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On the weight distribution of random binary linear codes

Nati Linial, Jonathan Mosheiff

TL;DR

The asymptotics of the moments of X of all orders o(nlogn) are determined and the weight distribution of random binary linear codes is investigated.

Abstract

We investigate the weight distribution of random binary linear codes. For $0<λ<1$ and $n\to\infty$ pick uniformly at random $λn$ vectors in $\mathbb{F}_2^n$ and let $C \le \mathbb{F}_2^n$ be the orthogonal complement of their span. Given $0<γ<1/2$ with $0< λ< h(γ)$ let $X$ be the random variable that counts the number of words in $C$ of Hamming weight $γn$. In this paper we determine the asymptotics of the moments of $X$ of all orders $o(\frac{n}{\log n})$.

On the weight distribution of random binary linear codes

TL;DR

The asymptotics of the moments of X of all orders o(nlogn) are determined and the weight distribution of random binary linear codes is investigated.

Abstract

We investigate the weight distribution of random binary linear codes. For and pick uniformly at random vectors in and let be the orthogonal complement of their span. Given with let be the random variable that counts the number of words in of Hamming weight . In this paper we determine the asymptotics of the moments of of all orders .

Paper Structure

This paper contains 22 sections, 23 theorems, 148 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Rough outline of the proof
  3. Preliminaries
  4. From moments to enumeration.
  5. Interpreting the central moments of $X$
  6. Computing $R_U$
  7. From $|\overline T_U|$ to $|T_U|$
  8. The intersection of $\mathbb V^{k}$ and the $\gamma n$-th layer
  9. Basic properties of $F(k,\gamma)$
  10. Proof of Lemma \ref{['lem:TProbability']}
  11. Bounding $|T_V|$ in general
  12. The function $F(m,\gamma,\delta)$
  13. Properties of $F(m,\gamma,\delta)$
  14. Derivation of the main theorems
  15. Moments of even order
  16. ...and 7 more sections

Key Result

theorem 1

Fix $\gamma <\frac{1}{2}$ and $0<\lambda<h(\gamma)$ and let Then, for $2 \le k \le o(\frac{n}{\log n})$,

Figures (5)

  • Figure 1: Illustration for Theorem \ref{['thm:mainNormalized']}. For $k < k_0= k_0(\gamma,\lambda)$ the $k$-th moment of $X$ is that of a normal distribution. The relevant range $\lambda < h(\gamma)$ is below the solid line. Note that $k_0=3$ for much of the parameters range.
  • Figure 2: The function $g(3,\frac{1}{5}, x)$ and its minimum (see Equation \ref{['eqn:FByMinimization']}).
  • Figure 3: ${F(k,\frac{1}{5}) - (k\cdot h(\frac{1}{5})-1)}$. (See Proposition \ref{['prop:FBounds']}).
  • Figure 4: Illustration for Lemma \ref{['lem:FMonotoneInDelta']} - $F(5,\frac{1}{5},\delta)$
  • Figure 5: . Illustration for Section \ref{['subsec:Extensions']} - Extending $F$ to $\gamma \in (\frac{1}{2},1)$. Solid:$F(5,\gamma)$Dashed:$F(5,\gamma,1)=F(5,1-\gamma)$Dotted:$5h(\gamma)-1$

Theorems & Definitions (46)

  • theorem 1
  • proposition 1
  • proof
  • definition 2
  • definition 3
  • definition 4
  • proposition 5
  • proof
  • definition 6
  • proposition 7
  • ...and 36 more