## I. INTRODUCTION Cyclic codes are among the most useful and well-studied code families for various reasons, such as effective encoding and decoding. A cyclic code can be viewed as an ideal in a certain quotient ring obtained from a polynomial ring with coefficients from a finite field [1, 2]. Quasi-Cyclic codes are a generalization of cyclic codes [6, 8]. Algebraically, Quasi-Cyclic codes are modules rather than ideals [10, 13]. A $m\Theta$ approach of the notion of sets has allowed to bring out the new classes of sets: $m\Theta$ sets. The notion of modal $\Theta$ -valent set ( $m\Theta$ set) noted $(\mathbb{F}_{p\mathbb{Z}}, F_{\alpha})$, $p$ prime, is defined by F. Ayissi Eteme in [12, 16, 7]. Research on modal algebra has evolved and led to the theory of $m\Theta$ codes [11, 15, 17]. The theory of error-correcting $m\Theta$ codes over finite fields has experienced tremendous growth since its inception [5]. Progress has been attained in the direction of determining the structural properties of $m\Theta$ codes over large families of $m\Theta$ fields. This paper is a contribution along those lines as we focus on codes over finite $m\Theta$ pseudo fields with a linear lattice of $m\Theta$ ideals (the so-called chain $m\Theta$ pseudo fields). The purpose of this paper is to obtain structure theorems for Quasi-Cyclic codes in more general setting. The structures of Quasi-Cyclic codes of length $r$ over finite chain $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)$ are established when $r$ is not divisible by the characteristic of the residue $m\Theta$ pseudo field $\overline{\mathbb{F}(p^k\mathbb{Z},1)}$. Some cases where $r$ is divisible by the characteristic of the residue $m\Theta$ pseudo field $\overline{\mathbb{F}(p^k\mathbb{Z},1)}$ are also considered. After presenting preliminary concepts and results on $m\Theta$ set in Section 2. Section 3 presents a canonical construction of the structures of modal $\Theta$ -valent field and modal $\Theta$ -valent field. Section 4 is devoted to the notion of modal $\Theta$ -valent extension of a finite field. Section 5 define the intrinsic polynomial representation of the $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},r)$. Section 6 presents the $m\Theta$ Quasi-Cyclic codes. Lastly, section 7 presents the structure of Quasi-Cyclic code over finite chain $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)$. ## II. PRELIMINARIES a) The modal $\Theta$ -valent set structure and the algebra of $(\mathbb{F}_{p\mathbb{Z}}, F_{\alpha})$ $m\Theta$ sets are considered to be non-classical sets which are compatible with a non-classical logic called the chrysippian $m\Theta$ logic. Definition 0.1. [14] Let $E$ be a non-empty set, $I$ be a chain whose first and last elements are 0 and 1 respectively, $(F_{\alpha})_{\alpha \in I_{*}}$ where $I_{*} = I \setminus \{0\}$ be a family of applications form $E$ to $E$. A $m\Theta$ set is the pair $(E, (F_{\alpha})_{\alpha \in I_{*}})$ simply denoted by $(E, F_{\alpha})$ satisfying the following four axioms: - $\bigcap_{\alpha}F_{\alpha}(E) = \bigcap_{\alpha \in I_{*}}\{F_{\alpha}(x):x\in E\} \neq \emptyset;$ - $\forall \alpha, \beta \in I_{*}$, if $\alpha \neq \beta$ then $F_{\alpha} \neq F_{\beta}$; - $\forall \alpha,\beta \in I_{*},F_{\alpha}\circ F_{\beta} = F_{\beta};$ - $\forall x, y \in E$, if $\forall \alpha \in I_{*}, F_{\alpha}(x) = F_{\alpha}(y)$ then $x = y$. Theorem 0.1. [16](The theorem of $m\Theta$ determination) Let $(E,F_{\alpha})$ be a $m\Theta$ set. $$ \forall x, y \in E, x =_{\Theta} y \text{if and only if} \forall \alpha \in I_*, F_{\alpha}(x) = F_{\alpha}(y). $$ Proof 0.1. [16] Definition 0.2. [5] Let $\mathrm{C}(E, F_{\alpha}) = \bigcap_{\alpha \in I_{*}} F_{\alpha}(E)$. We call $\mathrm{C}(E, F_{\alpha})$ the set of $m\Theta$ invariant elements of the $m\Theta$ set $(E, F_{\alpha})$. Proposition 0.1. [16] Let $(E,F_{\alpha})$ be a $m\Theta$ set. The following properties are equivalent: 1. $x\in \bigcap_{\alpha \in I_*}F_\alpha (E)$ 2. $\forall \alpha \in I_{*}$ $F_{\alpha}(x) = x$ 3. $\forall \alpha,\beta \in I_{*},F_{\alpha}(x) = F_{\beta}(x);$ 4. $\exists \mu \in I_{*}, x = F_{\mu}(x)$ Proof 0.2. [16] Definition 0.3. [12] Let $(E,F_{\alpha})$ and $(E^{\prime},F_{\alpha}^{\prime})$ be two $m\Theta$ sets. Let $X$ be a non-empty set. We shall call 1. $(E', F_{\alpha}')$ a modal $\Theta$ -valent subset of $(E, F_{\alpha})$ if the structure of $m\Theta$ set $(E', F_{\alpha}')$ is the restriction to $E'$ of the structure of the $m\Theta$ set $(E, F_{\alpha})$, this means: - $E^{\prime}\subseteq E$ - $\forall \alpha:\alpha \in I_{*},F^{\prime}_{\alpha} = F_{\alpha_{|E^{\prime}}}$ ### 2. $X$ a modal $\Theta$ -valent subset of $(E,F_{\alpha})$ if: (X,F_{\alpha|_X})\text{isa}m\Theta s\text{whichisamodal}\Theta\text{-valentsubsetof}(E,F_{\alpha}). In all what follows we shall write $F_{\alpha}x$ for $F_{\alpha}(x)$, $F_{\alpha}E$ for $F_{\alpha}(E)$, etc. Let $p \in \mathbb{N}$, a prime number. Let us recall that if $a \in \mathbb{F}_{p\mathbb{Z}}$. $$ \mathbb{F} _ {p \mathbb{Z}} = \mathbb{F} _ {p} \cup \{x _ {p \mathbb{Z}}: \neg (x \equiv 0 (\operatorname{mod} p)) \}; \quad \mathbb{F} _ {p} = \{0, 1, 2, \dots , p - 1 \}. $$ We define the $m\Theta$ support of $a$ denoted $s(a)$ as follows: $$ s \left(a\right) = \left\{ \begin{array}{l l} a & \text{if } a \in \mathbb{F}_{p}; \\ x & \text{if } a = x_{p \mathbb{Z}} \text{ with } x \nmid (x \equiv 0 \, (\text{mod} \, p)). \end{array} \right. $$ Thus $s(a)\in \mathbb{F}_p$ Definition 0.4. [14] Let $\bot$ be a binary operation on $\mathbb{F}_p$. So, $\forall a, b \in \mathbb{F}_p, a \bot b \in \mathbb{F}_p$. Let $x, y \in \mathbb{F}_{p\mathbb{Z}}$. We define a binary operation $\bot^*$ on $\mathbb{F}_{p\mathbb{Z}}$ as follows: $$ x \perp* y = \left\{ \begin{array}{l l} s (x) \perp s (y) & \text{if } \left\{ \begin{array}{l} x, y \in \mathbb{F} _ {p} \\ (s (x) \perp s (y)) \equiv 0 (\text{mod } p) \end{array} \right. \\ (s (x) \perp s (y)) _ {p \mathbb{Z}} & \text{otherwise} \end{array} \right. $$ $\perp^{*}$ as defined above on $\mathbb{F}_{p\mathbb{Z}}$ will be called a $m\Theta$ law on $\mathbb{F}_{p\mathbb{Z}}$ for $x,y\in \mathbb{F}_{p\mathbb{Z}}$ Thus we can define $x + y \in \mathbb{F}_{p\mathbb{Z}}$ and $x \times y \in \mathbb{F}_{p\mathbb{Z}}$ for every $x, y \in \mathbb{F}_{p\mathbb{Z}}$, where $+$ and $\times$ are $m\Theta$ addition and $m\Theta$ multiplication respectively. Theorem 0.2. $\left[12\right]\left(\mathbb{F}_{p\mathbb{Z}},F_{\alpha},+, \times\right)$ is a $m\Theta$ ring of unity 1 and of $m\Theta$ unity $\frac{1}{p\mathbb{Z}}$. Proof 0.3. [12] Remark 0.1. Since $p$ is prime, $(\mathbb{F}_{p\mathbb{Z}}, F_{\alpha})$ is a $m\Theta$ field. Definition 0.5. $\left[4\right]x$ is a divisor of zero in $(\mathbb{F}_{p\mathbb{Z}},F_{\alpha})$ if it exists $y\in \mathbb{F}_{p\mathbb{Z}}$ such that $x\times y = 0$ Example 0.1. [4] $$ p = 2, w e h a v e \mathbb {F} _ {2 \mathbb {Z}} = \{0, 1, 1 _ {2 \mathbb {Z}}, 3 _ {2 \mathbb {Z}} \} $$ The table of $m\Theta$ determination and tables laws of $\mathbb{F}_{2\mathbb{Z}}$. <table><tr><td>F2Z</td><td>0</td><td>1</td><td>12Z</td><td>32Z</td></tr><tr><td>F1</td><td>0</td><td>1</td><td>1</td><td>0</td></tr><tr><td>F2</td><td>0</td><td>1</td><td>0</td><td>1</td></tr></table> <table><tr><td>+ Θ</td><td>0</td><td>1</td><td>12Z</td><td>32Z</td></tr><tr><td>0</td><td>0</td><td>1</td><td>12Z</td><td>32Z</td></tr><tr><td>1</td><td>1</td><td>0</td><td>0</td><td>0</td></tr><tr><td>12Z</td><td>12Z</td><td>0</td><td>0</td><td>0</td></tr><tr><td>32Z</td><td>32Z</td><td>0</td><td>0</td><td>0</td></tr></table> <table><tr><td>xΘ</td><td>0</td><td>1</td><td>12Z</td><td>32Z</td></tr><tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>1</td><td>0</td><td>1</td><td>12Z</td><td>32Z</td></tr><tr><td>12Z</td><td>0</td><td>12Z</td><td>12Z</td><td>32Z</td></tr><tr><td>32Z</td><td>0</td><td>32Z</td><td>32Z</td><td>12Z</td></tr></table> <table><tr><td>xΘ</td><td>0</td><td>1</td><td>12Z</td><td>32Z</td></tr><tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>1</td><td>0</td><td>1</td><td>12Z</td><td>32Z</td></tr><tr><td>12Z</td><td>0</td><td>12Z</td><td>12Z</td><td>32Z</td></tr><tr><td>32Z</td><td>0</td><td>32Z</td><td>32Z</td><td>12Z</td></tr></table> #### Observation: $\mathbb{F}_{2\mathbb{Z}}$ has no divisor of zero, is a $m\Theta$ ring from four elements, that's a $m\Theta$ field of four elements. ## III. CANONICAL CONSTRUCTION OF MODAL Θ-VALENT FIELDS ( $m\Theta f$ ) AND MODAL Θ-VALENT PSEUDO FIELDS( $m\Theta pf$ ) Let $p$ be a prime number, $k \neq 0$ a positive integer, $q = p^k$ and $\mathbb{F}_q$ a finite field with $q$ elements. Two $m\Theta f$ $K_{1}$ and $K_{2}$ of same characteristic $p$ and of same cardinal $p^{2k}$ are $m\Theta$ isomorphic. ### a) Canonical construction of modal $\Theta$ -valent fields ( $m\Theta f$ )[9] Consider that $k = 1$, so $q = p$. $\mathbb{F}_p = \frac{\mathbb{Z}}{p\mathbb{Z}}$ is the prime field of characteristic $p$ and of $p$ elements. The modal $\Theta$ -valent quotient ring $(m\Theta qr)$ $\mathbb{F}_{p\mathbb{Z}}$ as the modal $\Theta$ -valent quotient $\frac{\mathbb{Z}_{p\mathbb{Z}}}{p\mathbb{Z}_{p\mathbb{Z}}}$. $$ Let \mathbb{F}_{p\mathbb{Z}}^* = \mathbb{F}_{p\mathbb{Z}} - \{0\}. \forall x \in \mathbb{F}_{p\mathbb{Z}}^*, \exists x' \in \mathbb{F}_{p\mathbb{Z}}^* / x \cdot x' = \frac{1_{p\mathbb{Z}}}{p\mathbb{Z}_{p\mathbb{Z}}}. $$ $\mathbb{F}_{p\mathbb{Z}}$ has $p^2$ elements but has no proper sub $m\Theta$ ring verifying the preceding property for $\mathbb{F}_{p\mathbb{Z}}^*$. For which reason, $\mathbb{F}_{p\mathbb{Z}}$ is the prime $m\Theta f$ with $p^2$ elements. $\mathbb{F}_p$ is the prime sub field of the $m\Theta$ invariants of $\mathbb{F}_{p\mathbb{Z}}$. Let $f$ be a polynomial with coefficients in $\mathbb{F}_p$. Clearly, it is all the same that: 1. $f_{p}(x)$ irreducible over $\mathbb{F}_p$ 2. $f_{p\mathbb{Z}}(x)$ irreducible over $\mathbb{F}_{p\mathbb{Z}}$ #### Observations: Let $\mathbb{F}(p\mathbb{Z},r) = \frac{\mathbb{F}_p\mathbb{Z}[X]}{(f(X))}$ be the $m\Theta r$ modulo $f(x)$, $(m\Theta r(f))$. $f(x)\in \mathbb{F}_p[X]$. $deg(f) = r$, $r\in \mathbb{N}^*$, $f$ irreducible over $\mathbb{F}_p$. $$ \mathbb{F}_{p\mathbb{Z}}[X]\longrightarrow \mathbb{F}(p\mathbb{Z}, r): g\longmapsto r_{g}; g = q_{g}\cdot f(x) + r_{g}, 0\leq dg(r_{g}) < dg(f). $$ $(\mathbb{F}_{p\mathbb{Z}})^r \longrightarrow \mathbb{F}(p\mathbb{Z}, r): (a_0, \dots, a_{r-1}) \longmapsto \sum_{i=0} a_i x^i$ is a bijection and therefore becomes a $m\Theta r$ isomorphism for the $m\Theta$ laws modulo $f(x)$. Since $f$ is irreducible over $\mathbb{F}_{p\mathbb{Z}}, \mathbb{F}(p\mathbb{Z}, r)$ is a $m\Theta f$. Theorem 0.3. 1. $\mathbb{F}(p\mathbb{Z},r)$ is a $m\Theta f$ of cardinal $p^{2r}$; 2. $\mathbb{F}_{p\mathbb{Z}}$ is its prime sub $m\Theta f$ of cardinal $p^2$; 3. $\mathbb{F}_{p\mathbb{Z}}$ and $\mathbb{F}(p\mathbb{Z},r)$ are booth of characteristic $p$ since $\forall i:$ $$ i = 0, \dots , p - 1; \underbrace {1 + 1 + \cdots + 1} _ {i t i m e s} + \underbrace {1 _ {p \mathbb {Z}} + \cdots + 1 _ {p \mathbb {Z}}} _ {(p - i) t i m e s} = 0 $$ Proof 0.4. [9] According to a previous notation, $$ \mathbb {F} (p \mathbb {Z}, 1) = \mathbb {F} _ {p \mathbb {Z}}, \mathbb {F} (p, 1) = \frac {\mathbb {Z}}{p \mathbb {Z}}, \mathbb {F} (p, r) = \mathbb {G} \mathbb {F} (p, r). $$ ### b) Canonical construction of modal $\Theta$ -valent pseudo fields ( $m\Theta p f$ ) Consider that $k \neq 1$, so $q = p^k$. Let then $\mathbb{F}(p^k\mathbb{Z}, 1)$ denote the quotient $m\Theta r \mathbb{F}_{p^k\mathbb{Z}} = \frac{\mathbb{Z}_{p\mathbb{Z}}}{p^k\mathbb{Z}_{p\mathbb{Z}}}$ and let $$ O (p ^ {k}, 1) = O _ {p ^ {k}} = \{\frac {a}{p ^ {k} \mathbb {Z} _ {p \mathbb {Z}}}: a \in \mathbb {Z} _ {p \mathbb {Z}}, s (a) / p ^ {k} \} = \{\frac {a}{p ^ {k} \mathbb {Z}}: a \in \mathbb {Z}, a / p ^ {k} \}. $$ Let $\mathbb{F}^* (p^k\mathbb{Z},1) = \mathbb{F}(p^k\mathbb{Z},1) - O(p^k,1); k\in \mathbb{N}^*$. Then $\forall x: x\in \mathbb{F}^{*}(p^{k}\mathbb{Z},1), \exists x^{\prime}: x^{\prime}\in \mathbb{F}^{*}(p^{k}\mathbb{Z},1): x\cdot x^{\prime} = \frac{1_{p\mathbb{Z}}}{p^{k}\mathbb{Z}_{p\mathbb{Z}}}$. So we call $\mathbb{F}_{p^k\mathbb{Z}}$ a $m\Theta$ pseudo field ( $m\Theta pf$ ). $\mathbb{F}_{p^k\mathbb{Z}}$ has $p^{k + 1}$ elements and is of characteristic $p^k$. It has no proper sub $m\Theta pf$ with the same as the preceding properties for $\mathbb{F}^* (p^k\mathbb{Z},1)$. Finally, $\mathbb{F}(p^k\mathbb{Z},1)$ is the prime $m\Theta pf$ with $p^{k + 1}$ elements. Let now $f \in \mathbb{Z}_{p^k}[X]$: $dg(f) = r$ and $f$ irreducible over $\mathbb{Z}_{p^k} = \frac{\mathbb{Z}}{p^k\mathbb{Z}}$. Let $\mathbb{F}(p^k\mathbb{Z},r) = \frac{\mathbb{F}(p^k\mathbb{Z},1)[X]}{(f(X))}$ $m\Theta r$ modulo $f(x)$. $\mathbb{F}(p^{k}\mathbb{Z},r)$ is a $m\Theta pf$. $(\mathbb{F}(p^k\mathbb{Z},1))^r\longrightarrow \mathbb{F}(p^k\mathbb{Z},r):(a_0,\dots,a_{r - 1})\longmapsto \sum_{i = 0}^{r - 1}a_ix^i$ is a bijection and therefore a $m\Theta$ ring modulo $f(X)$ isomorphism. Since card $\mathbb{F}(p^k\mathbb{Z},1) = p^{k + 1}$ card $\mathbb{F}(p^k\mathbb{Z},r) = p^{(k + 1)r}$ Theorem 0.4. [9] $\forall k\in \mathbb{N} - \{0\}$ 1. $\mathbb{F}(p^k\mathbb{Z},r)$ is a $m\Theta p f$ of cardinal $p^{(k + 1)r}$. 2. $\mathbb{F}(p^k\mathbb{Z},1)$ is its prime sub $m\Theta pf$ of $p^{k + 1}$ elements. 3. $\mathbb{F}(p^k\mathbb{Z},1)$ and $\mathbb{F}(p^k\mathbb{Z},r)$ are both of characteristic $p^k$ Proof 0.5. [9] $\mathbb{F}(p^k,r) = G\mathbb{F}(p^k,r)$ is the sub pseudo field of the $m\Theta$ invariants of the $m\Theta p f$ $\mathbb{F}(p^{k}\mathbb{Z},r)$. $\mathbb{F}_{p^k} = \frac{\mathbb{Z}}{p^k\mathbb{Z}}$ is the prime sub pseudo field of the $m\Theta$ invariants of $\mathbb{F}_{p^k\mathbb{Z}}$; the prime sub $m\Theta p f$ with $p^{k + 1}$ elements. Theorem 0.5. [9] 1. Any $m\Theta f$ $K$ of characteristic $p$ prime and then of cardinal $p^{2r}$, $r \in \mathbb{N}^*$ is $m\Theta$ isomorphic to the $m\Theta f$ $\mathbb{F}(p\mathbb{Z}, r)$; 2. Any $m\Theta f$ $K'$ of characteristic $p^k$, $p$ prime and then of cardinal $p^{(k+1)r}$, $r \in \mathbb{N}^*$ is $m\Theta$ isomorphic to the $m\Theta f$ $\mathbb{F}(p^k\mathbb{Z}, r)$. Proof 0.6. [9] ## IV. MODAL Θ-VALENT EXTENSION OF A FINITE FIELD Note that $K$ is a finite field of cardinal $p^n$, $p, n \in \mathbb{N}^*$ and then of characteristic $p$ prime. $\beta \in K$, of minimal polynomial $m_{\beta}(x) \in \mathbb{F}_p[x]$, $r = \deg_{\mathbb{F}_p}(m_{\beta}(x)) \in \mathbb{N}^*$, $m_{\beta}(x)$ is irreducible over $\mathbb{F}_p$. Observations: Let $I_{\beta} = \langle m_{\beta}(x) \rangle_{\mathbb{F}_p[x]}$ the principal ideal of $\mathbb{F}_p[x]$ generated by $m_{\beta}(x)$. Since $\mathbb{F}_p \subset \mathbb{F}_{p\mathbb{Z}}$, $\mathbb{F}_p[x] \subset \mathbb{F}_{p\mathbb{Z}}[x]$. Let $a \in \mathbb{F}^*_p\mathbb{Z}$: $\exists \mu, F_\mu a \neq 0$, then $F_\mu a \in \mathbb{F}^*_p$, thus $m_\beta(F_\mu a) \neq 0$ and since $F_\mu m_\beta(a) = m_\beta(F_\mu a)$, $F_\mu m_\beta(a) \neq 0$. Then $m_\beta(a) \neq 0$. Therefore, $m_{\beta}(x)$ is also irreducible over $\mathbb{F}_{p\mathbb{Z}}$. It is known that $\frac{\mathbb{F}_{p\mathbb{Z}}[x]}{\langle m_{\beta}(x) \rangle}$ is a $m\Theta$ field with $p^{2r}$ elements and then of characteristic $p$. $\frac{\mathbb{F}_p[x]}{m_\beta(x)}$ is its subfield of the $\Theta$ -invariants who has $p^r$ elements and characteristic $p$. Let $I_{\beta p\mathbb{Z}} = \langle m_{\beta}(x)\rangle_{\mathbb{F}_{p\mathbb{Z}}[x]}$ the principal $m\Theta$ ideal of $\mathbb{F}_{p\mathbb{Z}}[x]$ generated by $m_{\beta}(x)$. $\forall \alpha, F_{\alpha}I_{\beta p\mathbb{Z}} = I_{\beta}$ therefore $I_{\beta p\mathbb{Z}}$ is a $m\Theta$ maximal ideal of $\mathbb{F}_{p\mathbb{Z}}[x]$. Then define $\Phi_{\beta p\mathbb{Z}}:\mathbb{F}_{p\mathbb{Z}}[x]\longrightarrow \mathbb{F}(p\mathbb{Z},n)$ as follows; if $f(x) = \sum_{i = 0}^{q}a_{i}x^{i}\in \mathbb{F}_{p\mathbb{Z}}[x]$, $$ \Phi_ {\beta p \mathbb {Z}} (f (x)) = f (\beta) = \sum_ {i = 0} ^ {q} a _ {i} \beta^ {i} \in \mathbb {F} (p \mathbb {Z}, n). $$ By definition $\Phi_{\beta p\mathbb{Z}}$ is a $m\Theta$ ring morphism since then $\Phi_{\beta p\mathbb{Z}}(\mathbb{F}_{p\mathbb{Z}}[x]) = \{f(\beta) \mid f(x) \in \mathbb{F}_{p\mathbb{Z}}[x]\}$ is a sub $m\Theta$ field of $\mathbb{F}(p\mathbb{Z}, n)$. Therefore the following diagram $m\Theta$ commutes $$ \begin{array}{c} \mathbb{F}_{p\mathbb{Z}}[x] \xrightarrow{\Phi_{\beta} p\mathbb{Z}} \varphi_{\Phi_{\beta} p\mathbb{Z}}(\mathbb{F}_{p\mathbb{Z}}[x]) \xrightarrow{i_{p\mathbb{Z}}} \mathbb{F}(p\mathbb{Z}, n) \\\varphi_{\Phi_{\beta} p\mathbb{Z}} \Biggl\downarrow \\\frac{\mathbb{F}_{p\mathbb{Z}}[x]}{I_{\beta}} \xrightarrow{\Phi_{\beta} p\mathbb{Z} = \frac{\Phi_{\beta} p\mathbb{Z}}{\langle m_{\beta}(x)\rangle}} \end{array} $$ $\frac{\mathbb{F}_{p\mathbb{Z}}[x]}{I_{\beta}} = \frac{\mathbb{F}_{p\mathbb{Z}}[x]}{\langle m_{\beta}(x)\rangle}$ is a $m\Theta$ field of cardinal $p^{2r}$ and then of characteristic $p$. - Through the $m\Theta$ ring isomorphism $\widetilde{\Phi}_{\beta p\mathbb{Z}}$, $\widetilde{\Phi}_{\beta p\mathbb{Z}}(\mathbb{F}_{p\mathbb{Z}}[x])$ becomes a $m\Theta$ subfield of $\mathbb{F}(p\mathbb{Z}, n)$ with the $m\Theta$ field structure of $p^{2r}$ elements exported from $\frac{\mathbb{F}_{p\mathbb{Z}}[x]}{\langle m_{\beta}(x) \rangle}$ by $\widetilde{\Phi}_{\beta p\mathbb{Z}}$. Notation 0.1. $$ \mathbb {F} _ {p \mathbb {Z}} (\beta) = \Phi_ {p \mathbb {Z}} (\beta) = \widetilde {\Phi} _ {\beta p \mathbb {Z}} (\mathbb {F} _ {p \mathbb {Z}} [ x ]) = \{f (\beta) | f (x) \in \mathbb {F} _ {p \mathbb {Z} [ x ]} \} $$ Theorem 0.6. 1. $\mathbb{F}_{p\mathbb{Z}}[\beta]$ has $p^{2r}$ elements and characteristic $p$. 2. $\mathbb{F}_{p\mathbb{Z}}$ is the prime $m\Theta$ subfield of $\mathbb{F}_{p\mathbb{Z}}[\beta]$. 