The Discussion on Hulls of Cyclic Codes over the Ring $\Re =\Z_4+v\Z_4$,   $v^2=v$

Narendra Kumar; Abhay Kumar Singh

arXiv:1904.03600·cs.IT·April 9, 2019

The Discussion on Hulls of Cyclic Codes over the Ring $\Re =\Z_4+v\Z_4$, $v^2=v$

Narendra Kumar, Abhay Kumar Singh

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Abstract

For odd length $n$ , the cyclic codes construction over $ℜ = Z_{4} [v] / ⟨ v^{2} - v ⟩$ is provided. The hulls of cyclic codes over $ℜ$ are studied. The average $2$ -dimension $E (n)$ of the hulls of cyclic codes over $ℜ$ is also conferred. Among these, the various examples of generators of hulls of cyclic codes over $ℜ$ are provided, whose $Z_{4}$ -images are good $Z_{4}$ -linear codes with good parameters.

Tables1

$n$	$B (n)$	$E (n) = \frac{10 n - 4 B_{n}}{9}$	$n$	$B (n)$	$E (n) = \frac{10 n - 4 B_{n}}{9}$
$55$	$15$	$\frac{490}{9}$	$105$	$13$	$\frac{998}{9}$
$57^{†}$	$21$	$54$	$107^{†}$	$107$	$\frac{642}{9}$
$59^{†}$	$59$	$\frac{354}{9}$	$109^{†}$	$109$	$\frac{654}{9}$
$61^{†}$	$61$	$\frac{366}{9}$	$111$	$39$	$106$
$63$	$7$	$\frac{602}{9}$	$113^{†}$	$113$	$\frac{678}{9}$
$65^{†}$	$17$	$\frac{582}{9}$	$115$	$5$	$\frac{1130}{9}$
$67^{†}$	$67$	$\frac{402}{9}$	$117$	$19$	$\frac{1094}{9}$
$69$	$3$	$\frac{678}{9}$	$119$	$17$	$\frac{1122}{9}$
$71$	$1$	$\frac{706}{9}$	$121^{†}$	$111$	$\frac{766}{9}$
$73$	$1$	$\frac{726}{9}$	$123$	$43$	$\frac{1158}{9}$
$75$	$23$	$\frac{658}{9}$	$125^{†}$	$101$	$\frac{846}{9}$
$77$	$11$	$\frac{726}{9}$	$127$	$1$	$\frac{1266}{9}$
$79$	$1$	$\frac{726}{9}$	$129^{†}$	$45$	$\frac{1110}{9}$
$81^{†}$	$55$	$\frac{590}{9}$	$131^{†}$	$131$	$\frac{786}{9}$
$83^{†}$	$83$	$\frac{498}{9}$	$133$	$19$	$\frac{1250}{9}$
$85$	$21$	$\frac{766}{9}$	$135$	$23$	$\frac{1258}{9}$
$87$	$31$	$\frac{746}{9}$	$137^{†}$	$137$	$\frac{822}{9}$
$89$	$1$	$\frac{886}{9}$	$139^{†}$	$139$	$\frac{834}{9}$
$91$	$13$	$\frac{858}{9}$	$141$	$3$	$\frac{1398}{9}$
$93$	$3$	$102$	$143$	$23$	$\frac{1338}{9}$
$95$	$23$	$\frac{858}{9}$	$145^{†}$	$33$	$\frac{1318}{9}$
$97^{†}$	$97$	$\frac{582}{9}$	$147$	$45$	$\frac{1290}{9}$
$99^{†}$	$17$	$\frac{922}{9}$	$149^{†}$	$149$	$\frac{894}{9}$
$101^{†}$	$101$	$\frac{606}{9}$	$151$	$1$	$\frac{1506}{9}$
$103$	$1$	$114$	$153$	$23$	$\frac{1438}{9}$

Equations155

h \cdot k = h_{0} k_{0} + h_{1} k_{1} + \dots + h_{n - 1} k_{n - 1} .

h \cdot k = h_{0} k_{0} + h_{1} k_{1} + \dots + h_{n - 1} k_{n - 1} .

C^{⊥} = {h ∣ h \cdot k = 0, \forall k \in C} .

C^{⊥} = {h ∣ h \cdot k = 0, \forall k \in C} .

Hull(C) = C \cap C^{⊥} .

Hull(C) = C \cap C^{⊥} .

C = ⟨ f (x) g (x), 2 f (x) h (x)⟩ = ⟨ f (x) g (x), 2 f (x)⟩,

C = ⟨ f (x) g (x), 2 f (x) h (x)⟩ = ⟨ f (x) g (x), 2 f (x)⟩,

h^{*}(x)=a_{0}^{-1}x^{\deg h(x)}h\big{(}\dfrac{1}{x}\big{)}.

h^{*}(x)=a_{0}^{-1}x^{\deg h(x)}h\big{(}\dfrac{1}{x}\big{)}.

⟨ h^{*} (x) g^{*} (x), 2 h^{*} (x) f^{*} (x)⟩ = ⟨ h^{*} (x) g^{*} (x), 2 h^{*} (x)⟩

⟨ h^{*} (x) g^{*} (x), 2 h^{*} (x) f^{*} (x)⟩ = ⟨ h^{*} (x) g^{*} (x), 2 h^{*} (x)⟩

x^{n}-1=\prod_{j|n,j\in N_{2}}\big{(}\prod_{i=1}^{\gamma(j)}g_{ij}(x)\big{)}\prod_{j|n,j\notin N_{2}}\big{(}\prod_{i=1}^{\beta(j)}f_{ij}(x)f_{ij}^{*}(x)\big{)}~{}~{}\text{in}~{}\mathbb{Z}_{4}[x]

x^{n}-1=\prod_{j|n,j\in N_{2}}\big{(}\prod_{i=1}^{\gamma(j)}g_{ij}(x)\big{)}\prod_{j|n,j\notin N_{2}}\big{(}\prod_{i=1}^{\beta(j)}f_{ij}(x)f_{ij}^{*}(x)\big{)}~{}~{}\text{in}~{}\mathbb{Z}_{4}[x]

γ (j) = \frac{ϕ ( j )}{or d _{j} ( 2 )}, β (j) = \frac{ϕ ( j )}{2 or d _{j} ( 2 )}

γ (j) = \frac{ϕ ( j )}{or d _{j} ( 2 )}, β (j) = \frac{ϕ ( j )}{2 or d _{j} ( 2 )}

B_{n}=\deg\prod_{j|n,j\in N_{2}}\big{(}\prod_{i=1}^{\gamma(j)}g_{ij}(x)\big{)}=\sum_{j|n,j\in N_{2}}\dfrac{\phi(j)}{ord_{j}(2)}ord_{j}(2)=\sum_{j|n,j\in N_{2}}\phi(j).

B_{n}=\deg\prod_{j|n,j\in N_{2}}\big{(}\prod_{i=1}^{\gamma(j)}g_{ij}(x)\big{)}=\sum_{j|n,j\in N_{2}}\dfrac{\phi(j)}{ord_{j}(2)}ord_{j}(2)=\sum_{j|n,j\in N_{2}}\phi(j).

ξ (a + b v) = (a, 2 b + a) .

ξ (a + b v) = (a, 2 b + a) .

