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Left ideals of matrix rings and error-correcting codes

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Abstract

It is well-known that each left ideal in a matrix rings over a finite field is generated by an idempotent matrix. In this work we compute the number of left ideals in these rings, the number of different idempotents generating each left ideal, and give explicitly a set of idempotent generators of all left ideals of a given rank. We then apply these results to give examples of left group codes that have best possible minimum weight.

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References

  1. Anderson, F.W., Fuller, K.R.: Rings and Cathegories of Modules. Springer, New York (1973)

    Google Scholar 

  2. Brochero Martínez, F.E., Giraldo Vergara, C.R.: Explicit idempotents of finite group algebras. Finite Fields Appl. 28, 123–131 (2014)

    Article  MathSciNet  Google Scholar 

  3. Cornelissen, M.G., Polcino Milies, C.: Finitely generated groups G such that $G/{{\cal{Z}}}G \cong Cp \times Cp$. Commun. Algebra 42(1), 378–388 (2014)

    Article  Google Scholar 

  4. Ferraz, R.A., Polcino Milies, C.: Idempotents in group algebras and minimal abelian codes. Finite Fields Appl. 13, 382–393 (2007)

    Article  MathSciNet  Google Scholar 

  5. Ferraz, R.A., Dutra, F.S., Polcino Milies, C.: Semisimple group codes and dihedral codes. Algebra Discrete Math. 3, 28–48 (2009)

    MathSciNet  MATH  Google Scholar 

  6. Fischer, S.D., Alexander, M.N.: Matrices over finite fields. Am. Math. Monthly 73, 639–641 (1966)

    Article  MathSciNet  Google Scholar 

  7. García Pillado, C., González, S., Martínez, C., Markov, V., Nechaev, A.: R group codes over non-abelian groups. J. Algebra Appl. 12, 7 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Leal, G., Polcino Milies, C.: Isomorphic group (and loop) algebras. J. Algebra 155, 195–210 (1993)

    Article  MathSciNet  Google Scholar 

  9. Petrogradsky, V.M.: One-sided ideal growth of free associative algebras. Monatsh. Math. 149, 243–249 (2006)

    Article  MathSciNet  Google Scholar 

  10. Polcino Milies, C., Sehgal, S.K.: An Introduction to Group Rings, p. 371. Kluwer Academic Publishers, Dordrecht, The Netherlands (2002)

    Book  Google Scholar 

  11. van Lint, J.H., Wilson, R.M.: A Course in Combinatorics. Cambridge Univ Press, Cambridge (2001)

    Book  Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de São Paulo, CEP-05315-970, Caixa Postal, São Paulo, 66281, Brazil

    R. A. Ferraz & C. Polcino Milies

  2. Universidade Federal do ABC, Av. dos Estados 5000, CEP: 09210-580, Santo André, SP, Brazil

    C. Polcino Milies

  3. Instituto de Matemática, Estatística e Física, Universidade Federal do Rio Grande, Campus Carreros. Av. Itália km 8, CEP 96201-900, Rio Grande, RS, Brazil

    E. Taufer

Authors
  1. R. A. Ferraz
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  2. C. Polcino Milies
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  3. E. Taufer
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Correspondence to C. Polcino Milies.

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Ferraz, R.A., Milies, C.P. & Taufer, E. Left ideals of matrix rings and error-correcting codes. AAECC 32, 311–320 (2021). https://doi.org/10.1007/s00200-021-00498-4

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  • DOI: https://doi.org/10.1007/s00200-021-00498-4

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