3. Any sub $m\Theta$ field of $\mathbb{F}(p\mathbb{Z},n)$ containing $\mathbb{F}_{p\mathbb{Z}}$ and $\beta$ contains $\mathbb{F}_{p\mathbb{Z}}[\beta ]$ 4. $\forall a; a \in \mathbb{F}_{p\mathbb{Z}}[\beta], \exists a_i, i = 0,1, \dots, r - 1 / a_i \in \mathbb{F}_{p\mathbb{Z}}: a = \sum_{i=0}^{r-1} a_i \beta^i.$ Definition 0.6. Henceforth we call $\mathbb{F}_{p\mathbb{Z}}[\beta]$ the $m\Theta$ extension of $\mathbb{F}_p$ and $\mathbb{F}_{p\mathbb{Z}}$ to $\beta$. Definition 0.7. We call a $m\Theta$ primitive element of $\mathbb{F}(p\mathbb{Z},n)$ any generator if there exists one, noted $\alpha$, of $\mathbb{F}(p\mathbb{Z},n) - \mathbb{F}(p,n)$. This meaning that $\forall a: a \in \mathbb{F}(p\mathbb{Z},n) - \mathbb{F}(p,n), \exists m \in \mathbb{N}: 0 \leq m \leq \omega(\mathbb{F}^*(p\mathbb{Z},n); a = \alpha^m$. Example 0.2. $2_{3\mathbb{Z}}$ and $5_{3\mathbb{Z}}$ are two m3 generators of $\mathbb{F}_{3\mathbb{Z}}$. Proposition 0.2. If $\alpha \in \mathbb{F}(p\mathbb{Z}, n)$ is a $m\Theta$ primitive element then $\mathbb{F}(p\mathbb{Z}, n) = \mathbb{F}_{p\mathbb{Z}}(\alpha)$. Proof 0.7. Suppose $u \in \mathbb{F}(p\mathbb{Z}, n) - \mathbb{F}(p, n)$ and $\alpha$ is a $m\Theta$ primitive element: $\exists m, m \in \mathbb{N}: 0 \leq m \leq \omega(\mathbb{F}^*(p\mathbb{Z}, r), u = \alpha^m)$. Let $f(x) = x^m \in \mathbb{F}_{p\mathbb{Z}}[x]$, $\Phi_{\beta p\mathbb{Z}}(f(x)) = f(\alpha) = x^m$. Therefore $u = \alpha^m = f(\alpha) \in \mathbb{F}_{p\mathbb{Z}}(\alpha)$. Thus $\mathbb{F}(p\mathbb{Z}, n) = \mathbb{F}_{p\mathbb{Z}}(\alpha)$. ## V. THE INTRINSIC POLYNOMIAL REPRESENTATION OF THE $m\Theta$ PSEUDO FIELD $\mathbf{F}(p^k\mathbf{Z},r)$ Let $k \in \mathbb{N}$, $r, p \in \mathbb{N}^*$, $p$ prime $2 \leq p$. It is plain in [7] that: $$ \prod_{a\in\mathbb{F}^*(p^k\mathbb{Z},1)}(x-a) = \prod_{x\in\tilde{\mathbb{F}}_{p^k}^*}(x-a) \times \prod_{a\in\mathbb{F}^*(p^k\mathbb{Z},1)-\mathbb{F}_{p^k}^*}(x-a) = \left(x^{\varphi(p^k)}-1\right)\left(x^{\varphi(p^{k+1})}-1_{p\mathbb{Z}}\right). $$ $$ \mathbb {F} \left(p ^ {k}, 1\right) = \frac {\mathbb {Z}}{p ^ {k} \mathbb {Z}}. $$ Proposition 0.3. Let $< x^{\varphi(p^k)} - 1 >$ and $< x^{\varphi(p^{k+1})} - 1_{p\mathbb{Z}} >$ be the ideals of $\mathbb{F}(p^k, 1)[x]$ respectively generated by $x^{\varphi(p^k)} - 1$ and $x^{\varphi(p^{k+1})} - 1_{p\mathbb{Z}}$, then 1. $< x^{\varphi(p^k)} - 1 >$ is a maximal $m\Theta$ ideal of $\mathbb{F}(p^k, 1)[x]$; 2. $< x^{\varphi (p^{k + 1})} - 1_{p\mathbb{Z}} > _{\neq}^{\subset} < x^{\varphi (p^{k})} - 1 >.$ Proof 0.8.1. $< x^\varphi(p^k) - 1 >$ is a $m\Theta$ ideal since generated by the $m\Theta\Theta$ invariant polynomial $x^\varphi(p^k) - 1$; this $\Theta$ ideal is a maximal since $< x^\varphi(p^k) - 1 >_{\mathbb{F}_{p^k}[x]}$ is maximal in $\mathbb{F}_{p^k}[x]$ and $\forall \alpha \in I_*, F_\alpha < x^\varphi(p^k) - 1 > = < x^\varphi(p^k) - 1 >_{\mathbb{F}_{p^k}[x]}$. This is sufficient to claim that $\frac{\mathbb{F}(p^k, 1)[x]}{<x^\varphi(p^k) - 1>}$ is a $m\Theta$ pseudo field, and as such $m\Theta$ isomorphic to the $m\Theta$ pseudo field $\mathbb{F}(p^k, \varphi(p^k))$. 2. $x^{\varphi(p^{k+1})} - 1_{p\mathbb{Z}} \in < x^{\varphi(p^k)} - 1 >$. Since $\varphi(p^{k+1}) = p\varphi(p^k)$, $x^{\varphi(p^{k+1})} = x^{p\varphi(p^k)}$. Henceforth, $$ x^\varphi(p^{k+1}) - 1_{\mathbb{Z}} = (x^{\varphi(p^{k})})^p - 1_{\mathbb{Z}}^p = (x^{\varphi(p^{k})} - 1_{\mathbb{Z}})^p = (x^{\varphi(p^{k})} - 1)(x^{\varphi(p^{k})} - 1_{\mathbb{Z}})^{p-1} $$ This last expression shows that $x^{\varphi(p^{k+1})} - 1_{p\mathbb{Z}} \in <x^{\varphi(p^k)} - 1>$. Trivially, $x^{\varphi(p^k)} - 1 \notin <x^{\varphi(p^{k+1})} - 1_{p\mathbb{Z}} >$. Therefore $<x^{\varphi(p^{k+1})} - 1_{p\mathbb{Z}}>\subsetneq <x^{\varphi(p^k)} - 1>$, $\forall k \in \mathbb{N}^*$. Thus $<x^{p(p-1)} - 1_{p\mathbb{Z}}>\subsetneq <x^{p-1} - 1>$. Definition 0.8. The $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},\varphi (p^k)) = \frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{< x\varphi(p^k) - 1 >}$ is what we call the intrinsic polynomial representation of the $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},r)$. Corollary 0.1. $\mathbb{F}(p\mathbb{Z},\varphi (p)) = \frac{\mathbb{F}(p\mathbb{Z},1)[x]}{< x^{(p - 1)} - 1 >}$ is the intrinsic polynomial representation of $\mathbb{F}(p\mathbb{Z},r)$ with $r = \varphi (p) = p - 1$ $k = 1$ Proposition 0.4. For a finite commutative $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)$ the following conditions are equivalent: 1. $\mathbb{F}(p^k\mathbb{Z}, 1)$ is a local $m\Theta$ pseudo field and the maximal $m\Theta$ ideal $M$ of $\mathbb{F}(p^k\mathbb{Z}, 1)$ is principal; 2. $\mathbb{F}(p^k\mathbb{Z},1)$ is a local principal $m\Theta$ ideal pseudo field; 3. $\mathbb{F}(p^k\mathbb{Z},1)$ is a chain $m\Theta$ pseudo field. Proof 0.9. $i) \Rightarrow ii)$. Let $I$ be an $m\Theta$ ideal of $\mathbb{F}(p^k\mathbb{Z},1)$. If $I = \mathbb{F}(p^{k}\mathbb{Z},1)$ then $I$ is generated by the identity $1$. If $I \subsetneq \mathbb{F}(p^{k}\mathbb{Z},1)$, then $I \subseteq M$. By $i)$, $M$ is generated by an element, say $M = \langle a\rangle$. Therefore, $I = \langle a^i\rangle$, for some integer $k$. Hence, $\mathbb{F}(p^k\mathbb{Z},1)$ is a local principal $m\Theta$ ideal pseudo field. ii) $\Longrightarrow$ iii). Let $\mathbb{F}(p^k\mathbb{Z},1)$ be a local principal $m\Theta$ ideal pseudo field with the maximal ideal $M = \langle a\rangle$, and $A, B$ be proper ideals of $\mathbb{F}(p^k\mathbb{Z},1)$. Then $A, B \subseteq M$, whence there exist integers $l, m$ such that $A = \langle a^l\rangle$, $B = \langle a^m\rangle$ ( $l, m \leq$ the nilpotency of $a$ ). Hence, either $A \subseteq B$, or $B \subseteq A$. Thus, $\mathbb{F}(p^k\mathbb{Z},1)$ is a chain $m\Theta$ pseudo field. iii) $\Rightarrow$ i). Assume $\mathbb{F}(p^k\mathbb{Z},1)$ is a finite commutative chain $m\Theta$ pseudo field, then