A new construction of anticode-optimal Grassmannian codes

Year 2021, Volume: 8 Issue: 1, 31 - 39, 15.01.2021

Ben Paul Dela Cruz John Mark Lampos Herbert Palines Virgilio Sison

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.

Keywords

Ferrers diagram, Rank-metric code, Grassmannian, Constant dimension, Anticode bound

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.