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arxiv: 2604.21544 · v1 · submitted 2026-04-23 · 💻 cs.IT · math.IT

Design of MDP Convolutional Codes and Maximally Recoverable Codes Through the Lens of Matrix Completion

Sakshi Dang , Julia Lieb , Pedro Soto , Alex Sprintson This is my paper

Pith reviewed 2026-05-08 13:50 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords matrix completionmaximally recoverable codeslocally recoverable codesMDP convolutional codesgenerator matricessubfield elementserror-correcting codescode constructions
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The pith

Matrix completion problems yield general constructions for both maximally recoverable LRCs and MDP convolutional codes with sparse generator matrices over small subfields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper frames the design of maximally recoverable locally recoverable codes and maximum distance profile convolutional codes as instances of the matrix completion problem. It shows that solutions to these completion tasks produce generator matrices that are sparse and contain many entries from a small subfield of a larger extension field. This common structure unifies the construction methods for the two code families. A reader would care because the approach offers a systematic route to codes that achieve strong local recovery or distance guarantees while keeping encoding operations simple and field sizes modest.

Core claim

By casting the exact recovery conditions of MR-LRCs and the distance-profile requirements of MDP convolutional codes as matrix completion problems, the authors obtain constructions whose generator matrices are sparse and have a large number of entries belonging to a small subfield.

What carries the argument

The matrix completion problem, which formulates code design as the task of filling unspecified entries in a matrix so that the completed matrix satisfies prescribed rank or distance conditions.

If this is right

  • The resulting codes admit lower-complexity encoding and decoding because their generator matrices are sparse.
  • Implementation over smaller effective alphabets becomes feasible due to the predominance of small-subfield entries.
  • The same completion framework can be reused for other matrix-described coding problems that impose rank or distance constraints.
  • Explicit or algorithmic constructions become available once the relevant matrix completion instances are solved.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The shared matrix-completion view may expose previously unnoticed trade-offs between local recoverability and convolutional distance properties.
  • Efficient solvers for the underlying completion problems would immediately yield practical codes for large block lengths or high locality.
  • The sparsity and subfield structure could translate into hardware-friendly realizations for distributed storage or streaming applications.
  • Similar matrix-completion reductions might apply to network coding or index coding problems that also rely on rank conditions.

Load-bearing premise

Solutions to the matrix-completion problems can be found that simultaneously satisfy the exact recovery and distance-profile requirements of MR-LRCs and MDP convolutional codes without introducing additional constraints that destroy the desired properties.

What would settle it

A concrete parameter set for which no matrix completion exists that meets the recovery or distance-profile conditions while preserving sparsity and the small-subfield restriction would disprove the claimed generality of the technique.

read the original abstract

The matrix completion problem provides a unifying lens through which many fundamental problems in coding theory can be viewed. In this paper, we investigate Locally Recoverable Codes (LRCs) with Maximal Recoverability (MR) and Maximum Distance Profile (MDP) convolutional codes in the framework of matrix completion. In particular, we present techniques that are general enough to provide constructions for both types of codes. A common feature of our code constructions is the sparsity of their generator matrices and the property that a large number of the entries of the generator matrices are elements of a small subfield of a larger extension field.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript frames the design of maximally recoverable locally recoverable codes (MR-LRCs) and maximum distance profile (MDP) convolutional codes as matrix completion problems. It claims to offer general techniques for constructing such codes, characterized by sparse generator matrices with a large number of entries drawn from a small subfield of a larger extension field.

Significance. If the proposed techniques successfully yield valid constructions satisfying the recovery and distance properties, this work could offer a novel unified approach to code design in coding theory, potentially simplifying the construction of codes with desirable locality and distance properties. The emphasis on sparsity and subfield elements may have practical benefits for encoding and decoding complexity.

major comments (1)
  1. Abstract: The central claim that general techniques exist to solve the matrix-completion problems for both MR-LRCs and MDP convolutional codes while preserving exact recovery/distance-profile conditions rests on the assertion that the intersection of the algebraic variety (enforcing rank conditions) and the linear/sparsity/subfield constraints is non-empty. No existence argument, dimension count, or explicit construction verifying this compatibility is supplied, leaving the feasibility of the claimed constructions unestablished.
minor comments (1)
  1. The abstract and title would benefit from explicit parameter ranges (e.g., field extension degree, locality, or block length) for which the constructions are asserted to work.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to strengthen the abstract's claim regarding the feasibility of the matrix-completion constructions. We address this point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: Abstract: The central claim that general techniques exist to solve the matrix-completion problems for both MR-LRCs and MDP convolutional codes while preserving exact recovery/distance-profile conditions rests on the assertion that the intersection of the algebraic variety (enforcing rank conditions) and the linear/sparsity/subfield constraints is non-empty. No existence argument, dimension count, or explicit construction verifying this compatibility is supplied, leaving the feasibility of the claimed constructions unestablished.

    Authors: We agree that the abstract would benefit from an explicit reference to the existence of solutions. The body of the manuscript supplies general techniques that construct the desired codes by completing partial generator matrices with entries from a small subfield while enforcing the necessary rank conditions for maximal recoverability (MR-LRCs) and maximum distance profile (MDP convolutional codes). These techniques are presented in Sections 3 and 4, where we show how the sparsity pattern and subfield constraints are compatible with the algebraic variety by providing explicit completion rules that preserve the required ranks. To make this clearer, we will revise the abstract to include a brief statement noting that the constructions demonstrate the non-emptiness of the intersection via concrete matrix-completion procedures. revision: yes

Circularity Check

0 steps flagged

No circularity: constructions framed as independent matrix-completion problems

full rationale

The abstract presents the matrix-completion lens as a unifying framework for constructing MR-LRCs and MDP convolutional codes, emphasizing sparsity and small-subfield entries in generator matrices. No equations, parameter-fitting steps, or self-citations appear in the provided text that would reduce any claimed construction or existence result to a tautological input by definition. The techniques are described as general enough to satisfy the respective recovery and distance-profile conditions without any visible self-definitional loop, fitted-input prediction, or load-bearing self-citation chain. The derivation chain therefore remains self-contained against external algebraic and coding-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit axioms, free parameters, or invented entities are stated in the abstract; the matrix-completion framing itself is treated as an external tool rather than a new postulate.

pith-pipeline@v0.9.0 · 5397 in / 954 out tokens · 23661 ms · 2026-05-08T13:50:38.699237+00:00 · methodology

discussion (0)

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