August  2022, 16(3): 503-524. doi: 10.3934/amc.2020122

Codes over $ \frak m $-adic completion rings

Department of Mathematics, Bu Ali Sina University, Hamedan, Iran

*Corresponding author

Received  April 2020 Revised  September 2020 Published  August 2022 Early access  December 2020

The theory of linear codes over finite rings has been generalized to linear codes over infinite rings in two special cases; the ring of $ p $-adic integers and formal power series ring. These rings are examples of complete discrete valuation rings (CDVRs). In this paper, we generalize the theory of linear codes over the above two rings to linear codes over complete local principal ideal rings. In particular, we obtain the structure of linear and constacyclic codes over CDVRs. For this generalization, first we study linear codes over $ \hat{R}_{ \frak m} $, where $ R $ is a commutative Noetherian ring, $ \frak m = \langle \gamma\rangle $ is a maximal ideal of $ R $, and $ \hat{R}_{ \frak m} $ denotes the $ \frak m $-adic completion of $ R $. We call these codes, $ \frak m $-adic codes. Using the structure of $ \frak m $-adic codes, we present the structure of linear and constacyclic codes over complete local principal ideal rings.

Citation: Saadoun Mahmoudi, Karim Samei. Codes over $ \frak m $-adic completion rings. Advances in Mathematics of Communications, 2022, 16 (3) : 503-524. doi: 10.3934/amc.2020122
References:
[1]

M. F. Atiyah and I. G. Macdonald, Intrduction to Commutative Algebra, University of Oxford, 1969. doi: 10.1007/978-1-4612-0873-0.

[2]

A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Cryptogr., 6 (1995), 21-35.  doi: 10.1007/BF01390768.

[3]

I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc., 59 (1946), 54-106.  doi: 10.1090/S0002-9947-1946-0016094-3.

[4]

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.

[5]

S. T. DoughertyJ. Kim and H. Kulosman, MDS codes over finite principal ideal rings, MDS Codes Over Finite Principal Ideal Rings, 50 (2009), 77-92.  doi: 10.1007/s10623-008-9215-5.

[6]

S. T. DoughertyS. Y. Kim and Y. H. Park, Lifted codes and their weight enumerators, Discrete. Math., 305 (2005), 123-135.  doi: 10.1016/j.disc.2005.08.004.

[7]

S. T. DoughertyH. Liu and Y. H. Park, Lifted codes over finite chain rings, Math. J. Okayama Univ., 53 (2011), 39-53. 

[8]

S. T. Dougherty and H. Liu, Cyclic codes over formal power series rings, Acta Mathematica Scientia, 31B (2011), 331-343.  doi: 10.1016/S0252-9602(11)60233-6.

[9]

S. T. Dougherty and Y. H. Park, Codes over the $p$-adic integers, Des. Codes Cryptogr., 39 (2006), 65-80.  doi: 10.1007/s10623-005-2542-x.

[10]

D. Eisenbud, Commutative Algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995. doi: 10.1007/978-1-4612-0873-0.

[11]

E. E. Enochs and O. M. G. Jenda, Relative Homolodical Algebra, Walter de Gruyter, 2000. doi: 10.1007/978-1-4612-0873-0.

[12]

K. Guenda and T. A. Gulliver, MDS and self-dual codes over rings, Finite Fields Appl., 18 (2012), 1061-1075.  doi: 10.1016/j.ffa.2012.09.003.

[13]

S. Jean-Pierre, Local Fields, Berlin, New York, 1980. doi: 10.1007/978-1-4612-0873-0.

[14]

F. J. Makwilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, North-Holland, Amsterdam, 1977. doi: 10.1007/978-1-4612-0873-0.

[15]

H. Matsumura, Commutative Ring Theory, Cambridge, 1989. doi: 10.1007/978-1-4612-0873-0.

[16]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker, Inc., New York, 1974. doi: 10.1007/978-1-4612-0873-0.

[17]

K. R. McLean, Commutative Artinian principal ideal rings, Proc. London Math. Soc., 26 (1973), 249-272.  doi: 10.1112/plms/s3-26.2.249.

