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Sliding Window Adversarial Channels

Bikash Kumar Dey, Sidharth Jaggi, Michael Langberg, Anand D. Sarwate, Yihan Zhang

TL;DR

This paper introduces sliding window constraints for oblivious adversaries in arbitrarily varying channels (AVCs), defining windowed AVCs with per-window cost constraints and analyzing capacity under maximum error. The authors show that, for window lengths scaling as $x_{\mathrm{win}}(n), s_{\mathrm{win}}(n) = [\omega(\log n), o(n)]$, the windowed capacity equals the standard list-decoding capacity $\Caplist(\AVC)$ when the underlying AVC is non-symmetrizable, and they extend these results to cases where $x_{\mathrm{win}} > s_{\mathrm{win}}$ via a transformed auxiliary channel $\AVC'$. A three-phase achievability scheme (list-decodable Phase I, guard window Phase II, and hash-based Phase III) along with expurgation ensures per-window constraints, while the converse leverages an i.i.d. jammer that respects window constraints to bound rates by $\Caplist(\AVC)$. The paper also provides a binary case study showing capacity expressions and discusses the open problem of capacity when the base AVC is symmetrizable, highlighting the nuanced boundary between ECN-symmetrizability and stronger notions. Overall, the results connect sliding-window energy constraints to list-decoding limits and propose technique extensions such as interleaving and disambiguation for improved performance under asymmetric window constraints.

Abstract

In an arbitrarily varying channel (AVC), the channel has a state which is under the control of an adversarial jammer and the corresponding capacities are often functions of the "power" constraints on the transmitter and jammer. In this paper we propose a model in which the constraints must hold almost surely over contiguous subsequences of the codeword and state, which we call a sliding window constraint. We study oblivious jammers and codes with stochastic encoding under maximum probability of error. We show that this extra limitation on the jammer is beneficial for the transmitter: in some cases, the capacity for unique decoding with a sliding window constraint is equal to the capacity for list decoding in the standard model without sliding windows, roughly implying that the addition of window constraints reduces list decoding to unique decoding. The list decoding capacity in the standard model can be strictly larger than the unique decoding capacity.

Sliding Window Adversarial Channels

TL;DR

This paper introduces sliding window constraints for oblivious adversaries in arbitrarily varying channels (AVCs), defining windowed AVCs with per-window cost constraints and analyzing capacity under maximum error. The authors show that, for window lengths scaling as , the windowed capacity equals the standard list-decoding capacity when the underlying AVC is non-symmetrizable, and they extend these results to cases where via a transformed auxiliary channel . A three-phase achievability scheme (list-decodable Phase I, guard window Phase II, and hash-based Phase III) along with expurgation ensures per-window constraints, while the converse leverages an i.i.d. jammer that respects window constraints to bound rates by . The paper also provides a binary case study showing capacity expressions and discusses the open problem of capacity when the base AVC is symmetrizable, highlighting the nuanced boundary between ECN-symmetrizability and stronger notions. Overall, the results connect sliding-window energy constraints to list-decoding limits and propose technique extensions such as interleaving and disambiguation for improved performance under asymmetric window constraints.

Abstract

In an arbitrarily varying channel (AVC), the channel has a state which is under the control of an adversarial jammer and the corresponding capacities are often functions of the "power" constraints on the transmitter and jammer. In this paper we propose a model in which the constraints must hold almost surely over contiguous subsequences of the codeword and state, which we call a sliding window constraint. We study oblivious jammers and codes with stochastic encoding under maximum probability of error. We show that this extra limitation on the jammer is beneficial for the transmitter: in some cases, the capacity for unique decoding with a sliding window constraint is equal to the capacity for list decoding in the standard model without sliding windows, roughly implying that the addition of window constraints reduces list decoding to unique decoding. The list decoding capacity in the standard model can be strictly larger than the unique decoding capacity.
Paper Structure (10 sections, 2 theorems, 11 equations)

This paper contains 10 sections, 2 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Channel Model
  3. Main Results and Motivating Example
  4. Case study: The input-weight-constrained bit-flip AVC
  5. Proof Sketch for Theorem \ref{['thm:eq']}
  6. Theorem \ref{['thm:eq']}: Achievability
  7. Theorem \ref{['thm:eq']}: Converse
  8. Proof sketch for Theorem \ref{['thm:wxgtws']}
  9. Discussion
  10. Continuing the case study of Section \ref{['sec:case']}

Key Result

theorem 1

Let $\AVC = (W_{\ry|\rx,\rs}, \xtypes, \stypes)$ be a windowless non-symmetrizable AVC. Let $\winAVC=(W_{\ry|\rx,\rs}, \xtypes, \stypes,\xwin,\swin)$ be a corresponding windowed AVC for which $\xwin,\swin \in [\omega(\log{n}),o(n)]$. Then we have $\Capwin(\winAVC) = \Caplist(\AVC)$.

Theorems & Definitions (5)

  • definition 1: ECN-symmetrizability CN:88constraints:deterministic
  • theorem 1
  • remark 1
  • theorem 2: $\xwin(n) > \swin(n)$
  • definition 2: Strong-symmetrizability