Characteristic-dependent linear rank inequalities via complementary vector spaces

A theorem about linear rank inequalities that depend on the characteristic of the finite field

“…Characteristic-dependent linear rank inequalities have been presented by Blasiak, et al [1]; Dougherty et al [4]. In [7,8], we show some inequalities using the ideas of Blasiak and applications to network coding that improve some existing results.…”

“…We remark that in [1] two characteristic-dependent linear rank inequalities are produced in 7 variables; the first inequality is valid over finite fields with characteristic two and the second inequality is valid over finite fields with characteristic different from two. In [7], two inequalities are produced in n variables, for each n of the form 2t + 3, t ≥ 2; the first inequality is valid over finite fields whose characteristic divides t and the second inequality is valid over finite fields whose characteristic does not divide t. Obtaining, for each finite or co-finite set of prime numbers, an inequality that is valid over finite fields whose characteristic is in this set. In [8], three inequalities in 21 variables are produced.…”

How to Find New Characteristic-Dependent Linear Rank Inequalities using Secret Sharing

“…Information inequalities are a sub-class of linear rank inequalities [15]. A characteristic-dependent linear rank inequality is like a linear rank inequality but this is always satisfied by vector spaces over fields of certain characteristic and does not in general hold over other characteristics [3,11,12]. Information inequalities have been useful to estimate lower bounds on the optimal information ratio of access structures, and linear rank inequalities have been useful to estimate lower bounds on the optimal information ratio of access structures in linear secret sharing schemes, i.e.…”

“…To the best of our knowledge, characteristic-dependent linear rank inequalities have not been used for determining bounds in linear secret sharing schemes in specific finite fields, but due to the nature of distinguishing finite fields according to their characteristics, these inequalities can be useful. One area where these inequalities have been useful for determing bounds is in newtork coding [1,3,5,11,13].…”

“…These properties serve as a guide to define conditions and inequalities that must satisfy vector spaces of a specific field characteristic. Then, using the vector deletion technique of Blasiak et al in [1], which was improved in [11,12], we get a theorem that produces characteristic-dependent linear rank inequalities. We emphasize that this theorem produces a pair of inequalities as long as there is a binary matrix whose determinant is greater than 1.…”