On self-duality and hulls of cyclic codes over $\frac{\mathbb {F}_{2^m}[u]}{\langle u^k\rangle }$ with oddly even length

Abstract

Let $\mathbb {F}_{2^m}$ be a finite field of $2^m$ elements and denote $R=\mathbb {F}_{2^m}[u]/\langle u^k\rangle$ $=\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}+\cdots +u^{k-1}\mathbb {F}_{2^m}$ ($u^k=0$), where k is an integer satisfying $k\ge 2$. 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. In particular, a generator matrix is provided for the self-dual 2-quasi-cyclic code of length 4n over $\mathbb {F}_{2^m}$ derived by an arbitrary self-dual cyclic code of length 2n over $\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}$ and a Gray map from $\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}$ onto $\mathbb {F}_{2^m}^2$. Finally, the hull of each cyclic code with length 2n over $\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}$ is determined and all distinct self-orthogonal cyclic codes of length 2n over $\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}$ are listed.

Appendices

Appendix: All distinct self-dual cyclic codes of length 2n over the ring $R=\frac{\mathbb {F}_{2^m}[u]}{\langle u^k\rangle }$, where $3\le k\le 5$ and n is odd

By Lemma 2.3, Corollary 2.5, Theorem 3.1 and Theorem 3.3, we have the follows conclusions.

Case $k=4$

Using the notation in Theorem 3.3(ii), the number of self-dual cyclic codes of length 2n over $\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}+u^2\mathbb {F}_{2^m}+u^3\mathbb {F}_{2^m}$ $(u^4=0)$ is

$$\begin{aligned} (1+2^m+4^m)\cdot \prod _{j=2}^\lambda (1+2^{\frac{d_j}{2}m}+2^{d_jm})\cdot \prod _{j=\lambda +1}^{\lambda +\epsilon }(9+5\cdot 2^{d_jm}+4^{d_jm}). \end{aligned}$$

Precisely, all these codes are given by

$$\begin{aligned} \mathcal{C}=\left( \oplus _{j=1}^\lambda \varepsilon _j(x)C_j\right) \oplus \left( \oplus _{j=\lambda +1}^{\lambda +\epsilon }(\varepsilon _{j}(x)C_{j}\oplus \varepsilon _{j+\epsilon }(x)C_{j+\epsilon })\right) , \end{aligned}$$

where $C_j$ is an ideal of $\mathcal{K}_j+u\mathcal{K}_j+u^2\mathcal{K}_j+u^3\mathcal{K}_j$ $(u^4=0)$ listed as follows:

• (i) $C_1$ is one of the following $1+2^m+4^m$ ideals:

  $$\begin{aligned}&{\langle } u^2\rangle , \langle (x-1)\rangle ;\\&{\langle } u^2+u(x-1)\omega \rangle \text { where }\omega \in \mathbb {F}_{2^m}\text { and }\omega \ne 0;\\&{\langle } u^2+(x-1)\omega \rangle \text { where }\omega =a_0+ua_1, a_0,a_1\in \mathbb {F}_{2^m}\text { and }a_0\ne 0;\\&{\langle } u^3+(x-1)\omega \rangle \text { where }\omega \in \mathbb {F}_{2^m}\text { and }\omega \ne 0;\\&{\langle } u^3,u(x-1)\rangle . \end{aligned}$$

• (ii) Let $2\le j\le \lambda$. Then $C_j$ is one of the following $1+2^{\frac{d_j}{2}m}+2^{d_jm}$ ideals:

  $$\begin{aligned}&\langle u^2\rangle , \langle f_j(x)\rangle ;\\&\langle u^2+uf_j(x)\omega \rangle \hbox { where }\omega \in \varTheta _{j,1};\\&\langle u^2+f_j(x)\omega \rangle \hbox { where }\omega =a_0(x)+ua_1(x), a_0(x)\in \varTheta _{j,1}\hbox { and }a_1(x)\in \{0\}\cup \varTheta _{j,1};\\&\langle u^3+f_j(x)\omega \rangle \hbox { where }\omega \in \varTheta _{j,1};\\&\langle u^3,uf_j(x)\rangle . \end{aligned}$$

