interior points on the storage-repair-bandwidth tradeoff, and compare very well in comparison to scheme that employs space-sharing between MSR and MBR codes. For the parameter set $(n,k,d=k)$ with $n < 2k-1$, we construct a class of codes with an auxiliary parameter $w$, referred to as canonical codes. With $w$ in the range $n-k < w < k$, these codes operate in the region between the MSR point and the MBR point, and perform significantly better than the space-sharing line. They only require a field size greater than $w+n-k$. For the case of $(n,n-1,n-1)$, canonical codes can also be shown to achieve an interior point on the line-segment joining the MSR point and the next point of slope-discontinuity on the storage-repair-bandwidth tradeoff. Thus we establish the existence of exact-repair codes on a point other than the MSR and the MBR point on the storage-repair-bandwidth tradeoff. We also construct layered regenerating codes for general parameter set $(n,k

Motivation & Objective

• To construct explicit, high-rate exact-repair regenerating codes that outperform space-sharing between MSR and MBR points.
• To demonstrate the existence of exact-repair codes at interior points on the storage-repair-bandwidth tradeoff, challenging prior claims that such points are unachievable.
• To design codes with locality where local repair is achieved via layered regenerating codes, enhancing repair efficiency.
• To minimize field size requirements while maintaining high-rate performance and exact repair.
• To ensure no computation is required at helper nodes, enabling simple data transfer during repair.

Proposed method

• Introduces a layered code construction using combinatorial designs and orbit counting over finite fields.
• Employs a canonical code structure for parameters (n, k, d) with w ∈ (n−k, k), where w is an auxiliary parameter controlling code rate and repair efficiency.
• Uses linearized polynomials and field extensions to map message vectors to codewords via evaluation over elements in F_{q^N}.
• Applies Möbius inversion to count equivalence classes of specific sizes, enabling construction of codes with desired layering and symmetry.
• Constructs codes with locality by partitioning evaluation vectors and encoding each subset using canonical codes.
• Ensures no computation at helper nodes by transferring data directly, resulting in 'help-by-transfer' regenerating codes.

Experimental results

Research questions

• RQ1 Can exact-repair regenerating codes be constructed at interior points on the storage-repair-bandwidth tradeoff, beyond the MSR and MBR extremes?
• RQ2 How can high-rate regenerating codes be designed with low field size and no helper-node computation?
• RQ3 What is the achievable performance of layered codes compared to space-sharing between MSR and MBR codes?
• RQ4 Can codes with locality be constructed such that local repair is performed using layered regenerating codes?
• RQ5 What combinatorial and algebraic techniques enable the construction of such codes with guaranteed optimality in terms of minimum distance?

Key findings

• Canonical codes achieve an interior point on the storage-repair-bandwidth tradeoff for (n, k, d) = (n, n−1, n−1), proving the existence of exact-repair codes beyond MSR and MBR points.
• For parameters with w ∈ (n−k, k), the canonical codes operate significantly better than the space-sharing line, with field size greater than w + n − k.
• The (n, n−1, n−1)-canonical code with w = d−1 achieves optimality, matching the theoretical minimum distance bound.
• The accumulation profile of the canonical code shows α = w+1 and a_k = w, confirming it lies between the MSR point and the next slope-discontinuity point.
• Non-canonical layered codes are constructed for general (n, k, d) with d < k, achieving better performance than space-sharing but requiring higher field size.
• Codes with locality are constructed where local codes are based on canonical layered codes, achieving optimal minimum distance for given scalar dimension.