⟨ lcm (f (x) g (x), h^{*} (x) g^{*} (x)), 2 lcm (f (x), h^{*} (x))⟩

⟨ lcm (f (x) g (x), h^{*} (x) g^{*} (x)), 2 lcm (f (x), h^{*} (x))⟩

H (x) = gcd (h (x), f^{*} (x)) and G (x) = \frac{x ^{n} - 1}{gcd ( h ( x ) , f ^{*} ( x )) \cdot lcm ( f ( x ) , h ^{*} ( x ))} .

H (x) = gcd (h (x), f^{*} (x)) and G (x) = \frac{x ^{n} - 1}{gcd ( h ( x ) , f ^{*} ( x )) \cdot lcm ( f ( x ) , h ^{*} ( x ))} .

C = ⟨ v p_{1} (x) q_{1} (x), 2 v p_{1} (x), (1 - v) p_{2} (x) q_{2} (x), 2 (1 - v) p_{2} (x)⟩,

C = ⟨ v p_{1} (x) q_{1} (x), 2 v p_{1} (x), (1 - v) p_{2} (x) q_{2} (x), 2 (1 - v) p_{2} (x)⟩,

R_{1} (x) = gcd (r_{1} (x), p_{1}^{*} (x)), R_{2} (x) = gcd (r_{2} (x), p_{2}^{*} (x))

R_{1} (x) = gcd (r_{1} (x), p_{1}^{*} (x)), R_{2} (x) = gcd (r_{2} (x), p_{2}^{*} (x))

Q_{1} (x) = \frac{x ^{n} - 1}{gcd ( r _{1} ( x ) , p _{1}^{*} ( x )) . lcm ( p _{1} ( x ) , r _{1}^{*} ( x ))}, Q_{2} (x) = \frac{x ^{n} - 1}{gcd ( r _{2} ( x ) , p _{2}^{*} ( x )) . lcm ( p _{2} ( x ) , r _{2}^{*} ( x ))} .

Q_{1} (x) = \frac{x ^{n} - 1}{gcd ( r _{1} ( x ) , p _{1}^{*} ( x )) . lcm ( p _{1} ( x ) , r _{1}^{*} ( x ))}, Q_{2} (x) = \frac{x ^{n} - 1}{gcd ( r _{2} ( x ) , p _{2}^{*} ( x )) . lcm ( p _{2} ( x ) , r _{2}^{*} ( x ))} .

C = ⟨ v p_{1} (x) q_{1} (x), 2 v p_{1} (x), (1 - v) p_{2} (x) q_{2} (x), 2 (1 - v) p_{2} (x)⟩,

C = ⟨ v p_{1} (x) q_{1} (x), 2 v p_{1} (x), (1 - v) p_{2} (x) q_{2} (x), 2 (1 - v) p_{2} (x)⟩,

C^{⊥} = ⟨ v r_{1}^{*} (x) q_{1}^{*} (x), 2 v r_{1}^{*} (x), (1 - v) r_{2}^{*} (x) q_{2}^{*} (x), 2 (1 - v) r_{2}^{*} (x)⟩

C^{⊥} = ⟨ v r_{1}^{*} (x) q_{1}^{*} (x), 2 v r_{1}^{*} (x), (1 - v) r_{2}^{*} (x) q_{2}^{*} (x), 2 (1 - v) r_{2}^{*} (x)⟩

\overset{ˉ}{C} = ⟨ v P_{1} (x) Q_{1} (x), 2 v P_{1} (x), (1 - v) P_{2} (x) Q_{2} (x), 2 (1 - v) P_{2} (x)⟩,

\overset{ˉ}{C} = ⟨ v P_{1} (x) Q_{1} (x), 2 v P_{1} (x), (1 - v) P_{2} (x) Q_{2} (x), 2 (1 - v) P_{2} (x)⟩,

P_{1} (x) = lcm (p_{1} (x), r_{1}^{*} (x)), P_{2} (x) = lcm (p_{2} (x), r_{2}^{*} (x))

P_{1} (x) = lcm (p_{1} (x), r_{1}^{*} (x)), P_{2} (x) = lcm (p_{2} (x), r_{2}^{*} (x))

Q_{1} (x) = \frac{x ^{n} - 1}{gcd ( r _{1} ( x ) , p _{1}^{*} ( x )) . lcm ( p _{1} ( x ) , r _{1}^{*} ( x ))}, Q_{2} (x) = \frac{x ^{n} - 1}{gcd ( r _{2} ( x ) , p _{2}^{*} ( x )) . lcm ( p _{2} ( x ) , r _{2}^{*} ( x ))} .

Q_{1} (x) = \frac{x ^{n} - 1}{gcd ( r _{1} ( x ) , p _{1}^{*} ( x )) . lcm ( p _{1} ( x ) , r _{1}^{*} ( x ))}, Q_{2} (x) = \frac{x ^{n} - 1}{gcd ( r _{2} ( x ) , p _{2}^{*} ( x )) . lcm ( p _{2} ( x ) , r _{2}^{*} ( x ))} .

R_{1} (x) = \frac{x ^{n} - 1}{lcm ( p _{1} ( x ) q _{1} ( x ) , r _{1}^{*} ( x ) q _{1}^{*} ( x ))} . = gcd (r_{1} (x), p_{1}^{*} (x)),

R_{1} (x) = \frac{x ^{n} - 1}{lcm ( p _{1} ( x ) q _{1} ( x ) , r _{1}^{*} ( x ) q _{1}^{*} ( x ))} . = gcd (r_{1} (x), p_{1}^{*} (x)),

R_{2} (x) = \frac{x ^{n} - 1}{lcm ( p _{2} ( x ) q _{2} ( x ) , r _{2}^{*} ( x ) q _{2}^{*} ( x ))} . = gcd (r_{2} (x), p_{2}^{*} (x))

R_{2} (x) = \frac{x ^{n} - 1}{lcm ( p _{2} ( x ) q _{2} ( x ) , r _{2}^{*} ( x ) q _{2}^{*} ( x ))} . = gcd (r_{2} (x), p_{2}^{*} (x))

⟨ v L_{1} (x) M_{1} (x), 2 v L_{1} (x), (1 - v) L_{2} (x) M_{2} (x), 2 (1 - v) L_{2} (x)⟩,

⟨ v L_{1} (x) M_{1} (x), 2 v L_{1} (x), (1 - v) L_{2} (x) M_{2} (x), 2 (1 - v) L_{2} (x)⟩,

x^{n} - 1 = L_{1} (x) M_{1} (x) N_{1} (x) = M_{2} (x) L_{2} (x) N_{2} (x) .

x^{n} - 1 = L_{1} (x) M_{1} (x) N_{1} (x) = M_{2} (x) L_{2} (x) N_{2} (x) .