• (iii) Let $\lambda +1\le j\le \lambda +\epsilon$. Then the pair $(C_j,C_{j+\epsilon })$ of ideals is one of the following $9+5\cdot 2^{d_jm}+4^{d_jm}$ cases listed in the following table:

$\mathcal {L}$	$C_j$ (mod $f_j(x)^2$)	$|C_j|$	$C_{j+\epsilon }$ (mod $f_{j+\epsilon }(x)^2$)
5	$\bullet$ $\langle u^i\rangle$$(0\le i\le 4)$	$4^{(4-i)d_jm}$	$\diamond$ $\langle u^{k-i}\rangle$
4	$\bullet$ $\langle u^sf_j(x)\rangle$ ($0\le s\le 3$)	$2^{(4-s)d_jm}$	$\diamond$ $\langle u^{4-s},f_{j+\epsilon }(x)\rangle$
$3(2^{d_jm}-1)$	$\bullet$ $\langle u^{i}+u^{i-1} f_j(x)\omega \rangle$	$4^{(4-i)d_jm}$	$\diamond$ $\langle u^{4-i}+u^{3-i}f_{j+\epsilon }(x)\omega ^{\prime }\rangle$
	($i=1,2,3$)
$4^{d_jm}-2^{d_jm}$	$\bullet$ $\langle u^{2}+f_j(x)\vartheta \rangle$	$4^{2d_jm}$	$\diamond$ $\langle u^{2}+f_{j+\epsilon }(x)\vartheta ^{\prime }\rangle$
$2^{d_jm}-1$	$\bullet$ $\langle u^3+f_j(x)\omega \rangle$	$2^{4d_jm}$	$\diamond$ $\langle u^3+f_{j+\epsilon }(x)\omega ^{\prime }\rangle$
$2^{d_jm}-1$	$\bullet$ $\langle u^3+uf_j(x)\omega \rangle$	$2^{3d_jm}$	$\diamond$ $\langle u^2+f_{j+\epsilon }(x)\omega ^{\prime },$
			$uf_{j+\epsilon }(x)\rangle$
6	$\bullet$ $\langle u^i,u^sf_j(x)\rangle$	$2^{(8-(i+s))d_jm}$	$\diamond$ $\langle u^{4-s}, u^{4-i}f_{j+\epsilon }(x)\rangle$
	($0\le s<i\le 3$)
$2^{d_jm}-1$	$\bullet$ $\langle u^2+f_j(x)\omega , uf_j(x)\rangle$	$2^{5d_jm}$	$\diamond$ $\langle u^3+uf_{j+\epsilon }(x)\omega ^{\prime }\rangle$

where $\mathcal {L}$ is the number of pairs $(C_j,C_{j+\epsilon })$ in the same row, and

$$\begin{aligned} \omega= & {} \omega (x)\in \mathcal {F}_j=\frac{\mathbb {F}_{2^m}[x]}{\langle f_j(x)\rangle }, \omega \ne 0\;\hbox { and}\\ \omega ^{\prime }= & {} \delta _jx^{-d_j}\omega (x^{-1}) \ (\mathrm{mod} \ f_{j+\epsilon }(x)); \end{aligned}$$

$\vartheta =a_0(x)+ua_1(x)$ with $a_0(x),a_1(x)\in \mathcal {F}_j$ and $a_0(x)\ne 0$, and

$$\begin{aligned} \vartheta ^{\prime }=\delta _jx^{-d_j}\left( a_0(x^{-1})+ua_1(x^{-1})\right) \ (\mathrm{mod} \ f_{j+\epsilon }(x)). \end{aligned}$$

Case $k=3$

Using the notation in Theorem 3.3(ii), the number of self-dual cyclic codes of length 2n over the ring $\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}+u^2\mathbb {F}_{2^m}$ $(u^3=0)$ is

$$\begin{aligned} (1+2^m)\cdot \prod _{j=2}^\lambda (1+2^{\frac{d_j}{2}m}) \cdot \prod _{j=\lambda +1}^{\lambda +\epsilon }(7+3\cdot 2^{d_jm}). \end{aligned}$$

