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Binary cyclic codes from permutation polynomials over $\mathbb{F}_{2^m}$

Mrinal Kanti Bose, Udaya Parampalli, Abhay Kumar Singh

TL;DR

This work constructs binary cyclic codes of length $v=2^m-1$ from permutation polynomials over $ ext{F}_{2^m}$ using the trace-sequence framework of Ding. By analyzing the associated $2$-cyclotomic cosets and employing the Hartmann-Tzeng and BCH bounds, it yields infinite families with dimensions exceeding $v/2$ and minimum distances near the square-root bound, including a new optimal family with parameters $[2^m-1,2^m-2-3m,8]$ for odd $m\ge5$. The paper covers both odd and even $m$, providing explicit generator polynomials for various permutation monomials and trinomials, and demonstrates optimal or near-optimal codes in several cases (e.g., $[31,25,4]$, $[127,105,6]$, $[127,91,8]$, $[255,199,10]$). These constructions have potential implications for high-rate error correction, post-quantum cryptography, and related communications applications, and open avenues for tighter distance bounds and quantum-code design.

Abstract

Binary cyclic codes having large dimensions and minimum distances close to the square-root bound are highly valuable in applications where high-rate transmission and robust error correction are both essential. They provide an optimal trade-off between these two factors, making them suitable for demanding communication and storage systems, post-quantum cryptography, radar and sonar systems, wireless sensor networks, and space communications. This paper aims to investigate cyclic codes by an efficient approach introduced by Ding \cite{SETA5} from several known classes of permutation monomials and trinomials over $\mathbb{F}_{2^m}$. We present several infinite families of binary cyclic codes of length $2^m-1$ with dimensions larger than $(2^m-1)/2$. By applying the Hartmann-Tzeng bound, some of the lower bounds on the minimum distances of these cyclic codes are relatively close to the square root bound. Moreover, we obtain a new infinite family of optimal binary cyclic codes with parameters $[2^m-1,2^m-2-3m,8]$, where $m\geq 5$ is odd, according to the sphere-packing bound.

Binary cyclic codes from permutation polynomials over $\mathbb{F}_{2^m}$

TL;DR

This work constructs binary cyclic codes of length from permutation polynomials over using the trace-sequence framework of Ding. By analyzing the associated -cyclotomic cosets and employing the Hartmann-Tzeng and BCH bounds, it yields infinite families with dimensions exceeding and minimum distances near the square-root bound, including a new optimal family with parameters for odd . The paper covers both odd and even , providing explicit generator polynomials for various permutation monomials and trinomials, and demonstrates optimal or near-optimal codes in several cases (e.g., , , , ). These constructions have potential implications for high-rate error correction, post-quantum cryptography, and related communications applications, and open avenues for tighter distance bounds and quantum-code design.

Abstract

Binary cyclic codes having large dimensions and minimum distances close to the square-root bound are highly valuable in applications where high-rate transmission and robust error correction are both essential. They provide an optimal trade-off between these two factors, making them suitable for demanding communication and storage systems, post-quantum cryptography, radar and sonar systems, wireless sensor networks, and space communications. This paper aims to investigate cyclic codes by an efficient approach introduced by Ding \cite{SETA5} from several known classes of permutation monomials and trinomials over . We present several infinite families of binary cyclic codes of length with dimensions larger than . By applying the Hartmann-Tzeng bound, some of the lower bounds on the minimum distances of these cyclic codes are relatively close to the square root bound. Moreover, we obtain a new infinite family of optimal binary cyclic codes with parameters , where is odd, according to the sphere-packing bound.

Paper Structure

This paper contains 15 sections, 22 theorems, 79 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Essential results on 2-cyclotomic cosets modulo $v$
  4. Cyclic codes designed by periodic sequences
  5. Binary cyclic codes from polynomials over $\mathbb{F}_{2^m}$, $m$ is odd
  6. Binary code $\mathcal{C}_{s}$ from the trinomial $x + x^{2^{(m+1)/2}-1}+x^{2^m-2^{(m+1)/2}+1}$
  7. Binary code $\mathcal{C}_{s}$ from the trinomial $x^{3\cdot2^{(m+1)/2}+4}+x^{2^{(m+1)/2}+2}+x^{2^{(m+1)/2}}$
  8. Binary code $\mathcal{C}_{s}$ from the trinomial $x+x^3+x^{2^{(m+1)/2}+1}$
  9. Binary code $\mathcal{C}_{s}$ from the trinomial $x+x^3+x^{2^m-2^{(m+3)/2}+2}$
  10. Binary code $\mathcal{C}_{s}$ from the monomial $x^{2^{m-1}-2^{(m-1)/2}+1}$
  11. Binary cyclic codes from polynomials over $\mathbb{F}_{2^m}$, $m$ is even
  12. Binary code $\mathcal{C}_{s}$ from the trinomial of the form (\ref{['E21']}), where $(s,t)=(1,-1)$
  13. Binary code $\mathcal{C}_{s}$ from the trinomial of the form (\ref{['E21']}), where $(s,t)=(2,-2)$
  14. Binary code $\mathcal{C}_{s}$ from the trinomial of the form (\ref{['E21']}), where $(s,t)=(1,2^{\frac{m}{2}-1})$
  15. Summary and concluding remarks

Key Result

Lemma 1

For any coset leader $i \in \Gamma \setminus \{0\}$, $i$ is odd and $1 \leq i < 2^{m-1}$.

Theorems & Definitions (57)

  • Lemma 1: SETA7
  • Lemma 2: SETA3
  • remark 1
  • Lemma 3
  • remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Example 1
  • ...and 47 more