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Year 2021, Volume: 8 Issue: 1, 31 - 39, 15.01.2021
https://doi.org/10.13069/jacodesmath.858732

Abstract

References

  • [1] T. Etzion, Subspace codes 􀀀 bounds and constructions, 1st European Training School on Network Coding, Barcelona, Spain, (2013).
  • [2] T. Etzion, N. Silberstein, Error-Correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory 55(7) (2009) 2909–2919.
  • [3] T. Etzion, A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory 57(2) (2011) 1165–1173,
  • [4] B. Hernandez, V. Sison, Grassmannian codes as lifts of matrix codes derived as images of linear block codes over finite fields, Global Journal of Pure and Applied Mathematics 12(2) (2016) 1801–1820.
  • [5] A. Khaleghi, F. R. Kschischang, Projective space codes for the injection metric, In: Proc. 11th Canadian Workshop on Information Theory, Ottawa, 54(8) (2009) 9–12.
  • [6] A. Khaleghi, D. Silva, F. R. Kschischang, Subspace codes, IMA Int. Conf. 49(4) (2009) 1–21.
  • [7] R. Koetter, F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory 54(8) (2008) 3579–3591.
  • [8] F. Manganiello, E. Gorla, J. Rosenthal, Spread codes and spread decoding in network coding, In: Proc. 2008 IEEE ISIT, Toronto, Canada, (2008) 851–855.
  • [9] A. J. Menezes, I. F. Blake, X. Gao, R. C. Mullen, S. A. Vanstine, T. Yaghoobian, Applications of finite fields, Boston, MA: Kluwer Academic Publishers 1993.
  • [10] D. Silva, F. R. Kschischang, R. Koetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory 54(9) (2008) 3951–3967.
  • [11] H. Wang, C. Xing, R. Safavi-Naini, Linear authentication codes: bounds and constructions, IEEE Trans. Inform. Theory 49(4) (2003) 866–872.

A new construction of anticode-optimal Grassmannian codes

Year 2021, Volume: 8 Issue: 1, 31 - 39, 15.01.2021
https://doi.org/10.13069/jacodesmath.858732

Abstract

In this paper, we consider the well-known unital embedding from $\FF_{q^k}$ into $M_k(\FF_q)$ seen as a map of vector spaces over $\FF_q$ and apply this map in a linear block code of rate $\rho/\ell$ over $\FF_{q^k}$. This natural extension gives rise to a rank-metric code with $k$ rows, $k\ell$ columns, dimension $\rho$ and minimum distance $k$ that satisfies the Singleton bound. Given a specific skeleton code, this rank-metric code can be seen as a Ferrers diagram rank-metric code by appending zeros on the left side so that it has length $n-k$. The generalized lift of this Ferrers diagram rank-metric code is a Grassmannian code. By taking the union of a family of the generalized lift of Ferrers diagram rank-metric codes, a Grassmannian code with length $n$, cardinality $\frac{q^n-1}{q^k-1}$, minimum injection distance $k$ and dimension $k$ that satisfies the anticode upper bound can be constructed.

References

  • [1] T. Etzion, Subspace codes 􀀀 bounds and constructions, 1st European Training School on Network Coding, Barcelona, Spain, (2013).
  • [2] T. Etzion, N. Silberstein, Error-Correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory 55(7) (2009) 2909–2919.
  • [3] T. Etzion, A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory 57(2) (2011) 1165–1173,
  • [4] B. Hernandez, V. Sison, Grassmannian codes as lifts of matrix codes derived as images of linear block codes over finite fields, Global Journal of Pure and Applied Mathematics 12(2) (2016) 1801–1820.
  • [5] A. Khaleghi, F. R. Kschischang, Projective space codes for the injection metric, In: Proc. 11th Canadian Workshop on Information Theory, Ottawa, 54(8) (2009) 9–12.
  • [6] A. Khaleghi, D. Silva, F. R. Kschischang, Subspace codes, IMA Int. Conf. 49(4) (2009) 1–21.
  • [7] R. Koetter, F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory 54(8) (2008) 3579–3591.
  • [8] F. Manganiello, E. Gorla, J. Rosenthal, Spread codes and spread decoding in network coding, In: Proc. 2008 IEEE ISIT, Toronto, Canada, (2008) 851–855.
  • [9] A. J. Menezes, I. F. Blake, X. Gao, R. C. Mullen, S. A. Vanstine, T. Yaghoobian, Applications of finite fields, Boston, MA: Kluwer Academic Publishers 1993.
  • [10] D. Silva, F. R. Kschischang, R. Koetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory 54(9) (2008) 3951–3967.
  • [11] H. Wang, C. Xing, R. Safavi-Naini, Linear authentication codes: bounds and constructions, IEEE Trans. Inform. Theory 49(4) (2003) 866–872.
There are 11 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Ben Paul Dela Cruz This is me 0000-0002-0351-1231

John Mark Lampos This is me 0000-0002-8354-5772

Herbert Palines This is me 0000-0002-7407-2539

Virgilio Sison This is me 0000-0003-2955-2311

Publication Date January 15, 2021
Published in Issue Year 2021 Volume: 8 Issue: 1

Cite

APA Dela Cruz, B. P., Lampos, J. M., Palines, H., Sison, V. (2021). A new construction of anticode-optimal Grassmannian codes. Journal of Algebra Combinatorics Discrete Structures and Applications, 8(1), 31-39. https://doi.org/10.13069/jacodesmath.858732
AMA Dela Cruz BP, Lampos JM, Palines H, Sison V. A new construction of anticode-optimal Grassmannian codes. Journal of Algebra Combinatorics Discrete Structures and Applications. January 2021;8(1):31-39. doi:10.13069/jacodesmath.858732
Chicago Dela Cruz, Ben Paul, John Mark Lampos, Herbert Palines, and Virgilio Sison. “A New Construction of Anticode-Optimal Grassmannian Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications 8, no. 1 (January 2021): 31-39. https://doi.org/10.13069/jacodesmath.858732.
EndNote Dela Cruz BP, Lampos JM, Palines H, Sison V (January 1, 2021) A new construction of anticode-optimal Grassmannian codes. Journal of Algebra Combinatorics Discrete Structures and Applications 8 1 31–39.
IEEE B. P. Dela Cruz, J. M. Lampos, H. Palines, and V. Sison, “A new construction of anticode-optimal Grassmannian codes”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 8, no. 1, pp. 31–39, 2021, doi: 10.13069/jacodesmath.858732.
ISNAD Dela Cruz, Ben Paul et al. “A New Construction of Anticode-Optimal Grassmannian Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications 8/1 (January 2021), 31-39. https://doi.org/10.13069/jacodesmath.858732.
JAMA Dela Cruz BP, Lampos JM, Palines H, Sison V. A new construction of anticode-optimal Grassmannian codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2021;8:31–39.
MLA Dela Cruz, Ben Paul et al. “A New Construction of Anticode-Optimal Grassmannian Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 8, no. 1, 2021, pp. 31-39, doi:10.13069/jacodesmath.858732.
Vancouver Dela Cruz BP, Lampos JM, Palines H, Sison V. A new construction of anticode-optimal Grassmannian codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2021;8(1):31-9.