Precisely, all these codes are given by

$$\begin{aligned} \mathcal{C}=\left( \oplus _{j=1}^\lambda \varepsilon _j(x)C_j\right) \oplus \left( \oplus _{j=\lambda +1}^{\lambda +\epsilon }(\varepsilon _{j}(x)C_{j}\oplus \varepsilon _{j+\epsilon }(x)C_{j+\epsilon })\right) , \end{aligned}$$

where $C_j$ is an ideal of $\mathcal{K}_j+u\mathcal{K}_j+u^2\mathcal{K}_j$ $(u^3=0)$ listed as follows:

• (i) $C_1$ is one of the following $1+2^m$ ideals:

  $$\begin{aligned}&\langle (x-1)\rangle , \langle u^2,u(x-1)\rangle ;\\&\langle u^2+(x-1)\omega \rangle \hbox { where }\omega \in \mathbb {F}_{2^m}\hbox { and }\omega \ne 0. \end{aligned}$$

• (ii) Let $2\le j\le \lambda$. Then $C_j$ is one of the following $1+2^{\frac{d_j}{2}m}$ ideals:

  $$\begin{aligned}&\langle f_j(x)\rangle , \langle u^2, uf_j(x)\rangle ;\\&\langle u^2+f_j(x)\omega \rangle \hbox { where }\omega \in \varTheta _{j,1}. \end{aligned}$$

• (iii) Let $\lambda +1\le j\le \lambda +\epsilon$. Then the pair $(C_j,C_{j+\epsilon })$ of ideals is one of the following $7+3\cdot 2^{d_jm}$ cases listed in the following table:

$\mathcal {L}$	$C_j$ (mod $f_j(x)^2$)	$|C_j|$	$C_{j+\epsilon }$ (mod $f_{j+\epsilon }(x)^2$)
4	$\bullet$ $\langle u^i\rangle$$(i=0,1,2,3)$	$4^{(3-i)d_jm}$	$\diamond$ $\langle u^{3-i}\rangle$
3	$\bullet$ $\langle u^sf_j(x)\rangle$ ($0\le s\le 2$)	$2^{(3-s)d_jm}$	$\diamond$ $\langle u^{3-s},f_{j+\epsilon }(x)\rangle$
$2^{d_jm}-1$	$\bullet$ $\langle u+f_j(x)\omega \rangle$	$4^{2d_jm}$	$\diamond$ $\langle u^2+uf_{j+\epsilon }(x)\omega ^{\prime }\rangle$
$2^{d_jm}-1$	$\bullet$ $\langle u^2+uf_j(x)\omega \rangle$	$4^{d_jm}$	$\diamond$ $\langle u+f_{j+\epsilon }(x)\omega ^{\prime }\rangle$
$2^{d_jm}-1$	$\bullet$ $\langle u^2+f_j(x)\omega \rangle$	$2^{3d_jm}$	$\diamond$ $\langle u^2+f_{j+\epsilon }(x)\omega ^{\prime }\rangle$
3	$\bullet$ $\langle u^i,u^sf_j(x)\rangle$	$2^{(6-(i+s))d_jm}$	$\diamond$ $\langle u^{3-s}, u^{3-i}f_{j+\epsilon }(x)\rangle$
	$(0\le s<i\le 2)$

where $\mathcal {L}$ is the number of pairs $(C_j,C_{j+\epsilon })$ in the same row, $\omega =\omega (x)\in \mathcal {F}_j=\frac{\mathbb {F}_{2^m}[x]}{\langle f_j(x)\rangle }$ with $\omega \ne 0$, and $\omega ^{\prime }=\delta _jx^{-d_j}\omega (x^{-1}) \ (\mathrm{mod} \ f_{j+\epsilon }(x)).$

