[Paper Review] Codes over integers, and the singularity of random matrices with large entries

Sankeerth Rao Karingula, Shachar Lovett|arXiv (Cornell University)|Jan 1, 2020

Cooperative Communication and Network Coding2 citations

TL;DR

This paper introduces a novel framework for constructing Maximum Distance Separable (MDS) codes over the integers, demonstrating that such codes with linear rate and distance can be achieved using a constant-sized alphabet—contrasting sharply with finite fields, which require linear field size. The core technical contribution is a new bound showing that the singularity probability of an $ n \times n $ random matrix with i.i.d. entries uniformly chosen from $ \{-m, \ldots, m\} $ is at most $ m^{-cn} $ for some absolute constant $ c > 0 $, establishing a key probabilistic foundation for the code construction.

ABSTRACT

The prototypical construction of error correcting codes is based on linear codes over finite fields. In this work, we make first steps in the study of codes defined over integers. We focus on Maximum Distance Separable (MDS) codes, and show that MDS codes with linear rate and distance can be realized over the integers with a constant alphabet size. This is in contrast to the situation over finite fields, where a linear size finite field is needed. The core of this paper is a new result on the singularity probability of random matrices. We show that for a random $n imes n$ matrix with entries chosen independently from the range $\{-m,\ldots,m\}$, the probability that it is singular is at most $m^{-cn}$ for some absolute constant $c>0$.

Motivation & Objective

• To explore the feasibility of constructing MDS codes over the integers, a setting that differs fundamentally from classical finite-field-based coding theory.
• To determine whether MDS codes with linear rate and distance can be realized over the integers using a constant-sized alphabet, in contrast to the linear field size required over finite fields.
• To establish a probabilistic foundation for code construction by analyzing the singularity behavior of random integer matrices with large entries.

Proposed method

• Define MDS codes over the ring of integers, focusing on linear codes with integer coefficients and constant alphabet size.
• Use probabilistic methods to analyze the likelihood that an $ n \times n $ random matrix with entries in $ \{-m, \ldots, m\} $ is singular.
• Establish a non-trivial upper bound on the singularity probability, showing it decays as $ m^{-cn} $ for some absolute constant $ c > 0 $, using tools from random matrix theory and combinatorics.
• Leverage the singularity bound to prove that suitable generator matrices exist over $ \mathbb{Z} $, enabling the construction of MDS codes with desired rate and distance.
• Draw a parallel between the existence of such codes and the non-singularity of random integer matrices, using concentration and anti-concentration techniques.

Experimental results

Research questions

• RQ1 Can MDS codes with linear rate and linear distance be constructed over the integers using a constant-sized alphabet, without requiring field size to grow with block length?
• RQ2 What is the probability that a random $ n \times n $ matrix with i.i.d. entries uniformly chosen from $ \{-m, \ldots, m\} $ is singular, and how does this probability scale with $ m $ and $ n $?
• RQ3 How does the singularity probability of random integer matrices compare to that of random matrices over finite fields, and what implications does this have for coding theory over rings?
• RQ4 Can the singularity bound be used to guarantee the existence of full-rank generator matrices for MDS codes over $ \mathbb{Z} $?
• RQ5 What structural properties of integer matrices allow for the construction of MDS codes with optimal parameters?

Key findings

• MDS codes with linear rate and linear distance can be constructed over the integers using a constant-sized alphabet, which is a significant departure from the finite field case where field size must grow linearly with block length.
• The singularity probability of an $ n \times n $ random matrix with entries chosen independently and uniformly from $ \{-m, \ldots, m\} $ is at most $ m^{-cn} $ for some absolute constant $ c > 0 $, independent of $ n $.
• This bound implies that such matrices are non-singular with high probability when $ m $ is sufficiently large relative to $ n $, enabling the construction of full-rank generator matrices.
• The result establishes a new connection between coding theory over rings and the singularity of large random integer matrices.
• The existence of such codes over $ \mathbb{Z} $ is supported by the fact that the probability of encountering a singular matrix in the construction process is exponentially small in $ n $, ensuring high reliability.
• The technical contribution provides a quantitative guarantee on matrix non-singularity that underpins the feasibility of the code construction.