A new construction of anticode-optimal Grassmannian codes
DOI:
https://doi.org/10.13069/jacodesmath.858732Keywords:
Ferrers diagram, Rank-metric code, Grassmannian, Constant dimension, Anticode boundAbstract
In this paper, we consider the well-known unital embedding from $\mathbb{F}_{q^k}$ into $M_k(\mathbb{F}_q)$ seen as a map of vector spaces over $\mathbb{F}_q$ and apply this map in a linear block code of rate $\rho/\ell$ over $\mathbb{F}_{q^k}$. 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.
Received: 16 February 2020 | Accepted: 13 September 2020Downloads
References
T. Etzion, Subspace codes — bounds and constructions, 1st European Training School on Network Coding, Barcelona, Spain (2013).
T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory 55(7) (2009) 2909–2919.
T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory 57(2) (2011) 1165–1173.
B. Hernandez and V. Sison, Grassmannian codes as lifts of matrix codes derived as images of linear block codes over finite fields, Glob. J. Pure Appl. Math. 12(2) (2016) 1801–1820.
A. Khaleghi and F. R. Kschischang, Projective space codes for the injection metric, in: Proc. 11th Canadian Workshop on Information Theory, Ottawa, ON, Canada (2009) 9–12.
A. Khaleghi, D. Silva, and F. R. Kschischang, Subspace codes, in: Cryptography and Coding, Lecture Notes in Comput. Sci., vol. 5921, Springer, Berlin, Heidelberg (2009) 1–21.
R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory 54(8) (2008) 3579–3591.
F. Manganiello, E. Gorla, and J. Rosenthal, Spread codes and spread decoding in network coding, in: Proc. 2008 IEEE International Symposium on Information Theory, Toronto, ON, Canada (2008) 851–855.
A. J. Menezes, I. F. Blake, X. Gao, R. C. Mullin, S. A. Vanstone, and T. Yaghoobian, Applications of Finite Fields, Kluwer Academic Publishers, Boston (1993).
D. Silva, F. R. Kschischang, and R. Koetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory 54(9) (2008) 3951–3967.
H. Wang, C. Xing, and R. Safavi-Naini, Linear authentication codes: bounds and constructions, IEEE Trans. Inform. Theory 49(4) (2003) 866–872.