[18]

K. Samei and S. Mahmoudi, Cyclic $R$-additive codes, Discrete Math., 340 (2017), 1657-1668.  doi: 10.1016/j.disc.2016.11.007.

[19]

K. Samei and S. Mahmoudi, Singleton Bundes for $R$-additive codes, Adv. Math. Commun., 12 (2018), 107-114.  doi: 10.3934/amc.2018006.

[20]

P. Solé, Open problems 2: Cyclic codes over rings and $p$-adic fields, in (G. Cohen and J. Wolfmann eds.) Coding Theory and Applications, Lect. Notes Comp. Sci., 338, Springer-Verlag, 1988. doi: 10.1007/BFb0019872.

show all references

References:
[1]

M. F. Atiyah and I. G. Macdonald, Intrduction to Commutative Algebra, University of Oxford, 1969. doi: 10.1007/978-1-4612-0873-0.

[2]

A. R. Calderbank and N. J. A. Sloane, Modular and $p$-adic cyclic codes, Des. Codes Cryptogr., 6 (1995), 21-35.  doi: 10.1007/BF01390768.

[3]

I. S. Cohen, On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc., 59 (1946), 54-106.  doi: 10.1090/S0002-9947-1946-0016094-3.

[4]

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.

[5]

S. T. DoughertyJ. Kim and H. Kulosman, MDS codes over finite principal ideal rings, MDS Codes Over Finite Principal Ideal Rings, 50 (2009), 77-92.  doi: 10.1007/s10623-008-9215-5.

[6]

S. T. DoughertyS. Y. Kim and Y. H. Park, Lifted codes and their weight enumerators, Discrete. Math., 305 (2005), 123-135.  doi: 10.1016/j.disc.2005.08.004.

[7]

S. T. DoughertyH. Liu and Y. H. Park, Lifted codes over finite chain rings, Math. J. Okayama Univ., 53 (2011), 39-53. 

[8]

S. T. Dougherty and H. Liu, Cyclic codes over formal power series rings, Acta Mathematica Scientia, 31B (2011), 331-343.  doi: 10.1016/S0252-9602(11)60233-6.

[9]

S. T. Dougherty and Y. H. Park, Codes over the $p$-adic integers, Des. Codes Cryptogr., 39 (2006), 65-80.  doi: 10.1007/s10623-005-2542-x.

[10]

D. Eisenbud, Commutative Algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995. doi: 10.1007/978-1-4612-0873-0.

[11]

E. E. Enochs and O. M. G. Jenda, Relative Homolodical Algebra, Walter de Gruyter, 2000. doi: 10.1007/978-1-4612-0873-0.

[12]

K. Guenda and T. A. Gulliver, MDS and self-dual codes over rings, Finite Fields Appl., 18 (2012), 1061-1075.  doi: 10.1016/j.ffa.2012.09.003.

[13]

S. Jean-Pierre, Local Fields, Berlin, New York, 1980. doi: 10.1007/978-1-4612-0873-0.

[14]

F. J. Makwilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, North-Holland, Amsterdam, 1977. doi: 10.1007/978-1-4612-0873-0.

[15]

H. Matsumura, Commutative Ring Theory, Cambridge, 1989. doi: 10.1007/978-1-4612-0873-0.

[16]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker, Inc., New York, 1974. doi: 10.1007/978-1-4612-0873-0.

[17]

K. R. McLean, Commutative Artinian principal ideal rings, Proc. London Math. Soc., 26 (1973), 249-272.  doi: 10.1112/plms/s3-26.2.249.

[18]

K. Samei and S. Mahmoudi, Cyclic $R$-additive codes, Discrete Math., 340 (2017), 1657-1668.  doi: 10.1016/j.disc.2016.11.007.

[19]

K. Samei and S. Mahmoudi, Singleton Bundes for $R$-additive codes, Adv. Math. Commun., 12 (2018), 107-114.  doi: 10.3934/amc.2018006.

[20]

P. Solé, Open problems 2: Cyclic codes over rings and $p$-adic fields, in (G. Cohen and J. Wolfmann eds.) Coding Theory and Applications, Lect. Notes Comp. Sci., 338, Springer-Verlag, 1988. doi: 10.1007/BFb0019872.

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