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doi: 10.3934/amc.2020123

Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $

1. 

Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

2. 

Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

3. 

Department of Basic Sciences, Thai Nguyen University of Economics and Business Administration, Thai Nguyen province, Vietnam

4. 

Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Thailand

* Corresponding author: Bac T. Nguyen

Received  May 2020 Published  December 2020

For any odd prime $ p $, the structures and duals of $ \lambda $-constacyclic codes of length $ 8p^s $ over $ \mathcal R = \mathbb F_{p^m}+u\mathbb F_{p^m} $ are completely determined for all unit $ \lambda $ of the form $ \lambda = \xi^l\in \mathbb F_{p^m} $, where $ l $ is even. In addition, the algebraic structures of all cyclic and negacyclic codes of length $ 8p^s $ over $ \mathcal R $ are established in term of their generator polynomials. Dual codes of all cyclic and negacyclic codes of length $ 8p^s $ over $ \mathcal R $ are also investigated. Furthermore, we give the number of codewords in each of those cyclic and negacyclic codes. We also obtain the number of cyclic and negacyclic codes of length $ 8p^s $ over $ \mathcal R $.

Citation: Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $. Advances in Mathematics of Communications, doi: 10.3934/amc.2020123
References:
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T. Abualrub and R. Oehmke, On the generators of $\mathbb Z_ 4$ cyclic codes of length $2^ e$, IEEE Trans. Inform. Theory., 49 (2003), 2126-2133.  doi: 10.1109/TIT.2003.815763.  Google Scholar

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M. M. Al-Ashker, Simplex codes over the ring $\mathbb F_2+u\mathbb F_2$, Arab. J. Sci. Eng. Sect. A Sci., 30 (2005), 277-285.   Google Scholar

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E. BannaiM. HaradaT. IbukiyamaA. Munemasa and M. Oura, Type Ⅱ codes over $\Bbb F_ 2+u\Bbb F_ 2$ and applications to Hermitian modular forms, Abh. Math. Sem. Univ. Hamburg., 73 (2003), 13-42.  doi: 10.1007/BF02941267.  Google Scholar

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E. R. Berlekamp, Negacyclic Codes for the Lee Metric, Proceedings of the Conference on Combinatorial Mathematics and its Application, Chapel Hill, NC, 1968,298–316.  Google Scholar

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S. D. Berman, Semisimple cyclic and Abelian codes. Ⅱ, Cybernetics, 3 (1967), 17-23.  doi: 10.1007/BF01119999.  Google Scholar

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I. F. BlakeS. Gao and R. C. Mullin, Explicit factorization of $x^{2^k}+1$ over $\mathbb F_{p^m}$ with prime $p\equiv 3 \pmod 4$, Applicable Algebra in Engineering, Communication and Computing, 4 (1993), 89-94.  doi: 10.1007/BF01386832.  Google Scholar

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T. Blackford, Negacyclic codes over $\mathbb Z_ 4$ of even length, IEEE Trans. Inform. Theory, 49 (2003), 1417-1424.  doi: 10.1109/TIT.2003.811915.  Google Scholar

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T. Blackford, Cyclic codes over $\mathbb Z_4$ of oddly even length, Appl. Discr. Math., 128 (2003), 27-46.  doi: 10.1016/S0166-218X(02)00434-1.  Google Scholar

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A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over $\mathbb F_2 + u\mathbb F_2$, IEEE Trans. Inform. Theory, 45 (1999), 1250-1255.  doi: 10.1109/18.761278.  Google Scholar

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Y. CaoY. CaoH. Q. DinhF. FuJ. Gao and S. Sriboonchitta, Constacyclic codes of length $np^s$ over $\mathbb F_{p^m}+u\mathbb F_{p^m}$, Adv. Math. Commun., 12 (2018), 231-262.  doi: 10.3934/amc.2018016.  Google Scholar

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Y. Cao, Y. Cao, H. Q. Dinh, F. Fu, J. Gao and S. Sriboonchitta, A class of repeated-root constacyclic codes over $\mathbb F_{p^m} [u]/\langle u^e\rangle$ of Type 2, Finite Fields & Appl., (2019), 238–267. doi: 10.1016/j.ffa.2018.10.003.  Google Scholar

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Y. CaoY. CaoH. Q. Dinh and S. Jitman, An explicit representation and enumeration for self-dual cyclic codes over $\mathbb F_{2^m}+u\mathbb F_{2^m}$ of length $2^s$, Discrete Math., 342 (2019), 2077-2091.  doi: 10.1016/j.disc.2019.04.008.  Google Scholar

