American Institute of Mathematical Sciences

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August  2022, 16(3): 525-570. doi: 10.3934/amc.2020123

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

Hai Q. Dinh 1,2, , Bac T. Nguyen 3,, and  Paravee Maneejuk 4,

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  August 2022 Early access  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 $.

Keywords: Constacyclic codes, negacyclic codes, cyclic codes, dual codes, repeated-root codes, chain rings, local rings.
Mathematics Subject Classification: Primary 94B15, 94B05; Secondary 11T71.
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, 2022, 16 (3) : 525-570. doi: 10.3934/amc.2020123
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.

[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. 

[3]

E. Bannai, M. Harada, T. Ibukiyama, A. 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.

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

[6]

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

[7]

I. F. Blake, S. 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.

[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.

[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.

[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.

[11]

Y. Cao, Y. Cao, H. Q. Dinh, F. Fu, J. 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.

[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.

[13]

Y. Cao, Y. Cao, H. 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.

[14]

Y. Cao, Y. Cao, H. Q. Dinh, F. Fu, J. 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.

[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.

[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.

[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.

[18]

Y. Cao, Y. Cao, H. Q. Dinh, T. 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.

[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.

[20]

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

[21]

B. Chen, H. Q. Dinh, H. 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.

[22]

G. Castagnoli, J. L. Massey, P. A. Schoeller and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37 (1991), 337-342.  doi: 10.1109/18.75249.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

H. Q. Dinh, L. 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.

[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.

[31]

H. Q. Dinh, S. 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.

[32]

H. Q. Dinh, B. 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.

[33]

H. Q. Dinh, Y. Fan, H. Liu, X. 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.

[34]

G. Falkner, B. Kowol, W. Heise and E. Zehendner, On the existence of cyclic optimal codes, Atti Sem. Mat. Fis. Univ. Modena, 28 (1979), 326-341. 

[35]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. 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.

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

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

[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.

[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.

[40]

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

[41]

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

[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.

[43]

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

[44]

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

[45]

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

[46]

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

[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.

[48]

A. Sălăgean, Repeated-root cyclic and negacyclic codes over finite chain rings, Discrete Appl. Math., 154 (2006), 413-419.  doi: 10.1016/j.dam.2005.03.016.

[49]

M. Shi, S. Zhu and S. Yang, A class of optimal p-ary codes from one-weight codes over $\mathbb F_p[u]/\langle u^m\rangle$, J. Franklin Inst., 350 (2013), 729-737.  doi: 10.1016/j.jfranklin.2012.05.014.

[50]

M. Shi and Y. Zhang, Quasi-twisted codes with constacyclic constituent codes, Finite Fields and Their Applications, 39 (2016), 159-178.  doi: 10.1016/j.ffa.2016.01.010.

[51]

P. Udaya and A. Bonnecaze, Decoding of cyclic codes over $\mathbb F_2 + u\mathbb F_2$, IEEE Trans. Inform. Theory, 45 (1999), 2148-2157.  doi: 10.1109/18.782165.

[52]

J. H. van Lint, Repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37 (1991), 343-345.  doi: 10.1109/18.75250.

[53]

J. Wolfmann, Negacyclic and cyclic codes over $\mathbb Z_4$, IEEE Trans. Inform. Theory, 45 (1999), 2527-2532.  doi: 10.1109/18.796397.

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.

[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. 

[3]

E. Bannai, M. Harada, T. Ibukiyama, A. 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.

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

[6]

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

[7]

I. F. Blake, S. 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.

[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.

[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.

[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.

[11]

Y. Cao, Y. Cao, H. Q. Dinh, F. Fu, J. 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.

[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.

[13]

Y. Cao, Y. Cao, H. 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.

[14]

Y. Cao, Y. Cao, H. Q. Dinh, F. Fu, J. 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.

[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.

[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.

[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.

[18]

Y. Cao, Y. Cao, H. Q. Dinh, T. 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.

[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.

[20]

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

[21]

B. Chen, H. Q. Dinh, H. 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.

[22]

G. Castagnoli, J. L. Massey, P. A. Schoeller and N. von Seemann, On repeated-root cyclic codes, IEEE Trans. Inform. Theory, 37 (1991), 337-342.  doi: 10.1109/18.75249.

[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.

[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.

[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.

[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.

[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.

[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.

[29]

H. Q. Dinh, L. 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.

[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.

[31]

H. Q. Dinh, S. 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.

[32]

H. Q. Dinh, B. 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.

[33]

H. Q. Dinh, Y. Fan, H. Liu, X. 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.

[34]

G. Falkner, B. Kowol, W. Heise and E. Zehendner, On the existence of cyclic optimal codes, Atti Sem. Mat. Fis. Univ. Modena, 28 (1979), 326-341. 

[35]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. 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.

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

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

[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.

[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.

[40]

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

[41]

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

[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.

[43]

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

[44]

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

[45]

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

[46]

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

[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.

[48]

A. Sălăgean, Repeated-root cyclic and negacyclic codes over finite chain rings, Discrete Appl. Math., 154 (2006), 413-419.  doi: 10.1016/j.dam.2005.03.016.

[49]

M. Shi, S. Zhu and S. Yang, A class of optimal p-ary codes from one-weight codes over $\mathbb F_p[u]/\langle u^m\rangle$, J. Franklin Inst., 350 (2013), 729-737.  doi: 10.1016/j.jfranklin.2012.05.014.

[50]

M. Shi and Y. Zhang, Quasi-twisted codes with constacyclic constituent codes, Finite Fields and Their Applications, 39 (2016), 159-178.  doi: 10.1016/j.ffa.2016.01.010.

[51]

P. Udaya and A. Bonnecaze, Decoding of cyclic codes over $\mathbb F_2 + u\mathbb F_2$, IEEE Trans. Inform. Theory, 45 (1999), 2148-2157.  doi: 10.1109/18.782165.

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