Characteristic-dependent linear rank inequalities via complementary vector spaces. Issue 2 (17th February 2021)

• Record Type: Journal Article
• Authors: Peña-Macias, Victor; Sarria, Humberto
• Is Part Of: Journal of information & optimization sciences. Volume 42:Issue 2(2021)
• Page Start: 345
• Page End: 369
• Publication Date: 2021-02-17
• Subjects: 68P30; Entropy -- Complementary vector spaces -- Finite fields -- Index coding -- Lexicographic product -- Linear rank inequalities -- Matroids -- Network coding; Electronic data processing -- Periodicals; Information science -- Periodicals; Mathematical optimization -- Periodicals; 519.6
• DOI: 10.1080/02522667.2019.1668157
• Languages: English
• ISSNs: 0252-2667

Abstract

Abstract: A characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this paper, we produce new characteristic- dependent linear rank inequalities by an alternative technique to the usual Dougherty's inverse function method [9]. We take up some ideas of Blasiak [4], applied to certain complementary vector spaces, in order to produce them. Also, we present some applications to network coding. In particular, for each finite or co-finite set of primes P, we show that there exists a sequence of networks in which each member is linearly solvable over a field if and only if the characteristic of the field is in P, and the linear capacity, over fields whose characteristic is not in P, → 0 as k → ∞.