On cyclic codes of length $2^e$ over finite fields

TL;DR

This paper investigates cyclic codes of length 2^e over finite fields, introduces new generalized cyclotomy of order two, and characterizes the hull dimensions of these codes, expanding understanding beyond existing prime-length constructions.

Contribution

It introduces two new types of generalized cyclotomy of order two for length 2^e codes and characterizes hull dimensions, providing enumeration and construction methods for these codes.

Findings

All constructed codes are among the best cyclic codes.

Range of hull dimensions for these codes is determined.

Enumeration formulas for the codes are provided.

Abstract

Professor Cunsheng Ding gave cyclotomic constructions of cyclic codes with length being the product of two primes. In this paper, we study the cyclic codes of length $n=2^e$ and dimension $k=2^{e-1}$. Clearly, Ding's construction is not hold in this place. We describe two new types of generalized cyclotomy of order two, which are different from Ding's. Furthermore, we study two classes of cyclic codes of length $n$ and dimension $k$. We get the enumeration of these cyclic codes. What's more, all of the codes from our construction are among the best cyclic codes. Furthermore, we study the hull of cyclic codes of length $n$ over $\mathbb{F}_q$. We obtain the range of $\ell=\dim({\rm Hull}(C))$. We construct and enumerate cyclic codes of length $n$ having hull of given dimension.