[Paper Review] On the list recoverability of randomly punctured codes

Ben Lund, Aditya Potukuchi|arXiv (Cornell University)|May 4, 2020

Coding theory and cryptography10 references4 citations

TL;DR

This paper demonstrates that random puncturing of codes with good distance over large alphabets yields list recoverable codes beyond the Johnson bound, particularly showing that certain Reed-Solomon codes can achieve improved list recovery performance. The key result is that random puncturings of high-distance codes achieve input and output list sizes of order $\Omega(q)$ and $O(q)$, respectively, at rate $\Omega(q^{-1/2}/\log q)$, surpassing the Johnson bound's limits.

ABSTRACT

We show that a random puncturing of a code with good distance is list recoverable beyond the Johnson bound. In particular, this implies that there are Reed-Solomon codes that are list recoverable beyond the Johnson bound. It was previously known that there are Reed-Solomon codes that do not have this property. As an immediate corollary to our main theorem, we obtain better degree bounds on unbalanced expanders that come from Reed-Solomon codes.

Motivation & Objective

• To show that random puncturings of high-distance codes over large alphabets are list recoverable beyond the Johnson bound.
• To establish that Reed-Solomon codes can be punctured to achieve list recovery performance exceeding the Johnson bound.
• To provide improved degree bounds for unbalanced expanders constructed from Reed-Solomon codes.
• To identify the limitations of random puncturings in achieving high-rate, low-list-size list recoverability for fixed-degree Reed-Solomon codes.
• To propose a conjecture on the existence of high-rate, low-list-size list-recoverable Reed-Solomon codes via random puncturing.

Proposed method

• The authors analyze the list recoverability of randomly punctured codes using probabilistic and combinatorial techniques, focusing on codes with distance close to the maximum possible for their alphabet size.
• They apply concentration bounds and moment estimates to show that with high probability, a random puncturing preserves good list recoverability properties.
• The proof leverages the structure of random subsets of $\mathbb{F}_q$ and uses Markov's inequality to bound collision probabilities in sumset constructions.
• They define and analyze sets $A_i$ of values derived from rational functions of evaluation points to construct large families of codewords within bounded list sizes.
• The analysis includes bounding the size of sets $A_i$ derived from linear combinations of evaluation points, showing that they grow subexponentially under random selection.
• The main technical tool is a comparison to earlier results on list decodability of punctured codes, extending the framework to list recovery.

Experimental results

Research questions

• RQ1 Can random puncturings of high-distance codes achieve list recoverability beyond the Johnson bound?
• RQ2 Do there exist Reed-Solomon codes that are list recoverable with input and output list sizes $\Omega(q)$ and $O(q)$, respectively, at rate $\Omega(q^{-1/2}/\log q)$?
• RQ3 What is the fundamental limit of random puncturing in achieving high-rate, low-list-size list recoverability for fixed-degree Reed-Solomon codes?
• RQ4 Can the rate of list recoverable Reed-Solomon codes be improved beyond $\Omega(\ell^{-1})$ via random puncturing?
• RQ5 Is it possible to construct high-rate, low-list-size list-recoverable Reed-Solomon codes via random puncturing, even for constant-degree polynomials?

Key findings

• A random puncturing of a code with distance $n(1 - q^{-1} - \epsilon^2)$ achieves $(\rho, \ell, \ell(1+\alpha))$-list recoverability with high probability, where $\rho < 1 - (1+\alpha)^{-1/2}$ and $\epsilon = \Theta(q^{-1/2})$.
• The resulting code has rate $\Omega(\epsilon / \log q) = \Omega(q^{-1/2}/\log q)$, which is better than the Johnson bound's $\Omega(\ell^{-1})$ rate for the same list size.
• For $\rho = 0$ and $\alpha = 1$, the code is $(\ell, 2\ell)$-zero-error list recoverable with input and output list sizes $\Omega(q)$.
• The construction yields better degree bounds for unbalanced expanders derived from Reed-Solomon codes.
• The paper shows that random puncturings of degree-$d$ Reed-Solomon codes cannot achieve rate $\omega(1/\log q)$ for $\Omega(q)$-list recovery, indicating a fundamental barrier.
• The authors conjecture that degree-$d$ Reed-Solomon codes on $\mathbb{F}_q$ can be randomly punctured to rate $\Omega_d(1/\log q)$ while being $(\Omega(q), 1/2)$-list recoverable with high probability.