Case $k=5$

$\diamondsuit$ Using the notation in Theorem 3.3(ii), the number of self-dual cyclic codes of length 2n over the ring $\mathbb {F}_{2^m}+u\mathbb {F}_{2^m}+u^2\mathbb {F}_{2^m}+u^3\mathbb {F}_{2^m}+u^4\mathbb {F}_{2^m}$ $(u^5=0)$ is

$$\begin{aligned} (1+2^m+4^m)\cdot \prod _{j=2}^\lambda (1+2^{\frac{d_j}{2}m}+2^{d_jm}) \cdot \prod _{j=\lambda +1}^{\lambda +\epsilon }(11+7\cdot 2^{d_jm}+3\cdot 4^{d_jm}). \end{aligned}$$

Precisely, all these codes are given by

$$\begin{aligned} \mathcal{C}=\left( \oplus _{j=1}^\lambda \varepsilon _j(x)C_j\right) \oplus \left( \oplus _{j=\lambda +1}^{\lambda +\epsilon }(\varepsilon _{j}(x)C_{j}\oplus \varepsilon _{j+\epsilon }(x)C_{j+\epsilon })\right) , \end{aligned}$$

where $C_j$ is an ideal of $\mathcal{K}_j+u\mathcal{K}_j+u^2\mathcal{K}_j+u^3\mathcal{K}_j+u^4\mathcal{K}_j$ $(u^5=0)$ listed as follows:

• (i) Let $\lambda +1\le j\le \lambda +\epsilon$. Then the pair $(C_j,C_{j+\epsilon })$ of ideals is one of the following $11+7\cdot 2^{d_jm}+3\cdot 4^{d_jm}$ cases listed in the following table:

  $\mathcal {L}$	$C_j$ (mod $f_j(x)^2$)	$|C_j|$	$C_{j+\epsilon }$ (mod $f_{j+\epsilon }(x)^2$)
  6	$\bullet$ $\langle u^i\rangle$$(0\le i\le 5)$	$4^{(5-i)d_jm}$	$\diamond$ $\langle u^{5-i}\rangle$
  5	$\bullet$ $\langle u^sf_j(x)\rangle$	$2^{(5-s)d_jm}$	$\diamond$ $\langle u^{5-s},f_{j+\epsilon }(x)\rangle$
  	($0\le s\le 4$)
  $4(2^{d_jm}-1)$	$\bullet$ $\langle u^i+u^{i-1}f_j(x)\omega \rangle$	$4^{(5-i)d_jm}$	$\diamond$ $\langle u^{4-i}f_{j+\epsilon }(x)\omega ^{\prime }$
  	($i=1,2,3,4$)		$+u^{5-i}\rangle$
  $4^{d_jm}-2^{d_jm}$	$\bullet$ $\langle u^2+f_j(x)\vartheta \rangle$	$4^{3d_jm}$	$\diamond$ $\langle u^{3}+uf_{j+\epsilon }(x)\vartheta ^{\prime }\rangle$
  $4^{d_jm}-2^{d_jm}$	$\bullet$ $\langle u^3+uf_j(x)\vartheta \rangle$	$4^{2d_jm}$	$\diamond$ $\langle u^{2}+f_{j+\epsilon }(x)\vartheta ^{\prime }\rangle$
  $4^{d_jm}-2^{d_jm}$	$\bullet$ $\langle u^3+f_j(x)\vartheta \rangle$	$2^{5d_jm}$	$\diamond$ $\langle u^3+f_{j+\epsilon }(x)\vartheta ^{\prime }\rangle$
  $2^{d_jm}-1$	$\bullet$ $\langle u^4+f_j(x)\omega \rangle$	$2^{5d_jm}$	$\diamond$ $\langle u^4+f_{j+\epsilon }(x)\omega ^{\prime }\rangle$
  $2^{d_jm}-1$	$\bullet$ $\langle u^4+uf_j(x)\omega \rangle$	$2^{4d_jm}$	$\diamond$ $\langle u^3+f_{j+\epsilon }(x)\omega ^{\prime },$
  			$uf_{j+\epsilon }(x)\rangle$
  $2^{d_jm}-1$	$\bullet$ $\langle u^4+u^2f_j(x)\omega \rangle$	$2^{3d_jm}$	$\diamond$ $\langle u^2+f_{j+\epsilon }(x)\omega ^{\prime },$
  			$uf_{j+\epsilon }(x)\rangle$
  10	$\bullet$ $\langle u^i,u^sf_j(x)\rangle$	$2^{(10-(i+s))d_jm}$	$\diamond$ $\langle u^{5-s}, u^{5-i}f_{j+\epsilon }(x)\rangle$
  	$(0\le s<i\le 4)$
  $2^{d_jm}-1$	$\bullet$ $\langle u^2+f_j(x)\omega ,uf_j(x)\rangle$	$2^{7d_jm}$	$\diamond$ $\langle u^4+u^2f_{j+\epsilon }(x)\omega ^{\prime }\rangle$
  $2^{d_jm}-1$	$\bullet$ $\langle u^3+f_j(x)\omega ,uf_j(x)\rangle$	$2^{6d_jm}$	$\diamond$ $\langle u^4+uf_{j+\epsilon }(x)\omega ^{\prime }\rangle$
  $2^{d_jm}-1$	$\bullet$ $\langle u^3+uf_j(x)\omega ,$	$2^{5d_jm}$	$\diamond$ $\langle u^3+uf_{j+\epsilon }(x)\omega ^{\prime },$
  	$u^2f_j(x)\rangle$		$u^2f_{j+\epsilon }(x)\rangle$

