Skip to main content
Top
Published in: Applicable Algebra in Engineering, Communication and Computing 5/2021

31-01-2020 | OriginalPaper

Cyclic codes over \({\mathbb {F}}_2 +u{\mathbb {F}}_2+v{\mathbb {F}}_2 +v^2 {\mathbb {F}}_2 \) with respect to the homogeneous weight and their applications to DNA codes

Authors: Merve Bulut Yılgör, Fatmanur Gürsoy, Elif Segah Öztaş, Fatih Demirkale

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 5/2021

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, we study cyclic codes and their duals over the local Frobenius non-chain ring \(R={\mathbb {F}}_2[u,v] / \langle u^2=v^2,uv \rangle \), and we obtain optimal binary linear codes with respect to the homogeneous weight over R via a Gray map. Moreover, we characterize DNA codes as images of cyclic codes over R.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literature
1.
go back to reference Abualrub, T., Oehmke, R.: On the generators of Z4 codes of length 2e. IEEE Trans. Inform. Theory 49(9), 2126–2133 (2003)MathSciNetCrossRef Abualrub, T., Oehmke, R.: On the generators of Z4 codes of length 2e. IEEE Trans. Inform. Theory 49(9), 2126–2133 (2003)MathSciNetCrossRef
2.
go back to reference Abualrub, T., Siap, I.: Cyclic codes over \( Z_2+ uZ_2 \) and \( Z_2 + uZ_2 + u^2Z_2 \). Des. Codes Cryptogr. 42, 273–287 (2007)MathSciNetCrossRef Abualrub, T., Siap, I.: Cyclic codes over \( Z_2+ uZ_2 \) and \( Z_2 + uZ_2 + u^2Z_2 \). Des. Codes Cryptogr. 42, 273–287 (2007)MathSciNetCrossRef
3.
go back to reference Abulraub, T., Ghrayeb, A., Nian Zeng, X.: Construction of cyclic codes over \(GF(4)\) for DNA computing. J. Frankl. Inst. 343(4–5), 448–457 (2006)MathSciNetCrossRef Abulraub, T., Ghrayeb, A., Nian Zeng, X.: Construction of cyclic codes over \(GF(4)\) for DNA computing. J. Frankl. Inst. 343(4–5), 448–457 (2006)MathSciNetCrossRef
4.
go back to reference Adleman, L.: Molecular computation of solutions to combinatorial problems. Science 266(5187), 1021–1024 (1994)CrossRef Adleman, L.: Molecular computation of solutions to combinatorial problems. Science 266(5187), 1021–1024 (1994)CrossRef
5.
go back to reference Adleman, L., Rothemund, P.W.K., Roweis, S., Winfree, E.: On applying molecular computation to the data encryption standard. J. Comput. Biol. 6(1), 53–63 (1999)CrossRef Adleman, L., Rothemund, P.W.K., Roweis, S., Winfree, E.: On applying molecular computation to the data encryption standard. J. Comput. Biol. 6(1), 53–63 (1999)CrossRef
6.
go back to reference Bayram, A., Oztas, E.S., Siap, I.: Codes over \(F_4 + v F_4\) and some DNA applications. Des. Codes Cryptogr. 80(2), 379–393 (2015)CrossRef Bayram, A., Oztas, E.S., Siap, I.: Codes over \(F_4 + v F_4\) and some DNA applications. Des. Codes Cryptogr. 80(2), 379–393 (2015)CrossRef
7.
go back to reference Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997)MathSciNetCrossRef Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 235–265 (1997)MathSciNetCrossRef
10.
go back to reference Constantinescu, I., Heise, W.: A metric for codes over residue class rings of integers. Problemy Peredachi Informatsii 33(3), 22–28 (1997)MathSciNetMATH Constantinescu, I., Heise, W.: A metric for codes over residue class rings of integers. Problemy Peredachi Informatsii 33(3), 22–28 (1997)MathSciNetMATH
11.
go back to reference Darehmiraki, M.: A semi-general method to solve the combinatorial optimization problems based on nanocomputing. Int. J. Nanosci. 9(5), 391–398 (2010)CrossRef Darehmiraki, M.: A semi-general method to solve the combinatorial optimization problems based on nanocomputing. Int. J. Nanosci. 9(5), 391–398 (2010)CrossRef
