A class of repeated-root constacyclic codes over $\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ of Type $2$

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ where $p$ is an odd prime, $n$ be a positive integer satisfying ${\rm gcd}(n,p)=1$, and denote $R=\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ where $e\geq 4$ be an even integer. Let $\delta,\alpha\in \mathbb{F}_{p^m}^{\times}$. Then the class of $(\delta+\alpha u^2)$-constacyclic codes over $R$ is a significant subclass of constacyclic codes over $R$ of Type 2. For any integer $k\geq 1$, an explicit representation and a complete description for all distinct $(\delta+\alpha u^2)$-constacyclic codes over $R$ of length $np^k$ and their dual codes are given. Moreover, formulas for the number of codewords in each code and the number of all such codes are provided respectively. In particular, all distinct $(\delta+\alpha u^2)$-contacyclic codes over $\mathbb{F}_{p^m}[u]/\langle u^{e}\rangle$ of length $p^k$ and their dual codes are presented precisely.

[1]  Masaaki Harada,et al.  Type II Codes Over F2 + u F2 , 1999, IEEE Trans. Inf. Theory.

[2]  H. Dinh Constacyclic Codes of Length $2^s$ Over Galois Extension Rings of ${\BBF}_{2}+u{\BBF}_2$ , 2009, IEEE Transactions on Information Theory.

[3]  Taher Abualrub,et al.  Constacyclic codes over F2+uF2 , 2009, J. Frankl. Inst..

[4]  Shixin Zhu,et al.  (1+λu)-Constacyclic codes over Fp[u]/〈um〉 , 2010, J. Frankl. Inst..

[5]  Songsak Sriboonchitta,et al.  A class of linear codes of length 2 over finite chain rings , 2019, Journal of Algebra and Its Applications.

[6]  Shixin Zhu,et al.  (1+u) constacyclic and cyclic codes over F2+uF2 , 2006, Appl. Math. Lett..

[7]  Yonglin Cao On constacyclic codes over finite chain rings , 2013, Finite Fields Their Appl..

[8]  H. Dinh Constacyclic Codes of Length p^s Over Fpm + uFpm , 2010 .

[9]  Maria Carmen V. Amarra,et al.  On (1-u) -cyclic codes over Fpk + uFpk , 2008, Appl. Math. Lett..

[10]  A. Sălăgean,et al.  On the stucture of linear and cyclic codes over finite chain rings , 2018 .

[11]  Hongwei Liu,et al.  Self-dual codes over commutative Frobenius rings , 2010, Finite Fields Their Appl..

[12]  Wei Zhao,et al.  All α + uβ-constacyclic codes of length nps over Fpm+uFpm , 2018, Finite Fields Their Appl..

[13]  Reza Sobhani Complete classification of (δ+αu2)-constacyclic codes of length pk over Fpm+uFpm+u2Fpm , 2015, Finite Fields Their Appl..

[14]  Songsak Sriboonchitta,et al.  Cyclic and negacyclic codes of length 4ps over 𝔽pm + u𝔽pm , 2018, Journal of Algebra and Its Applications.

[15]  Graham H. Norton,et al.  On the Structure of Linear and Cyclic Codes over a Finite Chain Ring , 2000, Applicable Algebra in Engineering, Communication and Computing.

[16]  Sergio R. López-Permouth,et al.  Cyclic and negacyclic codes over finite chain rings , 2004, IEEE Transactions on Information Theory.

[17]  Q DinhHai,et al.  Repeated-root constacyclic codes of prime power length over F p m u { u a } and their duals , 2016 .

[18]  Q DinhHai Constacyclic codes of length 2sover Galois extension rings of F2 + uF2 , 2009 .

[19]  Li Dong,et al.  Complete classification of (δ + α u2)-constacyclic codes over 𝔽3m[u] of length 3n , 2018, Appl. Algebra Eng. Commun. Comput..

[20]  Songsak Sriboonchitta,et al.  On constacyclic codes of length 4ps over Fpm+uFpm , 2017, Discret. Math..

[21]  Shixin Zhu,et al.  Negacyclic codes of length 2p2 over 𝔽pm + u𝔽pm , 2015, Finite Fields Their Appl..

[22]  Jian Gao,et al.  On a Class of (δ+αu2)-Constacyclic Codes over Fq[u]/〈u4〉 , 2016, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..