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Let's Have Both! Optimal List-Recoverability via Alphabet Permutation Codes

Sergey Komech, Jonathan Mosheiff

TL;DR

This work introduces alphabet-permutation (AP) codes, a structured yet non-additive family generated by iterative coordinate-wise permutations. By identifying a $\mathcal{B}$-mixing property for permutation distributions, the authors prove that AP codes are almost surely $(\rho,\ell,L)$-list-recoverable with rates matching the Elias Bound while using only polynomial randomness. They show RACs fit inside the AP framework as a special case and extend the analysis beyond additivity to achieve optimal list-recovery via $\ell$-wise independence, with partial derandomization through near-independence techniques. The approach yields codes that combine strong list-recovery performance with structural efficiency, representing a significant step toward explicit constructions that rival fully random codes. The results have potential implications for pseudorandom object construction and related algorithmic applications that rely on robust list-recovery properties.

Abstract

We introduce alphabet-permutation (AP) codes, a new family of error-correcting codes defined by iteratively applying random coordinate-wise permutations to a fixed initial word. A special case recovers random additive codes and random binary linear codes, where each permutation corresponds to an additive shift over a finite field. We show that when these permutations are drawn from a suitably ``mixing'' distribution, the resulting code is almost surely list-recoverable with list size proportional to the inverse of the gap to capacity. Compared to any linear code, our construction achieves exponentially smaller list sizes at the same rate. Previously, only fully random codes were known to attain such parameters, requiring exponentially many random bits and offering no structure. In contrast, AP codes are structured and require only polynomially many random bits -- providing the first such construction to match the list-recovery guarantees of random codes.

Let's Have Both! Optimal List-Recoverability via Alphabet Permutation Codes

TL;DR

This work introduces alphabet-permutation (AP) codes, a structured yet non-additive family generated by iterative coordinate-wise permutations. By identifying a -mixing property for permutation distributions, the authors prove that AP codes are almost surely -list-recoverable with rates matching the Elias Bound while using only polynomial randomness. They show RACs fit inside the AP framework as a special case and extend the analysis beyond additivity to achieve optimal list-recovery via -wise independence, with partial derandomization through near-independence techniques. The approach yields codes that combine strong list-recovery performance with structural efficiency, representing a significant step toward explicit constructions that rival fully random codes. The results have potential implications for pseudorandom object construction and related algorithmic applications that rely on robust list-recovery properties.

Abstract

We introduce alphabet-permutation (AP) codes, a new family of error-correcting codes defined by iteratively applying random coordinate-wise permutations to a fixed initial word. A special case recovers random additive codes and random binary linear codes, where each permutation corresponds to an additive shift over a finite field. We show that when these permutations are drawn from a suitably ``mixing'' distribution, the resulting code is almost surely list-recoverable with list size proportional to the inverse of the gap to capacity. Compared to any linear code, our construction achieves exponentially smaller list sizes at the same rate. Previously, only fully random codes were known to attain such parameters, requiring exponentially many random bits and offering no structure. In contrast, AP codes are structured and require only polynomially many random bits -- providing the first such construction to match the list-recovery guarantees of random codes.

Paper Structure

This paper contains 14 sections, 11 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. Alphabet-Permutation Codes and the Road to List-Recovery
  4. Motivation and Overview
  5. Definition and Encoding
  6. Iterative Generation.
  7. Additive Codes as a Special Case
  8. Generalizing Beyond Additivity.
  9. From Additive Codes to List-Recoverable AP Codes
  10. List-Recovery via Mixing Ensembles
  11. Achieving Mixing via Independent Permutations
  12. Proof of \ref{['thm:introMainLR']}: Partial Derandomization via Near-Independence
  13. Mixing Implies $\mathcal{B}$-Avoidance---Proof of \ref{['prop:mixing']}
  14. Acknowledgement

Key Result

Theorem 1.1

Let $q,\ell,L,n \in \mathbb{N}$ with $q \ge 2$ and $\ell < \min\{q,L\}$. Suppose $0 \le \rho \le 1 - \frac{\ell}{q}$, and let $\eta \ge \frac{2\,\log_q\!\bigl(2n/\ln2\bigr)}{n}$ and $\delta > 0$. Define and assume $k := R \cdot \log_2 q \cdot n$ is an integer. Then there exists a random code ensemble $\mathcal{C} \subseteq \Sigma^n$ (with $|\Sigma| = q$) of rate $R$, using only $O(nk(\ell\log q +

Figures (1)

  • Figure 1: Computing $\operatorname{Enc}_\Pi(z)$ for $n=6$, $k=3$ and $z = (1,0,1)$.

Theorems & Definitions (34)

  • Theorem 1.1: Main Theorem
  • Definition 2.1: Alphabet-Permutation Code
  • Remark 2.2: Multiset semantics
  • Definition 2.3: Random AP Codes
  • Example 2.4
  • Definition 2.5: $\mathcal{B}$-mixing distribution
  • Proposition 2.5: $\cB$-mixing implies small intersections
  • proof : Proof sketch for \ref{['prop:mixing']}
  • Corollary 2.6: $\mathcal{B}_{\rho,\ell}$-mixing implies list-recoverability
  • proof : Proof of \ref{['cor:mixingToLR']} given \ref{['prop:mixing']}
  • ...and 24 more