12.
go back to reference Dinh, H.Q., Lopez-Permouth, S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inform. Theory 50(8), 1728–1744 (2004)MathSciNetCrossRef Dinh, H.Q., Lopez-Permouth, S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inform. Theory 50(8), 1728–1744 (2004)MathSciNetCrossRef
13.
go back to reference Dougherty, S.T., Kaya, A., Salturk, E.: Cyclic codes over local Frobenius rings of order 16. Adv. Math. Commun. 11(1), 99–114 (2017)MathSciNetCrossRef Dougherty, S.T., Kaya, A., Salturk, E.: Cyclic codes over local Frobenius rings of order 16. Adv. Math. Commun. 11(1), 99–114 (2017)MathSciNetCrossRef
14.
go back to reference Faria, L.C., Rocha, A.S., Kleinschmidt, J.H., Silva-Filho, M.C., Bim, E., Herai, R.H., Yamagishi, M.E., Palazzo Jr., R.: Is a genome a codeword of an error-correcting code? PLoS ONE 7(5), e36644 (2012)CrossRef Faria, L.C., Rocha, A.S., Kleinschmidt, J.H., Silva-Filho, M.C., Bim, E., Herai, R.H., Yamagishi, M.E., Palazzo Jr., R.: Is a genome a codeword of an error-correcting code? PLoS ONE 7(5), e36644 (2012)CrossRef
17.
go back to reference Greferath, M., Schmidt, S.E.: Gray isometries for finite chain rings and a nonlinear ternary \((36, 3^{12}, 15)\) code. IEEE Trans. Inform. Theory 45(7), 2522–2524 (1999)MathSciNetCrossRef Greferath, M., Schmidt, S.E.: Gray isometries for finite chain rings and a nonlinear ternary \((36, 3^{12}, 15)\) code. IEEE Trans. Inform. Theory 45(7), 2522–2524 (1999)MathSciNetCrossRef
18.
go back to reference Greferath, M., Schmidt, S.E.: Finite-ring combinatorics and MacWilliams’ equivalence theorem. J. Combin. Theory Ser. A 92, 17–28 (2000)MathSciNetCrossRef Greferath, M., Schmidt, S.E.: Finite-ring combinatorics and MacWilliams’ equivalence theorem. J. Combin. Theory Ser. A 92, 17–28 (2000)MathSciNetCrossRef
19.
go back to reference Guozhen, X., Mingxin, L., Lei, Q., Xuejia, L.: New field of cryptography: DNA cryptography. Chin. Sci. Bull. 51, 1413–1420 (2006)MathSciNetCrossRef Guozhen, X., Mingxin, L., Lei, Q., Xuejia, L.: New field of cryptography: DNA cryptography. Chin. Sci. Bull. 51, 1413–1420 (2006)MathSciNetCrossRef
20.
go back to reference Hammons, A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Solé, P.: The Z4 -linearity of Kerdock Preparata Goethals and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994)CrossRef Hammons, A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., Solé, P.: The Z4 -linearity of Kerdock Preparata Goethals and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994)CrossRef
21.
go back to reference Hesketh, E.E., Sayir, J., Goldman, N.: Improving communication for interdisciplinary teams working on storage of digital information in DNA. F1000Research 7, 39 (2018)CrossRef Hesketh, E.E., Sayir, J., Goldman, N.: Improving communication for interdisciplinary teams working on storage of digital information in DNA. F1000Research 7, 39 (2018)CrossRef
22.
go back to reference Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inform. Theory 52(11), 4892–4914 (2006)MathSciNetCrossRef Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inform. Theory 52(11), 4892–4914 (2006)MathSciNetCrossRef
23.