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Y. Cao, Y. Cao, H. Q. Dinh, F. Fu and F. Ma, Construction and enumeration for self-dual cyclic codes of even length over $\mathbb F_{2^m} + u\mathbb F_{2^m}$, Finite Fields & Appl., 61 (2020), in press. doi: 10.1016/j.ffa.2019.101598.  Google Scholar

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Y. Cao, Y. Cao, H. Q. Dinh, F. Fu and F. Ma, On matrix-product structure of repeated-root constacyclic codes over finite fields, Discrete Math., 343 (2020), 111768, in press. doi: 10.1016/j.disc.2019.111768.  Google Scholar

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Y. Cao, Y. Cao, H. Q. Dinh, R. Bandi and F. Fu, An explicit representation and enumeration for negacyclic codes of length $2^kn$ over $\mathbb Z_4+u\mathbb Z_4$, Adv. Math. Commun., 2020, in press. doi: 10.1007/s12095-020-00429-z.  Google Scholar

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Y. CaoY. CaoH. Q. DinhT. Bag and W. Yamaka, Explicit representation and enumeration of repeated-root $\delta^2+\alpha u^2$-constacyclic codes over $\mathbb F_{2^m}[u]/\langle u^{2\lambda}$, IEEE Access, 8 (2020), 55550-55562.  doi: 10.1016/j.jpaa.2017.11.007.  Google Scholar

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Y. Cao, Y. Cao, H. Q. Dinh and S. Jitman, An efcient method to construct self-dual cyclic codes of length $p^s$ over $\mathbb F_{p^m}+u\mathbb F_{p^m}$, Discrete Math., 343 (2020), 111868. doi: 10.1016/j.disc.2020.111868.  Google Scholar

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B. ChenY. Fan and L. Liu, Constacyclic codes over finite fields, Finite Fields & Appl., 18 (2012), 1217-1231.  doi: 10.1016/j.ffa.2012.10.001.  Google Scholar

[21]

B. ChenH. Q. DinhH. Liu and L. Wang, Constacyclic codes of length $2p^s$ over $\mathcal R$, Finite Fields & Appl., 36 (2016), 108-130.  doi: 10.1016/j.ffa.2015.09.006.  Google Scholar

[22]

G. CastagnoliJ. L. MasseyP. A. Schoeller and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37 (1991), 337-342.  doi: 10.1109/18.75249.  Google Scholar

[23]

H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings, IEEE Trans. Inform. Theory, 51 (2005), 4252-4262.  doi: 10.1109/TIT.2005.859284.  Google Scholar

[24]

H. Q. Dinh, Constacyclic codes of length $2^s$ over Galois extension rings of $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Trans. Inform. Theory, 55 (2009), 1730-1740.  doi: 10.1109/TIT.2009.2013015.  Google Scholar

[25]

H. Q. Dinh, Constacyclic codes of length $p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$, J. Algebra, 324 (2010), 940-950.  doi: 10.1016/j.jalgebra.2010.05.027.  Google Scholar

[26]

H. Q. Dinh, Repeated-root constacyclic codes of length $2p^s$, Finite Fields & Appl., 18 (2012), 133-143.  doi: 10.1016/j.ffa.2011.07.003.  Google Scholar

[27]

H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3p^s$ and their duals, Discrete Math., 313 (2013), 983-991.  doi: 10.1016/j.disc.2013.01.024.  Google Scholar

[28]

H. Q. Dinh, Structure of repeated-root cyclic and negacyclic codes of length $6p^s$ and their duals, AMS Contemporary Mathematics, 609 (2014), 69-87.  doi: 10.1090/conm/609/12150.  Google Scholar

[29]

H. Q. DinhL. Wang and S. Zhu, Negacyclic codes of length $2p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$, Finite Fields & Appl., 31 (2015), 178-201.  doi: 10.1016/j.ffa.2014.09.003.  Google Scholar

[30]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.  Google Scholar

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H. Q. DinhS. Dhompongsa and S. Sriboonchitta, On constacyclic codes of length $4p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$, Discrete Math., 340 (2017), 832-849.  doi: 10.1016/j.disc.2016.11.014.  Google Scholar

[32]

H. Q. DinhB. T. Nguyen and S. Sriboonchitta, Negacyclic codes of length $4p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$, Discrete Mathematics, 341 (2018), 1055-1071.  doi: 10.1016/j.disc.2017.12.019.  Google Scholar

[33]

H. Q. DinhY. FanH. LiuX. Liu and S. Sriboonchitta, On self-dual constacyclic codes of length $p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$, Discrete Math., 341 (2018), 324-335.  doi: 10.1016/j.disc.2017.08.044.  Google Scholar

