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On Binary Shadow Codes

Amir Tasbihi, Frank R. Kschischang

Abstract

We generalize the shadow codes of Cherubini and Micheli to include basic polynomials having arbitrary degree, and show that restricting basic polynomials to have degree one or less can result in improved lower bounds on the minimum distance of the code. However, even these improved lower bounds suggest that shadow codes have considerably inferior distance-rate characteristics compared with the concatenation of a Reed-Solomon outer code and a first-order Reed-Muller inner code.

On Binary Shadow Codes

Abstract

We generalize the shadow codes of Cherubini and Micheli to include basic polynomials having arbitrary degree, and show that restricting basic polynomials to have degree one or less can result in improved lower bounds on the minimum distance of the code. However, even these improved lower bounds suggest that shadow codes have considerably inferior distance-rate characteristics compared with the concatenation of a Reed-Solomon outer code and a first-order Reed-Muller inner code.
Paper Structure (12 sections, 8 theorems, 35 equations, 4 figures)

This paper contains 12 sections, 8 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Absolutely Irreducible Polynomials
  4. Delsarte--Goethals Codes
  5. Binary Shadow Codes
  6. Definition and Properties
  7. Shadow Codes of Degree 2
  8. Shadow Codes of Degree at Most 1
  9. RS-RM Concatenation
  10. RS-RM Concatenation versus Shadow Codes
  11. Results
  12. The Regime where $\delta \to \frac{1}{2}$ as $n \to \infty$

Key Result

Lemma 1

Any irreducible polynomial in $\mathbb{F}_q[x]$ has distinct roots in $\overline{\mathbb{F}}_q$. Moreover, distinct irreducible polynomials in $\mathbb{F}_q[x]$ do not have a common root in $\overline{\mathbb{F}}_q$.

Figures (4)

  • Figure 1: $k_0(n)$, the upper bound on the code dimension in a shadow code of degree at most 1 to have $\Delta(\mathcal{E}, \mathcal{B}_1) > 0$, and its approximation.
  • Figure 2: Encoder for a concatenated RS-RM code. Above each arrow is a vector belonging to the vector space written beneath the arrow. The block labeled RS is a linear RS encoder, and the blocks labeled RM are linear RM encoders. The S/P and P/S units are serial-to-parallel and parallel-to-serial converters, respectively.
  • Figure 3: Comparing binary shadow codes of degree at most 1 and degree 2, with the RS--RM concatenated, Delsarte--Goethals (DG), and first-order RM codes of the same length. The vertical axis denotes relative distance and the horizontal axis denotes the code rate. Also shown is the Gilbert-Varshamov (GV) bound MS77. For shadow codes, the indicated lines represent lower bounds on relative distance. The first- and second-order RM codes are shown with a single dot. For $n=2^{10}$, actual code parameters for shadow codes and some randomly generated linear codes are also shown.
  • Figure 4: Bounds on relative distance versus rate when both code families have dimension $k$ scaling with block length $n$ as $k = n^{0.49}$.

Theorems & Definitions (18)

  • Example 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Definition 1
  • ...and 8 more