clearly $\mathbb{F}(p^k\mathbb{Z},1)$ is local. To show the maximal $m\Theta$ ideal $M$ of $\mathbb{F}(p^k\mathbb{Z},1)$ is principal, suppose to the contrary that $M$ is generated by more than one element, say $b, c$ are in the generator set of $M$ and $b\notin c\mathbb{F}(p^k\mathbb{Z},1)$, $c\notin b\mathbb{F}(p^k\mathbb{Z},1)$. Then $\langle b\rangle \not\subseteq \langle c\rangle$ and $\langle c\rangle \not\subseteq \langle b\rangle$, a contradiction with the assumption that $\mathbb{F}(p^k\mathbb{Z},1)$ is a chain $m\Theta$ pseudo field. Thus, $M$ is principal, proving i). Let $a$ be a fixed generator of the maximal ideal $M$. Then $a$ is nilpotent and we denote its nilpotency index by $t$. The ideals of $\mathbb{F}(p^k\mathbb{Z},1)$ for a chain $$ \mathbb {F} \left(p ^ {k} \mathbb {Z}, 1\right) = \langle a ^ {0} \rangle \supseteq \langle a ^ {1} \rangle \supseteq \dots \supseteq \langle a ^ {t - 1} \rangle \supseteq \langle a ^ {t} \rangle = \langle 0 \rangle . $$ Let $\overline{\mathbb{F}(p^k\mathbb{Z},1)} = \frac{\mathbb{F}(p^k\mathbb{Z},1)}{M}$. By $-: \mathbb{F}(p^k\mathbb{Z},1)[x] \longrightarrow \overline{\mathbb{F}(p^k\mathbb{Z},1)}[x]$, we denote the natural $m\Theta$ pseudo field homomorphism that maps $\rho \longmapsto \rho + M$ and the variable $x$ to $x$. Proposition 0.5. Let $\mathbb{F}(p^k\mathbb{Z},1)$ be a finite commutative chain $m\Theta$ pseudo field, with maximal ideal $M = \langle a\rangle$, and let $t$ be a nilpotency $a$. Then we get the following statements. 1. For some prime $p$ and positive integers $k, l$ ( $k \geq l$ ), $|\mathbb{F}(p^k\mathbb{Z}, 1)| = p^{k+1}$, $|\overline{\mathbb{F}(p^k\mathbb{Z}, 1)}| = p^{l+1}$, and the characteristic of $\mathbb{F}(p^k\mathbb{Z}, 1)$ and $\mathbb{F}(p^k\mathbb{Z}, 1)$ are powers of $p$. 2. For $i = 0, \dots, t$, $|\langle a^i\rangle| = |\overline{\mathbb{F}(p^k\mathbb{Z},1)} |^{t - i}$. In particular, $|\mathbb{F}(p^{k}\mathbb{Z},1)| = |\overline{\mathbb{F}(p^{k}\mathbb{Z},1)} |^{t}$, so, $k = lt$. Two $m\Theta$ polynomials $f_1, f_2 \in \mathbb{F}(p^k\mathbb{Z}, 1)[x]$ are called $m\Theta$ coprime if $\langle f_1 \rangle + \langle f_2 \rangle = \mathbb{F}(p^k\mathbb{Z}, 1)[x]$. A $m\Theta$ polynomial $f \in \mathbb{F}(p^k\mathbb{Z}, 1)[x]$ is called basic $m\Theta$ irreducible if $\overline{f}$ is $m\Theta$ irreducible in $\overline{\mathbb{F}(p^k\mathbb{Z}, 1)}[x]$. A $m\Theta$ polynomial $f \in \mathbb{F}(p^k\mathbb{Z}, 1)[x]$ is called regular if it is not a zero divisor. ## VI. $m\Theta$ QUASI-CYCLIC CODES For a finite $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)$, consider the set $\mathbb{F}^r (p^k\mathbb{Z},1)$ of $n$ -tuples of elements from $\mathbb{F}(p^k\mathbb{Z},1)$ as a module over $\mathbb{F}(p^k\mathbb{Z},1)$ in the usual way. A subset $C \subseteq \mathbb{F}^r (p^k\mathbb{Z},1)$ is called a linear $m\Theta$ code of length $r$ over $\mathbb{F}(p^k\mathbb{Z},1)$ if $C$ is an $\mathbb{F}(p^k\mathbb{Z},1)$ -submodule of $\mathbb{F}^r (p^k\mathbb{Z},1)$. $C$ is called $m\Theta$ cyclic if, for every $m\Theta$ codeword $x = (x_0,x_1,\dots,x_{r - 1}) \in C$, its cyclic shift $(x_{n - 1},x_0,x_1,\dots,x_{n - 2})$ is also in $C$. An $n$ -tuple $c = (c_0,c_1,\dots,c_{r - 1}) \in \mathbb{F}^r (p^k\mathbb{Z},1)$ is identified with the $m\Theta$ polynomial $c_0 + c_1x + \dots +c_{r - 1}x^{r - 1}$ in $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{<x^r - 1 >}$, which is called the $m\Theta$ polynomial representation of $c = (c_0,c_1,\dots,c_{r - 1})$. It is well known that a code $C$ of length $r$ over $\mathbb{F}(p^k\mathbb{Z},1)$ is $m\Theta$ cyclic if and only if the $m\Theta$ set of polynomial representations of its $m\Theta$ codewords is an $m\Theta$ ideal of $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{<x^r - 1>}$. Given $x = (x_0, x_1, \dots, x_{r-1})$, $y = (y_0, y_1, \dots, y_{r-1}) \in \mathbb{F}^r(p^k\mathbb{Z}, 1)$, their scalar product is $$ x \cdot y = x _ {0} y _ {0} + x _ {1} y _ {1} + \dots + x _ {r - 1} y _ {r - 1}. $$ (evaluated in $\mathbb{F}(p^k\mathbb{Z},1)$ ). Two $m\Theta$ words $x, y$ are called orthogonal if $\forall \alpha \in I_{*}$, $F_{\alpha}(x) \cdot F_{\alpha}(y) = 0$. For a linear $m\Theta$ code $C$ over $\mathbb{F}(p^k\mathbb{Z},1)$, its dual code $C^\perp$ is the set of $m\Theta$ words over $\mathbb{F}(p^k\mathbb{Z},1)$ that are orthogonal to all $m\Theta$ codewords of $C$; $$ C^\perp = \{x \in \mathbb{F}(p^k\mathbb{Z}, 1) | \forall \alpha \in I_*, F_\alpha(x) \cdot F_\alpha(y) = 0, \forall y \in C\} $$ A $m\Theta$ code $C$ is called self-dual if $C = C^{\perp}$. For a finite $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)$ with maximal ideal $\langle a\rangle$ and the nilpotency $t$ of $a$ is even, the code $\langle a^{\frac{t}{2}}\rangle$ is self-dual and is called the trivial self-dual code. Proposition 0.6. Let $\mathbb{F}(p^k\mathbb{Z},1)$ be a finite commutative $m\Theta$ pseudo field and $$ a (x) = a _ {0} + a _ {1} x + \dots + a _ {r - 1} x ^ {r - 1}; $$ $$ b (x) = b _ {0} + b _ {1} x + \dots + b _ {r - 1} x ^ {r - 1} \in \mathbb {F} \left(p ^ {k} \mathbb {Z}, 1\right) [ x ]. $$ Then $a(x)b(x) = 0$ in $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{<x^r - 1>}$ if and only if $(a_0, a_1, \dots, a_{r-1})$ is $m\Theta$ orthogonal to $(b_{r-1}, b_{r-2}, \dots, b_0)$ and all its cyclic shifts. Proof 0.10. Let $\zeta$ denote the cyclic shift for $m\Theta$ codewords of length $r$, i.e., for each $(x_0, x_1, \dots, x_{r-1}) \in \mathbb{F}^r(p^k\mathbb{Z}, 1)$. $$ \zeta \left(x _ {0}, x _ {1}, \dots , x _ {r - 1}\right) = \left(x _ {r - 1}, x _ {0}, \dots , x _ {r - 2}\right). $$ Thus, $\zeta^i (b_{r - 1}, b_{r - 2}, \dots, b_0)$, $i = 1,2, \dots, r$ are all cyclic shifts of $(b_{r - 1}, b_{r - 2}, \dots, b_0)$. Let $c(x) = c_0 + c_1x + \dots + c_{r-1}x^{r-1} = a(x)b(x) \in \frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{<x^r - 1 >}$. Then for $k = 0,1, \dots, r - 1$, $$ c_k = \sum_{i+j=k\,or\,i+j=r-k} a_i b_j = (a_0,a_1,\dots,a_{r-1}) \cdot (b_k,b_{k-1},\dots,b_{k+1}) = (a_0,a_1,\dots,a_{r-1}) \cdot \zeta^{k+1} (b_{r-1},b_{r-2},\dots,b_0). $$ Therefore, $c(x) = 0$ if and only if $c_k = 0$ for $k = 0,1, \dots, r - 1$ if and only if $$ \left(a _ {0}, a _ {1}, \dots , a _ {r - 1}\right) \cdot \zeta^ {k + 1} \left(b _ {r - 1}, b _ {r - 2}, \dots , b _ {0}\right) = 0, $$ for $k = 0,1, \dots, r - 1$ if and only if $(a_0, a_1, \dots, a_{r-1})$ is orthogonal to $(b_{r-1}, b_{r-2}, \dots, b_0)$ and all its cyclic shifts, as desired. Definition 0.9. (quasi-cyclic $m\Theta$ code) A linear $m\Theta$ code $C$ of length $r = lk$ over a finite $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)$ is called a quasi-cyclic $m\Theta$ code of index $k$ if for every $m\Theta$ codeword $c\in C$ there exists a number $k$ such that the $m\Theta$ codeword obtained by $k$ cyclic shifts is also a $m\Theta$ codeword in $C$. That is, $$ c = (c_0, c_1, \dots, c_{r-1}) \in C \Longrightarrow c' = \zeta^k(c) = (c_{r-k}, \dots, c_0, \dots, c_{r-k-1}) \in C. $$ In the definition $k$ is defined as the smallest number of cyclic shifts where the $m\Theta$ code is invariant. Quasi-cyclic $m\Theta$ codes are a generalization of cyclic $m\Theta$ codes. ## VII. STRUCTURE OF QUASI-CYCLECT CODE OVER FINITE CHAIN $m\Theta$ PSEUDO FIELD $F(p^k \mathbf{Z}, 1)$ Let \mathbb{F}(p^k\mathbb{Z},1) be a finite chain $m\Theta$ pseudo field with the maximal $m\Theta$ ideal $\langle a\rangle$, and $t$ be the nilpotency of $a$. There exist a prime $p$ and an integer $l$ such that $|\overline{\mathbb{F}(p^k\mathbb{Z},1)}| = p^l$, $|\mathbb{F}(p^k\mathbb{Z},1)| = p^{lt}$, the characteristic of $\mathbb{F}(p^k\mathbb{Z},1)$ and $\overline{\mathbb{F}(p^k\mathbb{Z},1)}$ are powers of $p$. In this section, we assume $r$ to be a positive integer which is not divisible by $p$; that implies $r$ is not divisible by the characteristic of the residue $m\Theta$ pseudo field $\overline{\mathbb{F}(p^k\mathbb{Z},1)}$, so that $x^r - 1$ is square free in $\overline{\mathbb{F}(p^k\mathbb{Z},1)}[x]$. Therefore, $x^r - 1$ has a unique decomposition as a product of basic irreducible pairwise coprime $m\Theta$ polynomials in $\mathbb{F}(p^k\mathbb{Z},1)[x]$. Lemma 0.1. Let $\mathbb{F}(p^k\mathbb{Z},1)$ be a finite chain $m\Theta$ pseudo field with the maximal $m\Theta$ ideal $\langle a\rangle$, and $t$ be the nilpotency of $a$. If $f$ is a regular basic irreducible $m\Theta$ polynomial of the $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)[x]$, then $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{\langle f\rangle}$ is also a chain $m\Theta$ pseudo field with precisely the following ideals: $$ \langle 0 \rangle , \langle 1 \rangle , \langle 1 + \langle f \rangle \rangle , \langle a + \langle f \rangle \rangle , \dots , \langle a ^ {t - 1} + \langle f \rangle \rangle . $$ Proof 0.11. First we show that for distinct values of $i$, $j \in \{0,1, \dots, t-1\}$, $\langle a^i + \langle f \rangle \rangle \neq \langle a^j + \langle f \rangle \rangle$. Suppose $\langle a^i + \langle f \rangle \rangle = \langle a^j + \langle f \rangle \rangle$, for $0 \leq i < j \leq t-1$. Then, there exists $g(x) \in \mathbb{F}(p^k\mathbb{Z}, 1)[x]$ with $\deg(g) < \deg(f)$ such that $a^i + \langle f \rangle = a^j + \langle f \rangle$. That means $a^j g(x) - a^i \in \langle f \rangle$. As $$ deg(a^j g(x) - a^i) \leq deg(g) < deg(f) $$ it follows that $a^j g(x) - a^i = 0$. Multiplying by $a^{t-j}$ gives $a^{t+i-j} = 0$, which is a contradiction to our hypothesis that $a$ has nilpotency $t$ and $0 < t + i - j < t$. Let $I$ be a nonzero ideal of $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{\langle f\rangle}$ and $h + \langle f\rangle$ a nonzero element of $I$. By assumption, $f$ is a basic irreducible $m\Theta$ polynomial in $\mathbb{F}(p^k\mathbb{Z},1)[x]$, hence, $\bar{f}$ is irreducible in $\overline{\mathbb{F}(p^k\mathbb{Z},1)}[x]$. Therefore, $\gcd(\bar{h},\bar{f}) = 1$, or $\bar{f}$. If $\gcd(\bar{h},\bar{f}) = 1$, that is, $\bar{h},\bar{f}$ are coprime in $\overline{\mathbb{F}(p^k\mathbb{Z},1)}[x]$, then $h, f$ are coprime in $\mathbb{F}(p^k\mathbb{Z},1)[x]$. So there exist $u, v \in \mathbb{F}(p^k\mathbb{Z},1)[x]$ such that $uh +vf = 1$. That implies $$ (u + \langle f \rangle) (h + \langle f \rangle) = 1 + \langle f \rangle $$ whence $h + \langle f \rangle$ is invertible in $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{\langle f \rangle}$. Therefore, $$ I = \frac {\mathbb {F} (p ^ {k} \mathbb {Z} , 1) [ x ]}{\langle f \rangle} = \langle 1 + \langle f \rangle \rangle . $$ For the case $gcd(\bar{h}, \bar{f}) = \bar{f}$, for all $h + \langle f \rangle \in I$, which means $\bar{f}$ divides $\bar{h}$, hence, there exist $w, z \in \mathbb{F}(p^k\mathbb{Z}, 1)[x]$ such that $h = wf + az$. Whence $$ h + \langle f \rangle \in \langle a + \langle f \rangle \rangle , f o r a l l h + \langle f \rangle \in I $$ implying $I \subseteq \langle a + \langle f \rangle \rangle$. Let $k$ be the greatest integer $< t$ such that $I \subseteq \langle a^k + \langle f \rangle \rangle$. Then, as $I \not\subseteq \langle a^{k+1} \langle f \rangle \rangle$, there is a (nonzero) element $h' + \langle f \rangle \in I$ such that $h' + \langle f \rangle \notin \langle a^{k+1} + \langle f \rangle \rangle$. Since $h' + \langle f \rangle \in I \subseteq \langle a^k + \langle f \rangle \rangle$, there exist $w', z' \in \mathbb{F}(p^k\mathbb{Z}, 1)[x]$ such that $h' = w'f + a^kz'$. Now $gcd(\bar{z}', \bar{f}) = 1$ or $\bar{f}$. Suppose $gcd(\bar{z}', \bar{f}) = \bar{f}$, then $\bar{f}$ divides $\bar{z}'$ and so there exist $w'', z'' \in \mathbb{F}(p^k\mathbb{Z}, 1)[x]$ such that $z' = w''f + az''$. Hence, $$ \begin{array}{l} h ^ {\prime} = w ^ {\prime} f + a ^ {k} z ^ {\prime} = w ^ {\prime} f + a ^ {k} \left(w ^ {\prime \prime} f + a z ^ {\prime \prime}\right) \\= \left(w ^ {\prime} + a ^ {k} w ^ {\prime \prime}\right) f + a ^ {k + 1} z ^ {\prime \prime}. \\\end{array} $$ It follows that $h' + \langle f \rangle \in \langle a^{k+1} + \langle f \rangle \rangle$, a contradiction. Thus, $gcd(\bar{z}', \bar{f}) = 1$. The same argument as above yields that $z' + \langle f \rangle$ is invertible in $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{\langle f \rangle}$, which means that there exists $z_0 \in \mathbb{F}(p^k\mathbb{Z},1)[x]$ such that $$ \left(z ^ {\prime} + \langle f \rangle\right) \left(z _ {0} + \langle f \rangle\right) = 1 + \langle f \rangle . $$ Therefore $$ \begin{array}{l} a ^ {k} + \langle f \rangle = (z _ {0} + \langle f \rangle) (a ^ {k} z ^ {\prime} + \langle f \rangle) \\= \left(z _ {0} + \langle f \rangle\right) \left(h ^ {\prime} + \langle f \rangle\right) \in I. \\\end{array} $$ Consequently, $I = \langle a^{k} + \langle f\rangle \rangle$ Customarily, for a $m\Theta$ polynomial $f$ of degree $k$, its reciprocal $m\Theta$ polynomial $x^{k}f(x^{-1})$ will be denoted by $f^{*}$. Thus, for example, if $f(x) = a_{0} + a_{1}x + \dots +a_{k - 1}x^{k - 1} + a_{k}x^{k}$, then $$ \begin{array}{l} f ^ {*} (x) = x ^ {k} \left(a _ {0} + a _ {1} x ^ {- 1} + \dots + a _ {k - 1} x ^ {- (k - 1)} + a _ {k} x ^ {- k}\right) \\= a _ {k} + a _ {k - 1} x + \dots + a _ {1} x ^ {k - 1} + a _ {0} x ^ {k}. \\\end{array} $$ Moreover, if $f(x)$ is a factor of $x^r - 1$, we denote $\hat{f}(x) = \frac{x^r - 1}{f(x)}$. Theorem 0.7. Assume $\mathbb{F}(p^k\mathbb{Z},1)$ is a finite chain $m\Theta$ pseudo field with maximal $m\Theta$ ideal $\langle a\rangle$, and that $t$ is the nilpotency of $a$. Let $x^{r} - 1 = f_{1}f_{2}\dots f_{l}$ be a representation of $x^{r} - 1$ as a product of basic irreducible pairwise-coprime polynomials in $\mathbb{F}(p^k\mathbb{Z},1)[x]$. Then any ideal in $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{\langle f\rangle}$ is a sum of $m\Theta$ ideals of the form $\langle a^{j}\hat{f}_{i} + \langle x^{r} - 1\rangle \rangle$, where $0\leq j\leq t$, $1\leq i\leq r$. Proof 0.12. By the Chinese Reminder theorem, we have $$ \frac {\mathbb {F} (p ^ {k} \mathbb {Z} , 1) [ x ]}{\langle x ^ {r} - 1 \rangle} = \frac {\mathbb {F} (p ^ {k} \mathbb {Z} , 1) [ x ]}{\cap_ {i = 1} ^ {l} \langle f _ {i} \rangle} \cong \bigoplus \sum_ {i = 1} ^ {l} \frac {\mathbb {F} (p ^ {k} \mathbb {Z} , 1) [ x ]}{\langle f _ {i} \rangle}. $$ Thus, any $m\Theta$ ideal $I$ of $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{\langle x^r - 1\rangle}$ is of the form $\bigoplus \sum_{i=1}^{l} I_i$, where $I_i$ is an $m\Theta$ ideal of $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{\langle f_i\rangle}$. According to the previous lemma, for $1 \leq i \leq r$, $I_i = 0$ or $I_i = \langle a_k + \langle f_i\rangle \rangle$, for some $k \in \{0, \dots, t-1\}$. Then $I_i$ corresponds to $\langle a^k\hat{f}_i + \langle x^r - 1\rangle \rangle$ in $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{\langle x^r - 1\rangle}$. Consequently, $I$ is a sum of ideals of the form $\langle a^j\hat{f}_i + \langle x^r - 1\rangle \rangle$. Corollary 0.2. Let $\mathbb{F}(p^k\mathbb{Z},1)$ be a finite $m\Theta$ pseudo field with maximal $m\Theta$ ideal $\langle a\rangle$, and $t$ be the nilpotency of $a$. The numbers of quasi-cyclic $m\Theta$ codes over $\mathbb{F}(p^k\mathbb{Z},1)$ of length $r$ is $(t + 1)^l$, where $l$ is the number of factors in the unique factorization of $x^r - 1$ into a product of monic basic irreducible pairwise coprime $m\Theta$ polynomials. From now on, in order to simplify notation, we will just write $l_0 + l_1x + \dots + l_{r - 1}x^{r - 1}$ for the corresponding coset $l_0 + l_1x + \dots + l_{r - 1}x^{r - 1} + \langle x^r - 1 \rangle$ in $\frac{\mathbb{F}(p^k\mathbb{Z},1)[x]}{\langle x^r - 1 \rangle}$ Theorem 0.8. Let $C$ be a quasi-cyclic $m\Theta$ codes of length $r$ over a finite $m\Theta$ pseudo field with maximal $m\Theta$ ideal $\langle a\rangle$, and $t$ be the nilpotency of $a$. Then there exists a unique family of pairwise coprime monic $m\Theta$ polynomials $F_0, F_1, \dots, F_t$ in $\mathbb{F}(p^k\mathbb{Z}, 1)[x]$ such that $F_0F_1 \dots F_t = x^r - 1$ and $C = \langle \widehat{F_1}, a\widehat{F_2}, \dots, a^{t-1}\widehat{F_t} \rangle$. Moreover $$ | C | = \left(| \overline{{\mathbb{F} (p ^ {k} \mathbb{Z} , 1)}} |\right) ^ {\sum_ {i = 0} ^ {t - 1} (t - i) d e g (F _ {i + 1})}. $$ Proof 0.13. Let $x^r - 1 = f_1 \cdots f_l$ be the unique factorization of $x^r - 1$ into a product of monic basic irreducible pairwise coprime $m\Theta$ polynomials. $C$ is a direct sum of ideals of the form $\langle a^j \widehat{f}_i \rangle$, where $0 \leq j \leq t$, $1 \leq i \leq l$. After reordering if necessary, we can assume that $$ C = \langle \hat {f} _ {k _ {1} + 1} \rangle \oplus \dots \oplus \langle \hat {f} _ {k _ {1} + k _ {2}} \rangle \oplus \langle a \hat {f} _ {k _ {1} + k _ {2} + 1} \rangle \oplus \dots a \hat {f} _ {k _ {1} + k _ {2} + k _ {3}} \rangle \oplus $$ $$ \dots \oplus \left\langle a ^ {t - 1} \hat {f} _ {k _ {1} + \dots + k _ {t} + 1} \right\rangle \oplus \dots \oplus \left\langle a ^ {t - 1} \hat {f} _ {r} \right\rangle $$ where $k_{1}, \dots, k_{t} \geq 0$ and $k_{1} + \dots + k_{t} + 1 \leq r$. Let $k_{0} = 0$, and $k_{t+1}$ be a nonnegative integer such that $k_{1} + \dots + k_{t} + 1 \leq r$. For $i = 0, \dots, t$, define $$ F_{i} = f_{k_{0} + \dots + k_{i} + 1} \dots f_{k_{0} + \dots + k_{i} + 1} $$ Then by our construction, it is clear that $F_0, \dots, F_t$ are pairwise coprime, $F_0 \dots F_t = f_1 \dots f_r = x^r - 1$, and $$ C = \langle \widehat {F _ {1}} \rangle \oplus \langle a \widehat {F _ {2}} \rangle \oplus \dots \oplus \langle a ^ {t - 1} \widehat {F _ {t}} \rangle . $$ To prove the uniqueness, assume $G_0G_1 \cdots G_t = x^r - 1$ and $C = \langle \widehat{G_1}, a\widehat{G_2}, \dots, a^{t-1}\widehat{G_t} \rangle$. Then $$ \frac {\mathbb {F} (p ^ {k} \mathbb {Z} , 1) [ x ]}{\langle x ^ {r} - 1 \rangle} = \langle \widehat {G _ {0}} \rangle \oplus \langle \widehat {G _ {1}} \rangle \oplus \dots \oplus \langle \widehat {G _ {s}} \rangle $$ thus, $C = \langle \widehat{G_1} \rangle \oplus \langle a\widehat{G_2} \rangle \oplus \dots \oplus \langle a^{t - 1}\widehat{G_s} \rangle$. Now there exist nonnegative integers $l_0 = 0, l_1, \dots, l_{t + 1}$ with $l_0 + l_1 + \dots + l_{t + 1} = l$, and a permutation $\{f_1', \dots, f_r'\}$ of $\{f_1, \dots, f_r\}$ such that, for $i = 0,1, \dots, t$ $$ G _ {i} = f _ {l _ {0} + \dots + l _ {i} + 1} ^ {\prime} \dots f _ {l _ {0} + \dots + l _ {i} + 1} ^ {\prime}. $$ Hence, $$ C = \langle \hat{f}_{l_1+1} \rangle \oplus \dots \oplus \langle \hat{f}_{l_1+l_2} \rangle \oplus \langle a \hat{f}_{l_1+l_2+1} \rangle \oplus \dots \oplus \langle a \hat{f}_{l_1+l_2+l_3} \rangle \oplus \dots \oplus \left\langle a^{t-1} \hat{f}_{l_1+\dots+l_t+1} \right\rangle \oplus \dots \oplus \left\langle a^{t-1} \hat{f}_{r} \right\rangle $$ $$ \dots \oplus \left\langle a^{t-1} \hat{f}_{l_1+\dots+l_t+1}^\prime \right\rangle \oplus \dots \oplus \left\langle a^{t-1} \hat{f}_{r}^\prime \right\rangle $$ Now for $i = 0, \dots, t$, it follows that $l_i = k_i$, and, furthermore, $\{f_{l_0 + \dots + l_{i+1}}', \dots, f_{l_0 + \dots + l_{t+1}}'\}$ is a permutation of $\{f_{k_0 + \dots + k_{i+1}}, \dots, f_{k_0 + \dots + k_{t+1}}\}$. Therefore, $G_i = F_i$, for $i = 0, \dots, t$. To calculate the order $|C|$, note that $$ C = \langle \widehat {F _ {1}} \rangle \oplus \langle a \widehat {F _ {2}} \rangle \oplus \dots \oplus \langle a ^ {t - 1} \widehat {F _ {t}} \rangle $$ and for $i = 0,1,\dots,t - 1$ $$ \begin{array}{l} {| \langle a ^ {i} \widehat {F _ {i + 1}} \rangle |} = {(\frac {| \mathbb {F} (p ^ {k} \mathbb {Z} , 1) |}{| \langle a ^ {t - i} \rangle |}) ^ {(n - d e g \widehat {F _ {i + 1}})} = (\frac {| \overline {{\mathbb {F} (p ^ {k} \mathbb {Z} , 1)}} | ^ {t}}{| \overline {{\mathbb {F} (p ^ {k} \mathbb {Z} , 1)}} | ^ {i}}) ^ {d e g F _ {t + 1}}} \\= \left(| \overline {{\mathbb {F}}} (p ^ {k} \mathbb {Z}, 1) |\right) ^ {(t - i) d e g F _ {t + 1}}. \\\end{array} $$ Hence, $$ \begin{array}{l} {| C |} = {| \langle \widehat{F _ {1}} \rangle | \cdot | \langle a \widehat{F _ {2}} \rangle | \cdot \dots \cdot | \langle a ^ {t - 1} \widehat{F _ {t}} \rangle |} \\= \left(| \overline{{\mathbb{F} (p ^ {k} \mathbb{Z} , 1)}} |\right) ^ {t d e g F _ {1}} \cdot \left(| \overline{{\mathbb{F} (p ^ {k} \mathbb{Z} , 1)}} |\right) ^ {(t - 1) d e g F _ {2}} \dots \left(| \overline{{\mathbb{F} (p ^ {k} \mathbb{Z} , 1)}} |\right) ^ {d e g F _ {t}} \\= \left(| \overline{{\mathbb{F}}} (p ^ {k} \mathbb{Z}, 1) |\right) ^ {\sum_ {i = 0} ^ {t - 1} (t - i) d e g (F _ {i + 1})}. \\\end{array} $$ Theorem 0.9. Let $C$ be a quasi-cyclic code of length $r$ over a finite chain $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)$, which has maximal $m\Theta$ ideal $\langle a\rangle$ and $t$ is the nilpotency of $a$. Then there exist polynomials $g_0, g_1, \dots, g_{t-1}$ in $\mathbb{F}(p^k\mathbb{Z},1)[x]$ such that $C = \langle g_0, ag_1, \dots, a^{t-1}g_{t-1}\rangle$ and $g_{t-1}|g_{t-2}| \dots|g_1|g_0|(x^r - 1)$. Proof 0.14. According to previous theorem, there exists a family of pairwise coprime monic $m\Theta$ polynomials $F_0, F_1, \dots, F_t$ in $\mathbb{F}(p^k\mathbb{Z}, 1)[x]$ such that $F_0F_1 \dots F_t = x^r - 1$ and $C = \langle \widehat{F_1}, a\widehat{F_2}, \dots, a^{t-1}\widehat{F_t} \rangle$. Define $$ g _ {i} = \left\{ \begin{array}{l l} F _ {0} F _ {1} \dots F _ {t}, & \text{if } 0 \leq i \leq t - 2 \\ F _ {0}, & \text{if } i = t - 1. \end{array} \right. $$ Then clearly $g_{t-1}|g_{t-2}|\dots|g_1|g_0|(x^r - 1)$. Moreover, for $0 \leq i \leq t - 1$, we have $$ a ^ {i} \hat {F} _ {i + 1} = a ^ {i} F _ {0} \dots F _ {i} F _ {i + 2} \dots F _ {t} = a ^ {i} g _ {i} F _ {1} \dots F _ {i}. $$ Therefore, $C \subseteq \langle g_0, ag_1, \dots, a^{t-1}g_{t-1} \rangle$. On the other hand, $g_0 = F_0F_1 \cdots F_t \in C$. Since $F_1, F_2$ are coprime $m\Theta$ polynomials in $\mathbb{F}(p^k\mathbb{Z}, 1)[x]$, there exist polynomials $u, v \in \mathbb{F}(p^k\mathbb{Z}, 1)[x]$ such that $uF_1 + vF_2 = 1$. It follows that $$ \begin{array}{l} {g _ {1}} = {F _ {0} F _ {3} \dots F _ {t} = (u F _ {1} + v F _ {2}) F _ {0} F _ {3} \dots F _ {t}} \\= u F _ {0} F _ {1} F _ {3} \dots F _ {t} + c F _ {0} F _ {2} F _ {3} \dots F _ {t} = u \widehat {F _ {2}} + v g _ {0} \\\end{array} $$ whence $ag_{1} = au\widehat{F_{2}} + avg_{0} \in C$. Continuing this process, we obtain $a^{i}g_{i} \in C$ for $0 \leq i \leq t - 1$, which implies $$ \left\langle g _ {0}, a g _ {1}, \dots , a ^ {t - 1} g _ {t - 1} \right\rangle \subseteq C. $$ Consequently, $C = \langle g_0, ag_1, \dots, a^{t - 1}g_{t - 1}\rangle$. ## VIII. CONCLUSION This note studies the Quasi-Cyclic codes over a finite chain $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)$, which leads to the modal structure of the notion Quasi-Cyclic codes over a finite chain pseudo field [3]. It appears that the Structures of Quasi-Cyclic codes of length $r$ over a finite chain $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)$ are established when $r$ is not divisible by the characteristic of the residue $m\Theta$ pseudo field $\overline{\mathbb{F}(p^k\mathbb{Z},1)}$. Some cases where $r$ divisible by the characteristic of the residue $m\Theta$ field $\overline{\mathbb{F}(p^k\mathbb{Z},1)}$ are also considered. At the end of this study, some interesting problems remain to be solved: 1. We would like to construct the $m\Theta$ structure of cyclic dual codes and negacyclic codes over finite chain $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)$. 2. We would like to define a necessary and sufficient condition for the existence of self-dual cyclic $m\Theta$ codes over a $m\Theta$ pseudo field $\mathbb{F}(p^k\mathbb{Z},1)$.

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