L_{1} (x) M_{1} (x) .2 p_{1}^{*} (x) = 0, L_{2} (x) M_{2} (x) .2 p_{2}^{*} (x) = 0

L_{1} (x) M_{1} (x) .2 p_{1}^{*} (x) = 0, L_{2} (x) M_{2} (x) .2 p_{2}^{*} (x) = 0

2 L_{1} (x) . p_{1}^{*} (x) q_{1}^{*} (x) = 0, 2 L_{2} (x) . p_{2}^{*} (x) q_{2}^{*} (x) = 0

2 L_{1} (x) . p_{1}^{*} (x) q_{1}^{*} (x) = 0, 2 L_{2} (x) . p_{2}^{*} (x) q_{2}^{*} (x) = 0

L_{1} (x) M_{1} (x) .2 r_{1} (x) = 0 and L_{2} (x) M_{2} (x) .2 r_{2} (x) = 0,

L_{1} (x) M_{1} (x) .2 r_{1} (x) = 0 and L_{2} (x) M_{2} (x) .2 r_{2} (x) = 0,

k_{1} = j ∣ n, j \neq \in N_{2} \sum or d_{j} (2) . a_{1 j} + j ∣ n, j \neq \in N_{2} \sum or d_{j} (2) . a_{2 j} and

k_{1} = j ∣ n, j \neq \in N_{2} \sum or d_{j} (2) . a_{1 j} + j ∣ n, j \neq \in N_{2} \sum or d_{j} (2) . a_{2 j} and

k_{2} = j ∣ n, j \in N_{2} \sum or d_{j} (2) . b_{1 j} + j ∣ n, j \in N_{2} \sum or d_{j} (2) . b_{2 j} + j ∣ n, j \neq \in N_{2} \sum or d_{j} (2) . c_{1 j} + j ∣ n, j \neq \in N_{2} \sum or d_{j} (2) . c_{2 j},

k_{2} = j ∣ n, j \in N_{2} \sum or d_{j} (2) . b_{1 j} + j ∣ n, j \in N_{2} \sum or d_{j} (2) . b_{2 j} + j ∣ n, j \neq \in N_{2} \sum or d_{j} (2) . c_{1 j} + j ∣ n, j \neq \in N_{2} \sum or d_{j} (2) . c_{2 j},

4^{d e g R_{1} (x) + d e g R_{2} (x)} 2^{d e g Q_{1} (x) + d e g Q_{2} (x)} .

4^{d e g R_{1} (x) + d e g R_{2} (x)} 2^{d e g Q_{1} (x) + d e g Q_{2} (x)} .

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TopicsCoding theory and cryptography · graph theory and CDMA systems · Finite Group Theory Research

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11institutetext: Department of Applied Mathematics

Indian Institute of Technology(ISM), Dhanbad, Jharkhand , 826004, India

11email: [email protected]

and

Department of Applied Mathematics

Indian Institute of Technology(ISM), Dhanbad, Jharkhand , 826004, India

11email: [email protected]

The Discussion on Hulls of Cyclic Codes over the Ring $\Re=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ , $v^{2}=v$

Narendra Kumar

Abhay Kumar Singh

(Received: date / Accepted: date)

Abstract

For odd length $n$ , the cyclic codes construction over $\Re=\mathbb{Z}_{4}[v]/\langle v^{2}-v\rangle$ is provided. The hulls of cyclic codes over $\Re$ are studied. The average $2$ -dimension $E(n)$ of the hulls of cyclic codes over $\Re$ is also conferred. Among these, the various examples of generators of hulls of cyclic codes over $\Re$ are provided, whose $\mathbb{Z}_{4}$ -images are good $\mathbb{Z}_{4}$ -linear codes with good parameters.

Keywords:

Linear codes, Cyclic codes, Self-dual, Hulls.

MSC:

Primary: 94B05, Secondary: 94B15.

1 Introduction

The study of hulls of cyclic codes have crucial importance because of their applications. Therefore the hulls of linear and cyclic codes over finite fields have been well studied. Assmus et al. 15 introduced the theory of hull of linear codes to describe the finite projective planes. In this series, many authors discussed the properties of hulls to describe the complexity of some algorithms in 18 ; 19 ; 20 ; 21 ; 22 ; 23 . Again, Sendrier 17 described the hulls of linear codes of length $n$ over square order field $\mathbb{F}_{q}$ and proved that the average dimension of the hulls depend on the order of $\mathbb{F}_{q}$ . In 24 ,Skersys discussed the average dimension $E_{q}(n)$ of the hulls of cyclic codes over finite fields and studied the character of $\dfrac{E_{q}(n)}{n}$ , as $n$ tends to infinity. Later, Sangwisut et al. 9 described the generator polynomials and dimensions of the hulls of cyclic and negacyclic codes over finite field. Recently, Jitman and Sangwisut 7 provided the average dimension of the Hermitian hull of constacyclic codes over $\mathbb{F}_{q}$ and computed the upper and lower bounds. Furthermore, Jitman et al. 16 introduced the hulls of cyclic codes of odd length over $\mathbb{Z}_{4}$ and gave an algorithm to determine types of hulls of cyclic codes over $\mathbb{Z}_{4}$ . In this article, utilizing the results provided by Jitman et al. 16 , we study the construction of hulls for our purpose. The cyclic codes are important family of linear codes. The study of cyclic codes over finite rings have become a crucial topic of research. The cyclic codes over finite rings have been studied in a series of papers 4 ; 5 ; 11 ; 26 ; 27 . In particular, Hammons et al.27 done a brilliant work on codes over $\mathbb{Z}_{4}$ and discussed $\mathbb{Z}_{4}$ -linearity of different types of codes. After that many authors were attracted to do a lots of work over $\mathbb{Z}_{4}$ . The ring $\Re=\mathbb{Z}_{4}[v]/\langle v^{2}-v\rangle$ is an extension of ring $\mathbb{Z}_{4}$ . For the first time Bandi and Bhaintwal 13 introduced linear codes over finite ring $\Re$ . They also studied the MacWilliams identities for Lee weight and Gray enumerators for codes over $\Re$ . After that, Gao et al. 5 studied the different types of linear codes over $\Re$ and also obtained many new $\mathbb{Z}_{4}$ -linear codes with good parameters. Recently, Dinh et al. 11 introduced a new Gray map and they studied the all-constacyclic codes over $\Re$ . Moreover, they also obtained the various new $\mathbb{Z}_{4}$ -linear codes with better parameters. Further, Kumar and Singh 12 studied the DNA computing by using cyclic codes over $\Re$ . They provided various examples of reversible cyclic codes over $\Re$ and obtained many good $\mathbb{Z}_{4}$ -linear codes with good parameters with the help of Gray map. These works encourage us to study the Hulls of cyclic codes over $\Re=\mathbb{Z}_{4}+vZ_{4}$ , $v^{2}=v$ . The remaining part of the article is organised in such manner. In section $2$ , some basic properties and results related with cyclic codes over $\Re$ are conferred. Using the cyclic codes structure, the construction of hulls of cyclic codes over $\Re$ is studied in section $3$ . The types of hulls of cyclic codes over $\Re$ are discussed in section $4$ . Section $5$ is devoted to study of average $2$ -dimension of cyclic codes over $\Re$ . In section $6$ , various examples of hulls of cyclic codes over $\Re$ are given and we obtain the $\mathbb{Z}_{4}$ -linear codes with good parameters according to $\mathbb{Z}_{4}$ -database 25 .