go back to reference Kuekes, P.J., Robinett, W., Roth, R.M., Seroussi, G., Snider, G.S., Williams, R.S.: Resistor-logic demultiplexers for nano electronics based on constant-weight codes. Nanotechnology 17(4), 1052–1061 (2006)CrossRef Kuekes, P.J., Robinett, W., Roth, R.M., Seroussi, G., Snider, G.S., Williams, R.S.: Resistor-logic demultiplexers for nano electronics based on constant-weight codes. Nanotechnology 17(4), 1052–1061 (2006)CrossRef
24.
go back to reference Liebovitch, L.S., Tao, Y., Todorov, A.T., Levine, L.: Is there an error correcting code in the base sequence in DNA? Biophys. J. 71(3), 1539–1544 (1996)CrossRef Liebovitch, L.S., Tao, Y., Todorov, A.T., Levine, L.: Is there an error correcting code in the base sequence in DNA? Biophys. J. 71(3), 1539–1544 (1996)CrossRef
25.
go back to reference Martinez-Moro, E., Szabo, S.: On codes over local Frobenius non-chain rings of order 16. Contemp. Math. 634, 227–241 (2015)MathSciNetCrossRef Martinez-Moro, E., Szabo, S.: On codes over local Frobenius non-chain rings of order 16. Contemp. Math. 634, 227–241 (2015)MathSciNetCrossRef
26.
go back to reference Moon, J.N.J., Hughes, L.A., Smith, D.H.: Assignment of frequency lists in frequency hopping networks. IEEE Trans. Veh. Technol. 54(3), 1147–1159 (2005)CrossRef Moon, J.N.J., Hughes, L.A., Smith, D.H.: Assignment of frequency lists in frequency hopping networks. IEEE Trans. Veh. Technol. 54(3), 1147–1159 (2005)CrossRef
27.
go back to reference Oztas, E.S., Siap, I.: Lifted polynomials over \(F_{16}\) and their applications to DNA codes. Filomat 27(3), 459–466 (2013)MathSciNetCrossRef Oztas, E.S., Siap, I.: Lifted polynomials over \(F_{16}\) and their applications to DNA codes. Filomat 27(3), 459–466 (2013)MathSciNetCrossRef
28.
go back to reference Oztas, E.S., Siap, I.: On a generalization of lifted polynomials over finite fields and their applications to DNA codes. Int. J. Comput. Math. 92(9), 1976–1988 (2015)MathSciNetCrossRef Oztas, E.S., Siap, I.: On a generalization of lifted polynomials over finite fields and their applications to DNA codes. Int. J. Comput. Math. 92(9), 1976–1988 (2015)MathSciNetCrossRef
29.
go back to reference Oztas, E.S., Yildiz, B., Siap, I.: On DNA codes from a family of chain rings. J. Algebra Comb. Discrete Struct. Appl. 4(1), 93–102 (2017)MathSciNetMATH Oztas, E.S., Yildiz, B., Siap, I.: On DNA codes from a family of chain rings. J. Algebra Comb. Discrete Struct. Appl. 4(1), 93–102 (2017)MathSciNetMATH
30.
go back to reference Shi, M., Wu, R., Qian, L., Sok, L., Solé, P.: New classes of p-ary few weight codes. Bull. Malays. Math. Sci. Soc. 42(4), 1393–1412 (2019)MathSciNetCrossRef Shi, M., Wu, R., Qian, L., Sok, L., Solé, P.: New classes of p-ary few weight codes. Bull. Malays. Math. Sci. Soc. 42(4), 1393–1412 (2019)MathSciNetCrossRef
31.
go back to reference Shi, M., Zhang, Y.: Quasi-twisted codes with constacyclic constituent codes. Finite Fields Appl. 39, 159–178 (2016)MathSciNetCrossRef Shi, M., Zhang, Y.: Quasi-twisted codes with constacyclic constituent codes. Finite Fields Appl. 39, 159–178 (2016)MathSciNetCrossRef
32.
go back to reference Shi, M., Qian, L., Sok, L., Aydın, N., Solé, P.: On constacyclic codes over \(Z4[u]/ \langle u^2-1 \rangle \) and their Gray images. Finite Fields Appl. 45, 86–95 (2017)MathSciNetCrossRef Shi, M., Qian, L., Sok, L., Aydın, N., Solé, P.: On constacyclic codes over \(Z4[u]/ \langle u^2-1 \rangle \) and their Gray images. Finite Fields Appl. 45, 86–95 (2017)MathSciNetCrossRef
33.
