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Codes over $ \frak m $-adic completion rings
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 |
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 $.
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.
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