2 Preliminaries

As a linear code $C$ of length n over $\mathbb{Z}_{4}$ can be viewed as a vector space over $\mathbb{F}_{2}$ and the concept of 2-dimension of C was introduced in 28 and is given as $\dim_{2}(C)$ = log ${}_{2}(|C|)$ . Let $h=(h_{0},h_{1},\dots,h_{n-1}),k=(k_{0},k_{1},\dots,k_{n-1})\in\Re^{n}$ , the inner product is given as

[TABLE]

The dual code $C^{\perp}$ is conferred as

[TABLE]

Then $C$ is called self-orthogonal if $C\subseteq C^{\perp}$ and $C$ is self-dual if $C=C^{\perp}$ . The $hull$ of $C$ is defined as

[TABLE]

Let $C$ be a cyclic code over $\mathbb{Z}_{4}$ of odd length $n$ . Then generator of $C$ is given by

[TABLE]

where $f(x),g(x),h(x)$ are unique monic polynomials over $\mathbb{Z}_{4}$ such that $f(x)g(x)h(x)=x^{n}-1$ (14 , Theorem $12.3.13$ ) and $|C|=4^{\deg h(x)}2^{\deg g(x)}$ . Here, $C$ is said to be of type $4^{\deg h(x)}2^{\deg g(x)}$ . In this case, the $2$ -dimension of $C$ is dim ${}_{2}(C)$ = log ${}_{2}(|C|)=2\deg h(x)+\deg g(x)$ . A reciprocal polynomial of $h(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dots+x^{r}$ is defined to be

[TABLE]

Clearly, $(h^{*})^{*}(x)=h(x)$ . A polynomial $h(x)$ is called self-reciprocal if $h^{*}(x)=h(x)$ . From (14 , Theorem 12.3.20), the generator of dual cyclic code $C^{\bot}$ is given as

[TABLE]

Some results are taken from 16 , which are given below. Let $ord_{j}(i)$ denotes the mutiplicative order of $i$ modulo $j$ , where $i$ and $j$ are coprime integers. Let $N_{2}:=\{l\geq 1:l$ divides $2^{i}+1$ for some positive integer $i\}$ . The factorization of $x^{n}-1$ is given as

[TABLE]

where

[TABLE]

and $f_{ij}(x)$ , $f_{ij}^{*}(x)$ form a monic basic irreducible reciprocal polynomial pair and $g_{ij}(x)$ is a monic basic irreducible self-reciprocal self-reciprocal polynomial. Let $B_{n}=\deg\prod_{j|n,j\in N_{2}}\big{(}\prod_{i=1}^{\gamma(j)}g_{ij}(x)\big{)}$ . Then

[TABLE]

The Gray map $\xi$ is taken from 11 that transfers the elements of $\Re$ to elements of $\mathbb{Z}_{4}^{2}$ such that

[TABLE]

The Lee weight $w_{L}(a)$ in $\mathbb{Z}_{4}$ is given as $\min\{{a,4-a}\}$ . The Lee weight of any element of $\Re$ is defined as $w_{L}(a+bv)=w_{L}(a)+w_{L}(2b+a)$ .

Theorem 2.1

11 * The Gray map $\xi$ is $\mathbb{Z}_{4}$ -linear, and it is a distance-preserving map from $\Re^{n}$ (Lee distance) to $\mathbb{Z}^{2n}_{4}$ (Lee distance).*

3 Hulls of Cyclic Codes over $\Re$

In present section, the theory of hulls of cyclic codes over $\Re$ is discussed. First, we review some useful results, which will be used to describe the hulls of cyclic codes over $\Re$ .Theorem 3.1. 16 Let $C$ be a cyclic code of odd length $n$ over $\mathbb{Z}_{4}$ generated by $\langle f(x)g(x),2f(x)\rangle$ , where $x^{n}-1=f(x)g(x)h(x)$ and $f(x),g(x)$ and $h(x)$ are pairwise coprime. Then Hull(C) is generated by

[TABLE]

Furthermore, Hull(C) is of type $4^{\deg H(x)}2^{\deg G(x)}$ , where

[TABLE]

Some important results of cyclic codes over $\Re$ are given as follows.

Theorem 3.2. 11 A linear code over $\Re$ is given as $C=vC_{1}\oplus(1+3v)C_{2}$ . Then, $C$ is cyclic if and only if $C_{1}$ and $C_{2}$ are cyclic over $\mathbb{Z}_{4}$ .

Proposition 3.3. 11 Let n-tuple ${k}\in{\Re}^{n}$ , then $\xi\rho({k})={\rho}^{2}\xi({k})$ , where $\rho$ is the cyclic shift on $\mathbb{Z}_{4}^{2n}$ . Proposition 3.4. 11 Let $C$ be a cyclic code of length $n$ over $\Re$ . Then its Gray image $\xi(C)$ is a $2$ -quasi-cyclic code of length $2n$ over $\mathbb{Z}_{4}$ . Let $C$ be cyclic code over $\Re$ of odd length $n$ . By utilizing the equation $1$ , the generator of cyclic codes is described in next result. Theorem 3.5. Let $C=vC_{1}\oplus(1-v)C_{2}$ be a cyclic code of odd length $n$ over $\Re$ . Then

$C=\langle vp_{1}(x)q_{1}(x),2vp_{1}(x),(1-v)p_{2}(x)q_{2}(x),2(1-v)p_{2}(x)\rangle,$

where $p_{1}(x)q_{1}(x)r_{1}(x)=p_{2}(x)q_{2}(x)r_{2}(x)=x^{n}-1$ and $C_{1}=\langle p_{1}(x)q_{1}(x),2p_{1}(x)\rangle$ , $C_{2}=\langle p_{2}(x)q_{2}(x),2p_{2}(x)\rangle$ over $\mathbb{Z}_{4}$ , respectively.

Proof

Let $\hat{C}=\langle vp_{1}(x)q_{1}(x),2vp_{1}(x),(1-v)p_{2}(x)q_{2}(x),2(1-v)p_{2}(x)\rangle$ , then it is obvious that $\hat{C}\subseteq C$ . We have $(v)C_{1}=v\hat{C}$ since $v^{2}=v$ over $\mathbb{Z}_{4}$ . Moreover, $(1-v)^{2}=(1-v)$ , then $(1-v)C_{2}=(1-v)\hat{C}$ . Thus, $vC_{1}\oplus(1-v)C_{2}\subseteq\hat{C}$ . Hence, $C=\hat{C}$

Proposition 3.6.11 Let $C$ be a linear code of length $n$ over $\Re$ , then $C^{\perp}=vC_{1}^{\perp}\oplus(1-v)C_{2}^{\perp}$ .

By using the dual cyclic codes $C^{\bot}$ over $\mathbb{Z}_{4}$ [given in equation 2], we describe the structure of dual of cyclic codes over $\Re$ in next result.

Proposition 3.7. If $C=\langle vp_{1}(x)q_{1}(x),2vp_{1}(x),(1-v)p_{2}(x)q_{2}(x),2(1-v)p_{2}(x)\rangle$ is a cyclic code of odd length $n$ over $\Re$ . Then $C^{\perp}=\langle vr_{1}^{*}(x)q_{1}^{*}(x),2vr_{1}^{*}(x),(1-v)r_{2}^{*}(x)q_{2}^{*}(x),2(1-v)r_{2}^{*}(x)\rangle$ , where $p^{*}(x)=x^{\deg p(x)}$ $p(x^{-1})$ .