go back to reference Siap, I., Abulraub, T., Ghrayeb, A.: Cyclic DNA codes over the ring \(F_2[u]/(u^2-1)\) based on the deletion distance. J. Frankl. Inst. 346(8), 731–740 (2009)CrossRef Siap, I., Abulraub, T., Ghrayeb, A.: Cyclic DNA codes over the ring \(F_2[u]/(u^2-1)\) based on the deletion distance. J. Frankl. Inst. 346(8), 731–740 (2009)CrossRef
34.
go back to reference Sok, L., Shi, M., Solé, P.: Construction of optimal LCD codes over large finite fields. Finite Fields Appl. 50, 138–153 (2018)MathSciNetCrossRef Sok, L., Shi, M., Solé, P.: Construction of optimal LCD codes over large finite fields. Finite Fields Appl. 50, 138–153 (2018)MathSciNetCrossRef
35.
go back to reference UbaidurRahman, N.H., Balamurugan, C., Mariappan, R.: A novel DNA computing based encryption and decryption algorithm. Proc. Comput. Sci. 46, 463–475 (2015)CrossRef UbaidurRahman, N.H., Balamurugan, C., Mariappan, R.: A novel DNA computing based encryption and decryption algorithm. Proc. Comput. Sci. 46, 463–475 (2015)CrossRef
36.
go back to reference van Lint, J.H., Schrijver, A.: Construction of strongly regular graphs, two-weight codes and partial geometries by finite fields. Combinatorica 1(1), 63–73 (1981)MathSciNetCrossRef van Lint, J.H., Schrijver, A.: Construction of strongly regular graphs, two-weight codes and partial geometries by finite fields. Combinatorica 1(1), 63–73 (1981)MathSciNetCrossRef
37.
go back to reference Wang, X., Bao, Z., Hu, J., Wang, S., Zhan, A.: Solving the SAT problem using a DNA computing algorithm based on ligase chain reaction. BioSystems 91(1), 117–125 (2008)CrossRef Wang, X., Bao, Z., Hu, J., Wang, S., Zhan, A.: Solving the SAT problem using a DNA computing algorithm based on ligase chain reaction. BioSystems 91(1), 117–125 (2008)CrossRef
38.
go back to reference Yildiz, B., Karadeniz, S.: Cyclic codes over \({\mathbb{F}}_2+u {\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2\). Des. Codes Cryptogr. 58(3), 221–234 (2011)MathSciNetCrossRef Yildiz, B., Karadeniz, S.: Cyclic codes over \({\mathbb{F}}_2+u {\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2\). Des. Codes Cryptogr. 58(3), 221–234 (2011)MathSciNetCrossRef
39.
go back to reference Yildiz, B., Kelebek, I.G.: The homogeneous weight for \( R_k \), related Gray map a new binary quasi-cyclic codes. Filomat 31(4), 885–897 (2017)MathSciNetCrossRef Yildiz, B., Kelebek, I.G.: The homogeneous weight for \( R_k \), related Gray map a new binary quasi-cyclic codes. Filomat 31(4), 885–897 (2017)MathSciNetCrossRef
40.
go back to reference Yildiz, B., Siap, I.: Cyclic codes over \(F_2[u]/(u^4-1)\) and applications to DNA codes. Comput. Math. Appl. 63(7), 1169–1176 (2012)MathSciNetCrossRef Yildiz, B., Siap, I.: Cyclic codes over \(F_2[u]/(u^4-1)\) and applications to DNA codes. Comput. Math. Appl. 63(7), 1169–1176 (2012)MathSciNetCrossRef
41.
go back to reference Zhu, S., Wang, Y., Shi, M.: Some results on cyclic codes over \({F}_{2}+v{F}_{2}\). IEEE Trans. Inform. Theory 56(4), 1680–1684 (2010)MathSciNetCrossRef Zhu, S., Wang, Y., Shi, M.: Some results on cyclic codes over \({F}_{2}+v{F}_{2}\). IEEE Trans. Inform. Theory 56(4), 1680–1684 (2010)MathSciNetCrossRef
Metadata
Title
Cyclic codes over with respect to the homogeneous weight and their applications to DNA codes
Authors
Merve Bulut Yılgör
Fatmanur Gürsoy
Elif Segah Öztaş
Fatih Demirkale
Publication date
31-01-2020
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 5/2021
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00416-0

Other articles of this Issue 5/2021

Applicable Algebra in Engineering, Communication and Computing 5/2021 Go to the issue

Premium Partner