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Upper bounds on the rate of linear $q$-ary $k$-hash codes

Stefano Della Fiore, Marco Dalai

TL;DR

New upper bounds on the rate of linear $k$-hash codes in $\mathbb{F}_q^n$ are presented, that is, codes with the property that any $k$ distinct codewords are all simultaneously distinct in at least one coordinate.

Abstract

This paper presents new upper bounds on the rate of linear $k$-hash codes in $\mathbb{F}_q^n$, $q\geq k$, that is, codes with the property that any $k$ distinct codewords are all simultaneously distinct in at least one coordinate.

Upper bounds on the rate of linear $q$-ary $k$-hash codes

TL;DR

New upper bounds on the rate of linear -hash codes in are presented, that is, codes with the property that any distinct codewords are all simultaneously distinct in at least one coordinate.

Abstract

This paper presents new upper bounds on the rate of linear -hash codes in , , that is, codes with the property that any distinct codewords are all simultaneously distinct in at least one coordinate.
Paper Structure (4 sections, 7 theorems, 38 equations, 2 figures, 1 table)

This paper contains 4 sections, 7 theorems, 38 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. A simpler proof for $q=k=3$
  3. General Case $q\geq k\geq 3$
  4. New upper bounds

Key Result

Lemma 1

Let $q\geq 3$ be a prime power, and let $\mathcal{H}$ be a set of hyperplanes in $\mathbb{F}_q^m$ whose union is $\mathbb{F}_q^m\setminus\{0\}$. Then $|\mathcal{H}|\geq (q-1)m$.

Figures (2)

  • Figure 1: Codewords used in the proof of Theorem \ref{['th:main_th']}. Each codeword collides with one of the previous codewords in each coordinate of the dotted part.
  • Figure 2: Comparison between upper and lower bounds for $q \geq 5$ and $k=4$.

Theorems & Definitions (12)

  • Lemma 1: jamison-1977
  • Lemma 2: bruen-1992
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • Remark 1
  • ...and 2 more