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G. FalknerB. KowolW. Heise and E. Zehendner, On the existence of cyclic optimal codes, Atti Sem. Mat. Fis. Univ. Modena, 28 (1979), 326-341.   Google Scholar

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A. R. HammonsP. V. KumarA. R. CalderbankN. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar

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E. Kleinfeld, Finite Hjelmslev planes, Illinois J. Math., 3 (1959), 403-407.  doi: 10.1215/ijm/1255455261.  Google Scholar

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Y. Liu and M. Shi, Repeated-Root Constacyclic Codes of Length $k\ell p^s$, Bull. Malays. Math. Sci. Soc., 43 (2019). doi: 10.1007/s40840-019-00787-9.  Google Scholar

[39]

Y. Liu, M. Shi, H. Q. Dinh and S. Sriboonchitta, Repeated-root constacyclic codes of length $3l^mp^s$, Advances Math. Comm., 2019, to appear. doi: 10.3934/amc.2020025.  Google Scholar

[40]

J. L. MasseyD. J. Costello and J. Justesen, Polynomial weights and code constructions, IEEE Trans. Inform. Theory, 19 (1973), 101-110.  doi: 10.1109/tit.1973.1054936.  Google Scholar

[41]

B. R. McDonald, Finite rings with identity, Pure and Applied Mathematics, 28 (1974).  Google Scholar

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C. S. Nedeloaia, Weight distributions of cyclic self-dual codes, IEEE Trans. Inform. Theory, 49 (2003), 1582-1591.  doi: 10.1109/TIT.2003.811921.  Google Scholar

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E. Prange, Cyclic error-correcting codes in two symbols, Air Force Cambridge Research Center, (1957), 26 pp. Google Scholar

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E. Prange, Some cyclic error-correcting codes with simple decoding algorithms, Air Force Cambridge Research Center, (1958). Google Scholar

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E. Prange, The use of coset equivalence in the analysis and decoding of group codes, Air Force Cambridge Research Center, (1959). Google Scholar

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E. Prange, An algorithm for factoring $x^n-1$ over a finite field, TN-59-175, (1959). Google Scholar

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R. M. Roth and G. Seroussi, On cyclic MDS codes of length $q$ over ${\rm GF}(q)$, IEEE Trans. Inform. Theory, 32 (1986), 284-285.  doi: 10.1109/TIT.1986.1057151.  Google Scholar

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show all references

References:
[1]

T. Abualrub and R. Oehmke, On the generators of $\mathbb Z_ 4$ cyclic codes of length $2^ e$, IEEE Trans. Inform. Theory., 49 (2003), 2126-2133.  doi: 10.1109/TIT.2003.815763.  Google Scholar

[2]

M. M. Al-Ashker, Simplex codes over the ring $\mathbb F_2+u\mathbb F_2$, Arab. J. Sci. Eng. Sect. A Sci., 30 (2005), 277-285.   Google Scholar

[3]

E. BannaiM. HaradaT. IbukiyamaA. Munemasa and M. Oura, Type Ⅱ codes over $\Bbb F_ 2+u\Bbb F_ 2$ and applications to Hermitian modular forms, Abh. Math. Sem. Univ. Hamburg., 73 (2003), 13-42.  doi: 10.1007/BF02941267.  Google Scholar

[4] E. R. Berlekamp, Algebraic Coding Theory, Aegean Park Press, 1984.  doi: 10.1142/9407.  Google Scholar
[5]

E. R. Berlekamp, Negacyclic Codes for the Lee Metric, Proceedings of the Conference on Combinatorial Mathematics and its Application, Chapel Hill, NC, 1968,298–316.  Google Scholar

[6]

S. D. Berman, Semisimple cyclic and Abelian codes. Ⅱ, Cybernetics, 3 (1967), 17-23.  doi: 10.1007/BF01119999.  Google Scholar

[7]

I. F. BlakeS. Gao and R. C. Mullin, Explicit factorization of $x^{2^k}+1$ over $\mathbb F_{p^m}$ with prime $p\equiv 3 \pmod 4$, Applicable Algebra in Engineering, Communication and Computing, 4 (1993), 89-94.  doi: 10.1007/BF01386832.  Google Scholar

[8]

T. Blackford, Negacyclic codes over $\mathbb Z_ 4$ of even length, IEEE Trans. Inform. Theory, 49 (2003), 1417-1424.  doi: 10.1109/TIT.2003.811915.  Google Scholar

[9]

T. Blackford, Cyclic codes over $\mathbb Z_4$ of oddly even length, Appl. Discr. Math., 128 (2003), 27-46.  doi: 10.1016/S0166-218X(02)00434-1.  Google Scholar

