[Paper Review] Binary Cyclic Codes from Explicit Polynomials over $\gf(2^m)$

Cunsheng Ding, Zhengchun Zhou|arXiv (Cornell University)|Oct 5, 2013

Coding theory and cryptography4 citations

TL;DR

This paper constructs optimal and near-optimal binary cyclic codes using monomials and trinomials over $\mathrm{GF}(2^m)$, leveraging explicit polynomials to achieve flexible dimensions and tight bounds on minimum distance. The approach yields codes meeting or approaching known bounds on linear codes, with several examples demonstrating optimality, including $[31,10,12]$ and $[31,15,8]$ codes.

ABSTRACT

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, monomials and trinomials over finite fields with even characteristic are employed to construct a number of families of binary cyclic codes. Lower bounds on the minimum weight of some families of the cyclic codes are developed. The minimum weights of other families of the codes constructed in this paper are determined. The dimensions of the codes are flexible. Some of the codes presented in this paper are optimal or almost optimal in the sense that they meet some bounds on linear codes. Open problems regarding binary cyclic codes from monomials and trinomials are also presented.

Motivation & Objective

• To develop a systematic construction of binary cyclic codes using explicit monomials and trinomials over $\mathrm{GF}(2^m)$.
• To determine the minimum weight and dimension of the constructed codes for specific families.
• To identify conditions under which the codes are optimal or almost optimal relative to known bounds on linear codes.
• To explore the potential of these codes for applications in coding theory and cryptography due to their good distance properties and high linear span.
• To pose open problems for future research on the structure and parameters of cyclic codes from monomials and trinomials.

Proposed method

• Employ monomials and trinomials of the form $x^e$ and $x^e + x^f + x^g$ over $\mathrm{GF}(2^m)$ to define sequences with specific periodicity.
• Use the generator polynomial $g(x) = \frac{x^n - 1}{\gcd(S^n(x), x^n - 1)}$, where $S^n(x)$ is the generating function of the sequence.
• Leverage $q$-cyclotomic cosets modulo $n = 2^m - 1$ to analyze the structure of the cyclic codes.
• Apply the BCH bound and properties of minimal polynomials to derive lower bounds on the minimum distance.
• Compute the linear span of the sequences to assess their cryptographic strength, particularly in stream cipher applications.
• Use the trace function $\mathrm{Tr}(x)$ and the function $\mathbb{N}_q(i)$ to characterize the sequence structure and its algebraic properties.

Experimental results

Research questions

• RQ1 What are the dimension and minimum weight of the binary cyclic code $\mathcal{C}_s$ defined by the monomial $x^e$ with $e = 2^{(m-1)/2} + 2^{(3m-1)/4} - 1$ when $m \equiv 3 \pmod{4}$?
• RQ2 What are the dimension and minimum weight of $\mathcal{C}_s$ defined by $x^e$ with $e = 2^{4i} + 2^{3i} + 2^{2i} + 2^i - 1$ when $m = 5i$?
• RQ3 Can tight lower bounds on the minimum distance be established for the code $\mathcal{C}_s$ when $h > 0$ in the trinomial construction?
• RQ4 How do the linear spans of sequences derived from monomials and trinomials compare, and what implications do they have for cryptographic applications?
• RQ5 Under what conditions do the constructed binary cyclic codes achieve optimality or near-optimality in terms of minimum distance?

Key findings

• The code $\mathcal{C}_s$ with generator polynomial $x^4 + x + 1$ over $\mathrm{GF}(2^4)$ is a $[15,11,3]$ optimal binary cyclic code.
• For $m=4$, $h=2$, the code $\mathcal{C}_s$ with generator polynomial $x^8 + x^7 + x^6 + x^4 + 1$ is a $[15,7,5]$ optimal binary cyclic code.
• When $m=5$, $h=0$, the code $\mathcal{C}_s$ with generator polynomial $x^{21} + x^{18} + \cdots + x + 1$ is a $[31,10,12]$ optimal binary cyclic code.
• For $m=5$, $h=1$, the code $\mathcal{C}_s$ with generator polynomial $x^{16} + x^{14} + \cdots + x + 1$ is a $[31,15,8]$ optimal binary cyclic code.
• The linear span of sequences derived from certain monomials and trinomials is large, indicating strong cryptographic properties suitable for keystream generation.
• The BCH bound provides a lower bound of $d \geq 8$ for the code when $m$ is odd and $h=0$, based on the zeros of the reciprocal of the minimal polynomial.