Proof

If $C$ is a cyclic code, then dual $C^{\perp}$ is also cyclic code. Moreover, $C^{\perp}=vC_{1}^{\perp}\oplus(1-v)C_{2}^{\perp}$ by Proposition $3.6$ . Thus, equation $2$ and Proposition $3.5$ imply our result.

Lemma 3.8. Let $a=(a_{0},a_{1},\dots,a_{n-1})$ and $b=(b_{0},b_{1},\dots,b_{n-1})$ be vectors in $\Re^{n}$ with corresponding polynomials $a(x)$ and $b(x)$ , respectively. Then $a$ is orthogonal to $b$ and all its shifts if and only if $a(x)b^{*}(x)=0$ in $\Re[x]/\langle x^{n}-1\rangle$ .

Proof

Proof is similar to (14 , Theorem 12.3.18).

Next result provide the generator of the hulls of the cyclic codes over $\Re$ . Theorem 3.9. Let $C$ be cyclic code of odd length $n$ over $\Re$ such that $C=vC_{1}\oplus(1-v)C_{2}$ and $C$ is generated by

[TABLE]

where $p_{1}(x)q_{1}(x)r_{1}(x)=p_{2}(x)q_{2}(x)r_{2}(x)=x^{n}-1$ and $C_{1}=\langle p_{1}(x)q_{1}(x),2p_{1}(x)\rangle$ , $C_{2}=\langle p_{2}(x)q_{2}(x),2p_{2}(x)\rangle$ over $\mathbb{Z}_{4}$ , respectively. Then Hull(C) has generator of the form $\langle v~{}\text{lcm}~{}(p_{1}(x)q_{1}(x),r^{*}_{1}(x)q^{*}_{1}(x)),2v~{}\text{lcm}~{}(p_{1}(x),r^{*}_{1}(x)),(1-v)~{}\text{lcm}~{}(p_{2}(x)q_{2}(x),r^{*}_{2}(x)q^{*}_{2}(x)),2(1-v)~{}\text{lcm}~{}(p_{2}(x),r^{*}_{2}(x))\rangle$

In addition, Hull(C) is of the type $4^{\deg R_{1}(x)+\deg R_{2}(x)}2^{\deg Q_{1}(x)+\deg Q_{2}(x)}$ , where

[TABLE]

and

[TABLE]

Proof

From Theorem $3.5$ , the generator of a cyclic code $C$ over $\Re$ is conferred as

[TABLE]

and from Proposition $3.7$ , the dual of cyclic codes over $\Re$ is given as

[TABLE]

We consider $\bar{C}$ is a cyclic code over $\Re$ , which has generator of the form

[TABLE]

where

[TABLE]

and

[TABLE]

such that $x^{n}-1=P_{1}(x)Q_{1}(x)R_{1}(x)=P_{2}(x)Q_{2}(x)R_{2}(x)$ and $P_{1}(x),Q_{1}(x),R_{1}(x),P_{2}(x),Q_{2}(x),R_{2}(x)$ are pairwise coprime polynomials. Note that $\langle vP_{1}(x)Q_{1}(x),2vP_{1}(x),(1-v)P_{2}(x)Q_{2}(x),2(1-v)P_{2}(x)\rangle\subseteq\langle vp_{1}(x)q_{1}(x),2vp_{1}(x),(1-v)p_{2}(x)q_{2}(x),2(1-v)p_{2}(x)\rangle$ and also $\langle vP_{1}(x)Q_{1}(x),2vP_{1}(x),(1-v)P_{2}(x)Q_{2}(x),2(1-v)P_{2}(x)\rangle\subseteq\langle vr_{1}^{*}(x)q_{1}^{*}(x),2vr_{1}^{*}(x),(1-v)r_{2}^{*}(x)q_{2}^{*}(x),2(1-v)r_{2}^{*}(x)\rangle,$ therefore we get $\bar{C}\subseteq\text{Hull(C)}$ . Now, we have to show that $\text{Hull(C)}\subseteq\bar{C}$ . Note that Hull(C) is a cyclic code over $\Re$ and generated by

[TABLE]

where the polynomials $L_{1}(x),M_{1}(x),N_{1}(x),M_{2}(x),L_{2}(x),N_{2}(x)$ are pairwise coprime such that

[TABLE]

As the $\text{Hull(C)}\subseteq C^{\perp}$ is orthogonal to $C$ and from Lemma $3.8$ , we have

[TABLE]

that means $r^{*}_{1}(x)q^{*}_{1}(x)|L_{1}(x)M_{1}(x),~{}r^{*}_{2}(x)q^{*}_{2}(x)|L_{2}(x)M_{2}(x)$ and also

[TABLE]

which provided that $r^{*}_{1}(x)|L_{1}(x),r^{*}_{2}(x)|L_{2}(x)$ . The $Hull(C)\subseteq C$ is orthogonal to $C^{\perp}$ and from Lemma $3.8$ , we have

[TABLE]

that means $p_{1}(x)q_{1}(x)|L_{1}(x)M_{1}(x),~{}p_{2}(x)q_{2}(x)|L_{2}(x)M_{2}(x)$ and also

$2L_{1}(x).r_{1}(x)q_{1}(x)=0,~{}2L_{2}(x).r_{2}(x)q_{2}(x)=0$ , that gives $p_{1}(x)|L_{1}(x)$ and $p_{2}(x)|L_{2}(x)$ . Therefore, $\text{lcm}(p_{1}(x)q_{1}(x),r^{*}_{1}(x)q^{*}_{1}(x))|L_{1}(x)M_{1}(x),\text{lcm}(p_{2}(x)q_{2}(x),r^{*}_{2}(x)q^{*}_{2}(x))|L_{2}(x)M_{2}(x)$ and $\text{lcm}(r^{*}_{1}(x),p_{1}(x))|L_{1}(x),$ $~{}\text{lcm}(r^{*}_{2}(x),p_{2}(x))|L_{2}(x)$ . That means $P_{1}(x)Q_{1}(x)|L_{1}(x)M_{1}(x),~{}P_{2}(x)Q_{2}(x)|L_{2}(x)M_{2}(x)$ and $P_{1}(x)|L_{1}(x),~{}P_{2}(x)|L_{2}(x).$ Hence, $\text{Hull(C)}\subseteq\bar{C}$ . Thus, $\text{Hull(C)}=\bar{C}$ .

4 Types of Hulls of Cyclic Codes over $\Re$

Present section devotes to study the types of hulls of cyclic codes of odd length $n$ over $\Re$ . For this, first we discuss some results. Lemma 4.1 16 Let $\beta$ be a positive integer. For $1\leq i\leq\beta$ , let $(v_{i},z_{i}),(w_{i},d_{i})$ and $(u_{i},b_{i})$ be elements in $\{(0,0),(1,0),(0,1)\}$ . Let $a_{i}=\text{min}\{1-v_{i}-z_{i},w_{i}\}+\text{min}\{1-w_{i}-d_{i},v_{i}\}.$ Then $a_{i}\in\{0,1\}$ . Moreover, the following statements hold.

