Tight List-Sizes for Oblivious AVCs under Constraints

TL;DR

This paper determines the exact minimum list-size needed for reliable communication over oblivious adversarial channels with constraints, generalizing classical results and addressing open questions in the field.

Contribution

It provides a precise characterization of list-decoding capacity for oblivious AVCs with constraints, including new bounds and a subcode construction method.

Findings

Exact minimum list-size for reliable communication identified

Lower bound on list-decoding capacity established

Conditional upper bound proved under a conjecture

Abstract

We study list-decoding over adversarial channels governed by oblivious adversaries (a.k.a. oblivious Arbitrarily Varying Channels (AVCs)). This type of adversaries aims to maliciously corrupt the communication without knowing the actual transmission from the sender. For any oblivious AVCs potentially with constraints on the sender's transmitted sequence and the adversary's noise sequence, we determine the exact value of the minimum list-size that can support a reliable communication at positive rate. This generalizes a classical result by Hughes (IEEE Transactions on Information Theory, 1997) and answers an open question posed by Sarwate and Gastpar (IEEE Transactions on Information Theory, 2012). A lower bound on the list-decoding capacity (whenever positive) is presented. Under a certain combinatorial conjecture, we also prove a matching upper bound. En route to a tight characterization of the list-decoding capacity, we propose a method for subcode construction towards the resolution of the combinatorial conjecture.