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Capacity of an infinite family of networks related to the diamond network for fixed alphabet sizes

Sascha Kurz

TL;DR

The article studies the 1-shot capacity of the parameterized network family $\mathcal{N}_s$ under adversarial errors on a subset of edges, focusing on fixed alphabet sizes $|\mathcal{A}|$. It proves a structural result that reduces 1-error correction to a covering problem, enabling exact computations and ILP-based determinations of $\sigma(\mathcal{N}_s,|\mathcal{A}|)$ for several small parameter choices. It also develops linear-programming (LP) bounds via Delsarte's method and Krawtchouk polynomials to tighten upper bounds, showing that a previously conjectured closed form is not generally valid for small alphabets. The work provides a framework for exact and bounded capacity analysis in adversarial network channels and lays groundwork for extending to other 2-level networks like families $A$ and $E$ and their variations.

Abstract

We consider the problem of error correction in a network where the errors can occur only on a proper subset of the network edges. For a generalization of the so-called Diamond Network we consider lower and upper bounds for the network's (1-shot) capacity for fixed alphabet sizes.

Capacity of an infinite family of networks related to the diamond network for fixed alphabet sizes

TL;DR

The article studies the 1-shot capacity of the parameterized network family under adversarial errors on a subset of edges, focusing on fixed alphabet sizes . It proves a structural result that reduces 1-error correction to a covering problem, enabling exact computations and ILP-based determinations of for several small parameter choices. It also develops linear-programming (LP) bounds via Delsarte's method and Krawtchouk polynomials to tighten upper bounds, showing that a previously conjectured closed form is not generally valid for small alphabets. The work provides a framework for exact and bounded capacity analysis in adversarial network channels and lays groundwork for extending to other 2-level networks like families and and their variations.

Abstract

We consider the problem of error correction in a network where the errors can occur only on a proper subset of the network edges. For a generalization of the so-called Diamond Network we consider lower and upper bounds for the network's (1-shot) capacity for fixed alphabet sizes.
Paper Structure (7 sections, 7 theorems, 25 equations, 1 figure, 4 tables)

This paper contains 7 sections, 7 theorems, 25 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structure of network codes for generalized diamond networks $\mathcal{N}_s$
  4. Exact values and lower bounds for the maximum code sizes
  5. Upper bounds for the maximum code sizes
  6. Upper bounds for the code sizes for $\mathcal{N}_s$ based on the linear programming method
  7. Conclusion

Key Result

Lemma 5

Let $out(S)\subseteq \mathcal{U}$. If an outer code $C$ can be $t$-error corrected for a network $\mathcal{N}=(\mathcal{V},\mathcal{E},S,\mathbf{T})$ and an arbitrary alphabet $\mathcal{A}$, then we have $d(C)\ge 2t+1$.

Figures (1)

  • Figure 1: The network $\mathcal{N}_s$.

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • proof
  • Definition 6
  • Example 7
  • Definition 8
  • Lemma 9
  • ...and 13 more