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تسجيل مستخدم جديدOn cyclic codes of length $2^e$ over finite fields
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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, Dings construction is not hold in this place. We describe two new types of generalized cyclotomy of order two, which are different from Dings. Furthermore, we study two classes of cyclic codes of length $n$ and dimension $k$. We get the enumeration of these cyclic codes. Whats 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.
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Let $mathbb{F}_{2^m}$ be a finite field of $2^m$ elements, and $R=mathbb{F}_{2^m}[u]/langle u^krangle=mathbb{F}_{2^m}+umathbb{F}_{2^m}+ldots+u^{k-1}mathbb{F}_{2^m}$ ($u^k=0$) where $k$ is an integer satisfying $kgeq 2$. For any odd positive integer $
n$, an explicit representation for every self-dual cyclic code over $R$ of length $2n$ and a mass formula to count the number of these codes are given first. Then a generator matrix is provided for the self-dual and $2$-quasi-cyclic code of length $4n$ over $mathbb{F}_{2^m}$ derived by every self-dual cyclic code of length $2n$ over $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ and a Gray map from $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ onto $mathbb{F}_{2^m}^2$. Finally, the hull of each cyclic code with length $2n$ over $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ is determined and all distinct self-orthogonal cyclic codes of length $2n$ over $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ are listed.
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