  where $\mathcal {L}$ is the number of pairs $(C_j,C_{j+\epsilon })$ in the same row;

  $\omega =\omega (x)\in \mathcal {F}_j=\frac{\mathbb {F}_{2^m}[x]}{\langle f_j(x)\rangle }$, $\omega \ne 0$ and

  $$\begin{aligned} \omega ^{\prime }=\delta _jx^{-d_j}\omega (x^{-1}) \ (\mathrm{mod} \ f_{j+\epsilon }(x)); \end{aligned}$$

  $\vartheta =a_0(x)+ua_1(x)$ with $a_0(x),a_1(x)\in \mathcal {F}_j$ and $a_0(x)\ne 0$, and

  $$\begin{aligned} \vartheta ^{\prime }=\delta _jx^{-d_j}\left( a_0(x^{-1})+ua_1(x^{-1})\right) \ (\mathrm{mod} \ f_{j+\epsilon }(x)). \end{aligned}$$

• (ii) $C_1$ is one of the following $1+2^m+4^m$ ideals:

  $\langle (x-1)\rangle$;

  $\langle u^3+(x-1)\omega \rangle$ where $\omega =a_0+ua_1$, $a_0,a_1\in \mathbb {F}_{2^m}$ and $a_0\ne 0$;

  $\langle u^4+(x-1)\omega \rangle$ where $\omega \in \mathbb {F}_{2^m}$ and $\omega \ne 0$;

  $\langle u^3, u^2(x-1)\rangle$, $\langle u^4, u(x-1)\rangle$;

  $\langle u^3+u(x-1)\omega , u^2(x-1)\rangle$ where $\omega \in \mathbb {F}_{2^m}$ and $\omega \ne 0$.

• (iii) Let $2\le j\le \lambda$. Then $C_j$ is one of the following $1+2^{\frac{d_j}{2}m}+2^{d_jm}$ ideals:

  $\langle f_j(x)\rangle$;

  $\langle u^3+f_j(x)\omega \rangle$ where $\omega =a_0(x)+ua_1(x)$, $a_0(x)\in \varTheta _{j,1}$ and $a_1(x)\in \{0\}\cup \varTheta _{j,1}$;

  $\langle u^4+f_j(x)\omega \rangle$ where $\omega \in \varTheta _{j,1}$;

  $\langle u^3, u^2f_j(x)\rangle$, $\langle u^4, uf_j(x)\rangle$;

  $\langle u^3+uf_j(x)\omega , u^2f_j(x)\rangle$ where $\omega \in \varTheta _{j,1}$.