$2-\text{min}\{1-v_{i}-z_{i},w_{i}\}-\text{max}\{v_{i},1-w_{i}-d_{i}\}-\text{min}\{1-w_{i}-d_{i},v_{i}\}-\text{max}\{w_{i},1-v_{i}-z_{i}\}=z_{i}+d_{i}$ . 2. 2.

If $a_{i}=0$ , then $z_{i}+d_{i}\in\{0,1,2\}$ . 3. 3.

If $a_{i}=1$ , then $z_{i}+d_{i}=0$ . 4. 4.

Let $a=\sum_{i=1}^{\beta}a_{ij},$ then $\sum_{i=1}^{\beta}(z_{i}+d_{i})=c$ for some $0\leq c\leq 2(\beta-a)$ .

Theorem 4.2 The types of the hull of a cyclic code of odd length n over $\Re$ are $4^{k_{1}}2^{k_{2}}$ , where

[TABLE]

$0\leq a_{1j},a_{2j}\leq\beta(j),0\leq b_{1j},b_{2j}\leq\gamma(j)$ and $0\leq c_{1j}\leq 2(\beta(j)-a_{1j})$ , $0\leq c_{2j}\leq 2(\beta(j)-a_{2j})$ .

Proof

From Theorem $3.9$ , Hull(C) has type

[TABLE]

From equation $(3)$ , we get

[TABLE]

and

[TABLE]

where $k=1,2$ and $(u_{kij},b_{kij}),(v_{kij},z_{kij}),(w_{kij},d_{kij})\in\{(0,0),(1,0)(0,1)\}$ . According to above relations, we have

[TABLE]

that means

[TABLE]

In the similar way, we determine the $\deg Q_{k}(x)$ , where $k=1,2$ .

[TABLE]

Thus,

[TABLE]

From Theorem $3.9$ , Hull(C) is of the type $4^{\deg R_{1}(x)+\deg R_{2}(x)}2^{\deg Q_{1}(x)+\deg Q_{2}(x)}$ , where

[TABLE]

where $0\leq a_{1j},a_{2j}\leq\beta(j),0\leq b_{1j},b_{2j}\leq\gamma(j)$ and $0\leq c_{1j}\leq 2(\beta(j)-a_{1j})$ , $0\leq c_{2j}\leq 2(\beta(j)-a_{2j})$

Corollary 4.3 For odd length $n$ such that $n\in N_{2}$ . Then the types of the hull of a cyclic codes over $\Re$ are of the form $4^{0}2^{k_{2}}$ , where

[TABLE]

Proof

From the Theorem $4.2$ , if $n\in N_{2}$ and $j|n,j\in N_{2}$ , we get the required result.

Algorithm: The types of the hull of a cyclic code of odd length $n$ over $\Re.$

For each $j|n$ , consider the following cases.

(a)

If $j\in N_{2}$ , then compute $ord_{j}(2)$ and $\gamma(j)$ . 2. (b)

If $j\notin N_{2}$ , then compute $ord_{j}(2)$ and $\beta(j)$ . 2. 2.

Compute

[TABLE]

where $0\leq a_{1j},a_{2j}\leq\beta(j)$ . 3. 3.

Next, we determine $k_{2}$ for fixed $a_{1j},a_{2j}$ in $2$ ,

$k_{2}=\sum\limits_{j|n,j\in N_{2}}ord_{j}(2)\cdot b_{1j}+\sum\limits_{j|n,j\notin N_{2}}ord_{j}(2)\cdot c_{1j}+\sum\limits_{j|n,j\in N_{2}}ord_{j}(2)\cdot b_{2j}+\sum\limits_{j|n,j\notin N_{2}}ord_{j}(2)\cdot c_{2j}$ ,

where $0\leq b_{1j},b_{2j}\leq\gamma(j)$ and $0\leq c_{1j}\leq 2(\beta(j)-a_{1j})$ , $0\leq c_{2j}\leq 2(\beta(j)-a_{2j}).$

**Example $1$ : **

For $n=7$ , we discuss the types of hulls of cyclic codes over $\Re$ . If $1\in N_{2}$ , then $ord_{1}(2)=1$ , and $\gamma(1)=1$ and if $7\notin N_{2}$ , then $ord_{7}(2)=3$ and $\beta(7)=1$ . Which implies $k_{1}=3a_{17}+3a_{27}$ , with $0\leq(a_{17},a_{27})\leq 1$ . Then we have following cases.

Let $(a_{17},a_{27})=(0,0)$ , i.e. $k_{1}=0$ and

$k_{2}=b_{11}+b_{21}+3c_{17}+3c_{27}$ , with $0\leq(b_{11},b_{21})\leq 1$ and $0\leq c_{17},c_{27}\leq 2$ . Then $k_{2}=(0,1,2,3,4,5,6,7,8,9,10,11,12,13,14)$ 2. 2.

Let $(a_{17},a_{27})=(1,0)$ , i.e. $k_{1}=3$ and

$k_{2}=b_{11}+b_{21}+3c_{17}+3c_{27}$ , with $0\leq(b_{11},b_{21})\leq 1$ and $c_{17}=0,0\leq c_{27}\leq 2$ . Then $k_{2}=(0,1,2,3,4,5,6,7,8)$ 3. 3.

Let $(a_{17},a_{27})=(0,1)$ , i.e. $k_{1}=3$ and

$k_{2}=b_{11}+b_{21}+3c_{17}+3c_{27}$ , with $0\leq(b_{11},b_{21})\leq 1$ and $c_{27}=0,0\leq c_{17}\leq 2$ . Then $k_{2}=(0,1,2,3,4,5,6,7,8)$ 4. 4.

Let $(a_{17},a_{27})=(1,1)$ , i.e. $k_{1}=6$ and

$k_{2}=b_{11}+b_{21}+3c_{17}+3c_{27}$ , with $0\leq(b_{11},b_{21})\leq 1$ and $c_{17},c_{27}=0$ . Then $k_{2}=(0,1,2)$

**Example 2: **

For $n=15$ , we discuss the types of hulls of cyclic codes over $\Re$ . Let $1,3,5\in N_{2}$ , we get $ord_{1}(2)=1,ord_{3}(2)=2,ord_{5}(2)=4$ , and $\gamma(1)=\gamma(3)=\gamma(5)=1$ and $15\notin N_{2}$ , then $ord_{15}(2)=4$ and $\beta(15)=1$ . Which implies $k_{1}=4a_{115}+4a_{215}$ , with $0\leq(a_{115},a_{215})\leq 1$ . Then, we have following cases.

Let $(a_{115},a_{215})=(0,0)$ , i.e. $k_{1}=0$ and

$k_{2}=b_{11}+b_{21}+2b_{13}+2b_{23}+4b_{15}+4b_{25}+4c_{115}+4c_{215}$ , with $0\leq(b_{11},b_{21},b_{13},b_{23},b_{15},b_{25})\leq 1$ and $0\leq c_{115},c_{215}\leq 2$ . Then $k_{2}=(0,1,2,\dots,30)$ 2. 2.

Let $(a_{115},a_{215})=(1,0)$ , i.e. $k_{1}=4$ and

$k_{2}=b_{11}+b_{21}+2b_{13}+2b_{23}+4b_{15}+4b_{25}+4c_{115}+4c_{215}$ , with $0\leq(b_{11},b_{21},b_{13},b_{23},b_{15},b_{25})\leq 1$ and $c_{115}=0,~{}0\leq c_{215}\leq 2$ . Then $k_{2}=(0,1,2,\dots,22)$ 3. 3.