[10]

A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over $\mathbb F_2 + u\mathbb F_2$, IEEE Trans. Inform. Theory, 45 (1999), 1250-1255.  doi: 10.1109/18.761278.  Google Scholar

[11]

Y. CaoY. CaoH. Q. DinhF. FuJ. Gao and S. Sriboonchitta, Constacyclic codes of length $np^s$ over $\mathbb F_{p^m}+u\mathbb F_{p^m}$, Adv. Math. Commun., 12 (2018), 231-262.  doi: 10.3934/amc.2018016.  Google Scholar

[12]

Y. Cao, Y. Cao, H. Q. Dinh, F. Fu, J. Gao and S. Sriboonchitta, A class of repeated-root constacyclic codes over $\mathbb F_{p^m} [u]/\langle u^e\rangle$ of Type 2, Finite Fields & Appl., (2019), 238–267. doi: 10.1016/j.ffa.2018.10.003.  Google Scholar

[13]

Y. CaoY. CaoH. Q. Dinh and S. Jitman, An explicit representation and enumeration for self-dual cyclic codes over $\mathbb F_{2^m}+u\mathbb F_{2^m}$ of length $2^s$, Discrete Math., 342 (2019), 2077-2091.  doi: 10.1016/j.disc.2019.04.008.  Google Scholar

[14]

Y. CaoY. CaoH. Q. DinhF. FuJ. Gao and S. Sriboonchitta, A class of linear codes of length 2 over finite chain rings, Journal of Algebra and its Applications, 19 (2020), 1-15.  doi: 10.1142/S0219498820501030.  Google Scholar

[15]

Y. Cao, Y. Cao, H. Q. Dinh, F. Fu and F. Ma, Construction and enumeration for self-dual cyclic codes of even length over $\mathbb F_{2^m} + u\mathbb F_{2^m}$, Finite Fields & Appl., 61 (2020), in press. doi: 10.1016/j.ffa.2019.101598.  Google Scholar

[16]

Y. Cao, Y. Cao, H. Q. Dinh, F. Fu and F. Ma, On matrix-product structure of repeated-root constacyclic codes over finite fields, Discrete Math., 343 (2020), 111768, in press. doi: 10.1016/j.disc.2019.111768.  Google Scholar

[17]

Y. Cao, Y. Cao, H. Q. Dinh, R. Bandi and F. Fu, An explicit representation and enumeration for negacyclic codes of length $2^kn$ over $\mathbb Z_4+u\mathbb Z_4$, Adv. Math. Commun., 2020, in press. doi: 10.1007/s12095-020-00429-z.  Google Scholar

[18]

Y. CaoY. CaoH. Q. DinhT. Bag and W. Yamaka, Explicit representation and enumeration of repeated-root $\delta^2+\alpha u^2$-constacyclic codes over $\mathbb F_{2^m}[u]/\langle u^{2\lambda}$, IEEE Access, 8 (2020), 55550-55562.  doi: 10.1016/j.jpaa.2017.11.007.  Google Scholar

[19]

Y. Cao, Y. Cao, H. Q. Dinh and S. Jitman, An efcient method to construct self-dual cyclic codes of length $p^s$ over $\mathbb F_{p^m}+u\mathbb F_{p^m}$, Discrete Math., 343 (2020), 111868. doi: 10.1016/j.disc.2020.111868.  Google Scholar

[20]

B. ChenY. Fan and L. Liu, Constacyclic codes over finite fields, Finite Fields & Appl., 18 (2012), 1217-1231.  doi: 10.1016/j.ffa.2012.10.001.  Google Scholar

[21]

B. ChenH. Q. DinhH. Liu and L. Wang, Constacyclic codes of length $2p^s$ over $\mathcal R$, Finite Fields & Appl., 36 (2016), 108-130.  doi: 10.1016/j.ffa.2015.09.006.  Google Scholar

[22]

G. CastagnoliJ. L. MasseyP. A. Schoeller and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37 (1991), 337-342.  doi: 10.1109/18.75249.  Google Scholar

[23]

H. Q. Dinh, Negacyclic codes of length $2^s$ over Galois rings, IEEE Trans. Inform. Theory, 51 (2005), 4252-4262.  doi: 10.1109/TIT.2005.859284.  Google Scholar

[24]

H. Q. Dinh, Constacyclic codes of length $2^s$ over Galois extension rings of $\mathbb{F}_2+u\mathbb{F}_2$, IEEE Trans. Inform. Theory, 55 (2009), 1730-1740.  doi: 10.1109/TIT.2009.2013015.  Google Scholar

