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Lower Bounds on Conversion Bandwidth for MDS Convertible Codes in Split Regime

Lewen Wang, Sihuang Hu

TL;DR

This work introduces a linear-algebraic framework to bound the bandwidth required when converting between MDS codes in the split regime of convertible codes. By establishing an inclusion relation between restricted generator-matrix column spaces, the authors transform the bandwidth problem into a linear-programming task that yields closed-form lower bounds on read bandwidth. The bounds improve on prior results in several parameter regimes and are tight in the regime $r^F \le r^I \le k^F$, matching existing construction performance. An explicit example demonstrates a bound-achieving conversion, underscoring the practical relevance of the theoretical framework.

Abstract

We propose several new lower bounds on the bandwidth costs of MDS convertible codes using a linear-algebraic framework. The derived bounds improve previous results in certain parameter regimes and match the bandwidth cost of the construction proposed by Maturana and Rashmi (2022 IEEE International Symposium on Information Theory) for $r^F\le r^I\le k^F$, implying that our bounds are tight in this case.

Lower Bounds on Conversion Bandwidth for MDS Convertible Codes in Split Regime

TL;DR

This work introduces a linear-algebraic framework to bound the bandwidth required when converting between MDS codes in the split regime of convertible codes. By establishing an inclusion relation between restricted generator-matrix column spaces, the authors transform the bandwidth problem into a linear-programming task that yields closed-form lower bounds on read bandwidth. The bounds improve on prior results in several parameter regimes and are tight in the regime , matching existing construction performance. An explicit example demonstrates a bound-achieving conversion, underscoring the practical relevance of the theoretical framework.

Abstract

We propose several new lower bounds on the bandwidth costs of MDS convertible codes using a linear-algebraic framework. The derived bounds improve previous results in certain parameter regimes and match the bandwidth cost of the construction proposed by Maturana and Rashmi (2022 IEEE International Symposium on Information Theory) for , implying that our bounds are tight in this case.

Paper Structure

This paper contains 8 sections, 5 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Preliminaries
  4. Organization of This Paper
  5. An Inclusion Relation Between Generator Matrices
  6. Lower Bounds on Bandwidth Cost
  7. Conclusion
  8. An Example Achieving Our Bound in Theorem 3

Key Result

Lemma 1

Let $(\mathcal{C}^I,\mathcal{C}^F)$ be an $[n^{I},k^{I};n^{F},k^{F};\ell]$ MDS convertible code in the split regime with $k^I=\lambda k^F$. Then the number of unchanged symbols is at most $k^I$, and this bound is achievable.

Figures (1)

  • Figure 1: The red line corresponds to the boundary of the first constraint, and the blue line corresponds to that of the second constraint.

Theorems & Definitions (15)

  • Definition 1: MDS Convertible Codes maturana2022convertible
  • Definition 2: Stable Convertible Codematurana2022convertible
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • ...and 5 more