Let $(a_{115},a_{215})=(0,1)$ , i.e. $k_{1}=4$ and

$k_{2}=b_{11}+b_{21}+2b_{13}+2b_{23}+4b_{15}+4b_{25}+4c_{115}+4c_{215}$ , with $0\leq(b_{11},b_{21},b_{13},b_{23},b_{15},b_{25})\leq 1$ and $0\leq c_{115}\leq 2,c_{215}=0$ . Then $k_{2}=(0,1,2,\dots,22)$ 4. 4.

Let $(a_{115},a_{215})=(1,1)$ , i.e. $k_{1}=8$ and

$k_{2}=b_{11}+b_{21}+2b_{13}+2b_{23}+4b_{15}+4b_{25}+4c_{115}+4c_{215}$ , with $0\leq(b_{11},b_{21},b_{13},b_{23},b_{15},b_{25})\leq 1$ and $c_{115}=c_{215}=0$ . Then $k_{2}=(0,1,2,\dots,14)$

5 The Average $2$ -Dimension $E(n)$

In this section, the average $2$ -dimension of the cyclic codes of length $n$ over $\Re$ is discussed. The average $2$ -dimension of the cyclic codes over $\mathbb{Z}_{4}$ is given as

[TABLE]

where $C(n,4)$ denotes the set of all cyclic codes over $\mathbb{Z}_{4}$ . Lemma 5.1 16 Let $(v,z),(w,d),(u,b)\in\{(0,0),(1,0),(0,1)\}$ . Then

$E(1-\text{max}\{u,1-u-b\})=\dfrac{1}{3}$ 2. 2.

$E(2+\text{min}\{1-v-z,w\}-\text{max}\{v,1-w-d\}+\text{min}\{1-w-d,v\}-\text{max}\{w,1-v-z\})=\dfrac{10}{9}$

Next, we study the formula of average $2$ - dimension of cyclic codes over $\Re$ by utilizing above result and the expectation $E(Y)$ , where $Y$ is the random variable of the $2$ -dimension $dim_{2}(\text{Hull(C)})$ and $C$ is chosen randomly from $C(n,4)$ with uniform probability. Theorem 5.2 The value of $E(n)$ of hulls of cyclic codes of odd length $n$ over $\Re$ is conferred as

[TABLE]

where $B_{n}$ is given in Equation $(4)$ .

Proof

The structure of cyclic codes $C$ of odd length $n$ over $\Re$ is determined as

[TABLE]

where $p_{1}(x)q_{1}(x)r_{1}(x)=p_{2}(x)q_{2}(x)r_{2}(x)=x^{n}-1$ , where for $i=1,2$ , $p_{i}(x),q_{i}(x)$ and $r_{i}(x)$ are pairwise coprime polynomials over $\mathbb{Z}_{4}$ . The Hull(C) has type

[TABLE]

and $2$ - dimension of Hull(C) is

[TABLE]

Next, utilizing the Lemma $5.1$ , we have

[TABLE]

Corollary 5.3 In the Theorem 5.2, we have $E(n)<\dfrac{10n}{9}$ . In Table $1$ , we are given the average $2$ -dimension $E(n)$ of the hull of cyclic codes of odd length $n$ from $55$ upto $151$ . The $(\dagger)$ denotes the condition the $n\in N_{2}$ and otherwise $n\not\in N_{2}$ in the Table $1$ . The Table $1$ is given after the references.

Table $1$ .

6 Computational work

In this section, we have provided the various examples of hulls of cyclic codes of odd length over $\Re$ . Among these, the generator polynomials of hulls of cyclic codes over $\Re$ are obtained. We know that cyclic codes over $\Re$ are mapped via Gray map into $\mathbb{Z}_{4}^{2}$ and these Gray images are $\mathbb{Z}_{4}$ -linear codes. Thus, the Lee weights of the $\mathbb{Z}_{4}$ -images are obtained. Various good $\mathbb{Z}_{4}$ -linear codes with good parameters are obtained according to data-base25 . Here, the monic polynomials are considered in ascending order such that $x^{4}+3x^{3}+2x^{2}+x+1$ as $11231$ in Tables $2,3.$ The generator of hulls of cyclic codes over $\Re$ are given in following manner such that $(3+2v)+(1+2v)x+vx^{2}+(2+v)x^{3}+2vx^{4}=(31020)+\text{v}(22112)$ in Tables $2,3.$ The $({*})$ denotes good $\mathbb{Z}_{4}$ -parameter.

Example $3$ . Let $n=15$ and

[TABLE]

over $\mathbb{Z}_{4}$ . In Table 2, the generator of hulls of cyclic codes over $\Re$ are obtained.

Table 2.