[25]

H. Q. Dinh, Constacyclic codes of length $p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$, J. Algebra, 324 (2010), 940-950.  doi: 10.1016/j.jalgebra.2010.05.027.  Google Scholar

[26]

H. Q. Dinh, Repeated-root constacyclic codes of length $2p^s$, Finite Fields & Appl., 18 (2012), 133-143.  doi: 10.1016/j.ffa.2011.07.003.  Google Scholar

[27]

H. Q. Dinh, Structure of repeated-root constacyclic codes of length $3p^s$ and their duals, Discrete Math., 313 (2013), 983-991.  doi: 10.1016/j.disc.2013.01.024.  Google Scholar

[28]

H. Q. Dinh, Structure of repeated-root cyclic and negacyclic codes of length $6p^s$ and their duals, AMS Contemporary Mathematics, 609 (2014), 69-87.  doi: 10.1090/conm/609/12150.  Google Scholar

[29]

H. Q. DinhL. Wang and S. Zhu, Negacyclic codes of length $2p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$, Finite Fields & Appl., 31 (2015), 178-201.  doi: 10.1016/j.ffa.2014.09.003.  Google Scholar

[30]

H. Q. Dinh and S. R. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728-1744.  doi: 10.1109/TIT.2004.831789.  Google Scholar

[31]

H. Q. DinhS. Dhompongsa and S. Sriboonchitta, On constacyclic codes of length $4p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$, Discrete Math., 340 (2017), 832-849.  doi: 10.1016/j.disc.2016.11.014.  Google Scholar

[32]

H. Q. DinhB. T. Nguyen and S. Sriboonchitta, Negacyclic codes of length $4p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$, Discrete Mathematics, 341 (2018), 1055-1071.  doi: 10.1016/j.disc.2017.12.019.  Google Scholar

[33]

H. Q. DinhY. FanH. LiuX. Liu and S. Sriboonchitta, On self-dual constacyclic codes of length $p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$, Discrete Math., 341 (2018), 324-335.  doi: 10.1016/j.disc.2017.08.044.  Google Scholar

[34]

G. FalknerB. KowolW. Heise and E. Zehendner, On the existence of cyclic optimal codes, Atti Sem. Mat. Fis. Univ. Modena, 28 (1979), 326-341.   Google Scholar

[35]

A. R. HammonsP. V. KumarA. R. CalderbankN. J. A. Sloane and P. Solé, The $\mathbb Z_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar

[36] W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[37]

E. Kleinfeld, Finite Hjelmslev planes, Illinois J. Math., 3 (1959), 403-407.  doi: 10.1215/ijm/1255455261.  Google Scholar

[38]

Y. Liu and M. Shi, Repeated-Root Constacyclic Codes of Length $k\ell p^s$, Bull. Malays. Math. Sci. Soc., 43 (2019). doi: 10.1007/s40840-019-00787-9.  Google Scholar

[39]

Y. Liu, M. Shi, H. Q. Dinh and S. Sriboonchitta, Repeated-root constacyclic codes of length $3l^mp^s$, Advances Math. Comm., 2019, to appear. doi: 10.3934/amc.2020025.  Google Scholar

[40]

J. L. MasseyD. J. Costello and J. Justesen, Polynomial weights and code constructions, IEEE Trans. Inform. Theory, 19 (1973), 101-110.  doi: 10.1109/tit.1973.1054936.  Google Scholar

[41]

B. R. McDonald, Finite rings with identity, Pure and Applied Mathematics, 28 (1974).  Google Scholar

[42]

C. S. Nedeloaia, Weight distributions of cyclic self-dual codes, IEEE Trans. Inform. Theory, 49 (2003), 1582-1591.  doi: 10.1109/TIT.2003.811921.  Google Scholar

[43]

E. Prange, Cyclic error-correcting codes in two symbols, Air Force Cambridge Research Center, (1957), 26 pp. Google Scholar

[44]

E. Prange, Some cyclic error-correcting codes with simple decoding algorithms, Air Force Cambridge Research Center, (1958). Google Scholar

[45]

E. Prange, The use of coset equivalence in the analysis and decoding of group codes, Air Force Cambridge Research Center, (1959). Google Scholar

[46]

E. Prange, An algorithm for factoring $x^n-1$ over a finite field, TN-59-175, (1959). Google Scholar

[47]

R. M. Roth and G. Seroussi, On cyclic MDS codes of length $q$ over ${\rm GF}(q)$, IEEE Trans. Inform. Theory, 32 (1986), 284-285.  doi: 10.1109/TIT.1986.1057151.  Google Scholar

[48]

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