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A New Class of Linear Codes

Giacomo Cherubini, Giacomo Micheli

Abstract

Let $n$ be a prime power, $r$ be a prime with $r\mid n-1$, and $\varepsilon\in (0,1/2)$. Using the theory of multiplicative character sums and superelliptic curves, we construct new codes over $\mathbb F_r$ having length $n$, relative distance $(r-1)/r+O(n^{-\varepsilon})$ and rate $n^{-1/2-\varepsilon}$. When $r=2$, our binary codes have exponential size when compared to all previously known families of linear and non-linear codes with relative distance asymptotic to $1/2$, such as Delsarte--Goethals codes. Moreover, concatenating with a Reed--Solomon code gives a family of codes of length $n$, asymptotic distance $1/2$ and rate $Ω(n^{-\varepsilon})$ for any fixed small $\varepsilon>0$, improving our initial construction. Such rate is also asymptotically better than the one by Kschischang and Tasbihi obtained by concatenating a Reed--Solomon with Reed--Muller, improving by a factor in $Ω(n^{1/2}/\log(n))$.

A New Class of Linear Codes

Abstract

Let be a prime power, be a prime with , and . Using the theory of multiplicative character sums and superelliptic curves, we construct new codes over having length , relative distance and rate . When , our binary codes have exponential size when compared to all previously known families of linear and non-linear codes with relative distance asymptotic to , such as Delsarte--Goethals codes. Moreover, concatenating with a Reed--Solomon code gives a family of codes of length , asymptotic distance and rate for any fixed small , improving our initial construction. Such rate is also asymptotically better than the one by Kschischang and Tasbihi obtained by concatenating a Reed--Solomon with Reed--Muller, improving by a factor in .
Paper Structure (10 sections, 11 theorems, 43 equations)

This paper contains 10 sections, 11 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Shadow Codes Construction
  3. Comparison with existing bounds
  4. Upper bounds
  5. Lower bounds: Gilbert--Varshamov bound, Delsarte--Goethals codes
  6. An application of Coding Theory to Number Theory: the maximum value of certain character sums
  7. Code Boosts
  8. Degree One Boost
  9. Binary Concatenation Boost
  10. Binary Non-Linear Boost

Key Result

Theorem 1.1

Let $q\geq 3$ be a prime power and $r$ be a prime dividing $q-1$. Then, for any $\varepsilon\in(0,1/2)$ and for any $q$ larger than an explicit constant $C=C(\varepsilon)$ depending only on $\varepsilon$, there is an ${ \mathbb F }_r$-linear code having parameters $[q,\lfloor q^{1/2-\varepsilon}\rfl

Theorems & Definitions (23)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 13 more