$\mathbb{Z}_{4}-\text{polynomials}$

$\mathbb{Z}_{4}-\text{ polynomials}$

${\tiny\text{Generator of hulls over}~{}\Re}$

$\mathbb{Z}_{4}$ -Parameters

$p_{1}=q_{2}=1131201$

$q_{1}=r_{2}=31$

$r_{1}=p_{2}=102232311$

$(220022200200000)+v(313301033221000)$

$(30,4^{0}2^{10},16)^{*}$

$p_{1}=q_{2}=102232311$

$q_{1}=r_{2}=111$

$r_{1}=p_{2}=312321$

$(200020220000000)+v(322312301111000)$

$(30,4^{0}2^{10},16)^{*}$

$p_{1}=q_{2}=13201$

$q_{1}=r_{2}=33300111$

$r_{1}=p_{2}=10231$

$(222202022002000)+v(300132303313000)$

$(30,4^{0}2^{9},2)^{*}$

$p_{1}=q_{2}=10231$

$q_{1}=r_{2}=3001$

$r_{1}=p_{2}=102232311$

$(200220202222000)+v(313121231023000)$

$(30,4^{0}2^{9},12)^{*}$

$p_{1}=r_{2}=10231$

$r_{1}=q_{2}=11111$

$q_{1}=p_{2}=3233221121$

$(122110301311000)+v(122112101333000)$

$(30,4^{4}2^{4},16)^{*}$

$p_{1}=r_{2}=111$

$q_{1}=p_{2}=11111$

$r_{1}=q_{2}=3233221121$

$(200222020000000)+v(022222000022000)$

$(30,4^{0}2^{9},16)^{*}$

$p_{1}=r_{2}=13201$

$q_{1}=p_{2}=10231$

$r_{1}=q_{2}=33300111$

$(131101031201000)+v(113123211021000)$

$(30,4^{4}2^{7},12)^{*}$

$p_{1}=r_{2}=312321$

$q_{1}=p_{2}=1021311$

$r_{1}=q_{2}=11111$

$(102130123131000)+v(102132323111000)$

$(30,4^{4}2^{0},24)^{*}$

$p_{1}=r_{2}=31$

$q_{1}=q_{2}=111$

$r_{1}=p_{2}=1001001001001$

$(220220220220220)+v(000000000000000)$

$(30,4^{0}2^{2},40)^{*}$

$p_{1}=r_{2}=1021211$

$q_{1}=q_{2}=113232201$

$r_{1}=p_{2}=31$

$(133323233021000)+v(131321231023000)$

$(30,4^{4}2^{1},20)$

$p_{1}=r_{2}=312321$

$q_{1}=q_{2}=102232311$

$r_{1}=p_{2}=111$

$(122332121111000)+v(122132103333000)$

$(30,4^{4}2^{2},16)^{*}$

$p_{1}=r_{2}=3001$

$q_{1}=q_{2}=11111$

$r_{1}=p_{2}=130131031$

$(220002200022000)+v(000000000000000)$

$(30,4^{0}2^{4},24)^{*}$

$p_{1}=r_{2}=11111$

$q_{1}=q_{2}=3001$

$r_{1}=p_{2}=130131031$

$(200200200200200)+v(000000000000000)$

$(30,4^{0}2^{3},20)^{*}$

$p_{1}=r_{2}=10231$

$q_{1}=q_{2}=13201$

$r_{1}=p_{2}=32200111$

$(200220202222000)+v(022022220220000)$

$(30,4^{0}2^{4},32)^{*}$

$p_{1}=r_{2}=(10231)(13201)$

$q_{1}=p_{2}=(31)$

$r_{1}=q_{2}=(111)(11111)$

$(202200220200000)+v(020022002022222)$

$(30,4^{0}2^{6},24)^{*}$

Example $4$ . Let $n=21$ and $x^{21}-1=(x-1)(x^{2}+x+1)(x^{6}+2x^{5}+3x^{4}+3x^{2}+x+1)(x^{6}+x^{5}+3x^{4}+3x^{2}+2x+1)(x^{3}+2*x^{2}+x+3)(x^{3}+3x^{2}+2x+3)$ over $\mathbb{Z}_{4}$ . In Table 3, the generator of hulls of cyclic codes over $\Re$ are obtained.

**Table 3.

$\mathbb{Z}_{4}$ -polynomials

$\mathbb{Z}_{4}$ - polynomials

$\text{Generator of hulls over}~{}\Re$

$\mathbb{Z}_{4}$ -Parameters

$p_{1}=q_{2}=1322121$

$q_{1}=r_{2}=1301301031031$

$r_{1}=p_{2}=3231$

$(222202220022020200000)+v(121332032333021031100)$

$(42,4^{0}2^{9},24)^{*}$

$p_{1}=q_{2}=132120121$

$q_{1}=r_{2}=30000001$

$r_{1}=p_{2}=1130321$

$(202022220202220000000)+v(100120102002320000000)$

$(42,4^{0}2^{28},4)$

$p_{1}=r_{2}=333222333111$

$q_{1}=p_{2}=3001002001$

$r_{1}=q_{2}=31$

$(300022322300122000000)+v(300222322100122000000)$

$(42,4^{9}2^{1},12)$

$p_{1}=q_{2}=3231$

$q_{1}=r_{2}=3100000310000031$

$r_{1}=p_{2}=3121$

$(300022322300122000000)+v(300222322100122000000)$

$(42,4^{0}2^{3},24)^{*}$

$p_{1}=q_{2}=1230311$

$q_{1}=r_{2}=3330000111$

$r_{1}=p_{2}=1130321$

$(000000000000000000000)+v(312320332212012100000)$

$(42,4^{0}2^{6},16)^{*}$

$p_{1}=q_{2}=3001$

$q_{1}=p_{2}=3001002001$

$r_{1}=r_{2}=3002003001$

$(300000300300100000000)+v(300000300300100000000)$

$(42,4^{10}2^{3},2)$

$p_{1}=q_{2}=12311$

$q_{1}=r_{2}=111$

$r_{1}=p_{2}=3311211102331321$

$(222202220022020200000)+v(123132030331221011100)$

$(42,4^{0}2^{27},2)^{*}$

$p_{1}=r_{2}=111$

$q_{1}=p_{2}=3121$

$r_{1}=q_{2}=10203100123013111$

$(200022000000000000000)+v(002202020222002022200)$

$(42,4^{0}2^{16},12)^{*}$

$p_{1}=q_{2}=1233031201$

$q_{1}=r_{2}=111$

$r_{1}=p_{2}=10213030321$

$(202000022002000000000)+v(132103000313100000000)$

$(42,4^{0}2^{19},16)^{*}$

$p_{1}=r_{2}=302031$

$q_{1}=q_{2}=31$

$r_{1}=p_{2}=3311211102331321$

$(311010031101003110100)+v(032300003230000323000)$

$(42,4^{3}2^{4},24)^{*}$

$p_{1}=r_{2}=13031022031$

$q_{1}=q_{2}=111$

$r_{1}=p_{2}=3002003001$

$(320100120020100000000)+v(000320220200000000000)$

$(42,4^{9}2^{23},2)^{*}$

$p_{1}=r_{2}=1212231$

$q_{1}=q_{2}=1230311$

$r_{1}=p_{2}=3002003001$

$(323310212311201231100)+v(030300003210000301000)$

$(42,4^{3}2^{9},16)$

$p_{1}=r_{2}=1001001001001001001$

$q_{1}=q_{2}=111$

$r_{1}=p_{2}=31$

$(323310212311201231100)+v(030300003210000301000)$

$(42,4^{0}2^{2},56)$

$p_{1}=q_{2}=123333321$

$q_{1}=r_{2}=1301301031031$

$r_{1}=p_{2}=31$

$(202202222022020000000)+v(020202200222020000000)$

$(42,4^{0}2^{8},24)^{*}$

$p_{1}=p_{2}=10213030321$

$q_{1}=r_{2}=121021231$

$r_{1}=q_{2}=3231$

$(130120312010212100000)+v(202032303021233001100)$

$(42,4^{6}2^{9},12)$

**

7 Conclusion

In this article, the construction of cyclic codes of odd length over $\Re$ is conferred. The generators of hulls of cyclic codes over $\Re$ are studied. Moreover, the types of hulls of cyclic codes are discussed. Further, the average $2$ -Dimension $E(n)$ are also conferred. Among, these many new $\mathbb{Z}_{4}$ -linear codes are obtained, which have good parameters. The discussion of hulls of cyclic codes of even length over $\Re$ will be another open problem.

Bibliography28

The reference list from the paper itself. Each links out to its DOI / PubMed record.

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The Discussion on Hulls of Cyclic Codes over the Ring ℜ=Z4+vZ4\Re=\mathbb{Z}_{4}+v\mathbb{Z}_{4}ℜ=Z4​+vZ4​, v2=vv^{2}=vv2=v

Abstract

Keywords:

MSC:

1 Introduction

2 Preliminaries

Theorem 2.1

3 Hulls of Cyclic Codes over ℜ\Reℜ

Proof

Proof

Proof

Proof

4 Types of Hulls of Cyclic Codes over ℜ\Reℜ

Proof

Proof

5 The Average 222-Dimension E(n)E(n)E(n)

Proof

6 Computational work

7 Conclusion

The Discussion on Hulls of Cyclic Codes over the Ring $\Re=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ , $v^{2}=v$

3 Hulls of Cyclic Codes over $\Re$

4 Types of Hulls of Cyclic Codes over $\Re$

5 The Average $2$ -Dimension $E(n)$