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In Linear Algebra over finite fields, a characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of spans of vector subspaces of a finite dimensional vector space over a finite field of determined characteristic, and does not in general hold over fields with other characteristic. This paper shows a preliminary result in the production of these inequalities. We produce three new inequalities in 21 variables using as guide a particular binary matrix, with entries in a finite field, whose rank is 8, with characteristic 2; 9 with characteristic 3; or 10 with characteristic neither 2 nor 3. The first inequality is true over fields whose characteristic is 2; the second inequality is true over fields whose characteristic is 2 or 3; the third inequality is true over fields whose characteristic is neither 2 nor 3.

En Álgebra Lineal sobre cuerpos finitos, una desigualdad rango lineal dependiente de la característica es una desigualdad lineal que es válida para dimensiones de sumas de subspacios vectoriales de un espacio vectorial de dimensión finita sobre un cuerpo finito de determinada característica, y no es válida en general sobre cualquier cuerpo de otra característica. Este documento presenta un resultado preliminar referente a la producción de estas desigualdades. Nosotros producimos tres desigualdades nuevas en 21 variables usando como guía una matriz binaria particular, con entradas en un cuerpo finito, cuyo rango es 8, 9 o 10 dependiendo de que la característica sea 2, 3 o distinta de 2 y 3. La primera desigualdad es válida sobre cuerpos de característica 2; la segunda es válida sobre cuerpos de característica 2 o 3; la tercera es válida sobre cuerpos de característica distinta de 2 y 3.

In Linear Algebra over finite fields, a linear rank inequality is a linear inequality that is always satisfied by ranks (dimensions) of subspaces of a vector space over any field. Information inequalities are a sub-class of linear rank inequalities (A.

Characteristic-dependent linear rank inequalities have been presented in (

In this paper, we answer affirmatively. Following a particular case, we show a method to produce characteristic dependent linear rank inequalities using as guide a suitable binary matrix; we use the dependency relationships of its columns which are naturally associated with matroid representations. The rank of the desired matrix is 8 if the entries are in a field whose characteristic is 2; the rank is 9 if the characteristic is 3; and the rank is 10 if the characteristic is neither 2 nor 3. We "convert" this property in three inequalities: the first inequality is true over fields whose characteristic is 2; the second inequality is true over fields whose characteristic is 2 or 3; the third inequality is true over fields whose characteristic is neither 2 nor 3. The inequalities do not in general hold over fields with other characteristic. We hope that the techniques presented in this paper can be applied to other types of matrices whose rank behaves in a similar way to the described matrix.

The paper is organized as follows. We introduce some mathematical tools of information theory. After, we show the theorem that produces the described inequalities; before presenting the proof, we give some propositions and lemmas that will be helpful. Finally, we show the proof and some conclusions.

In the following, we introduce the necessary concepts to understand this paper. Let A, _{
1
}
_{
n
} be vector sub-spaces of a finite dimensional vector space _{
i
}
_{
i
}
_{
n
} define the random variables _{
1
} = _{
a1
} , _{
n
}
_{
An
} . For

The random variables _{
1
}
_{
n
} are called _{1}: _{i}, _{A}

We give the following definition in order to fix ideas about inequalities.

Definition 1. Let _{
k
} be subsets of [n]. Let _{1} Є ℝ, for 1

- is called a _{1}, _{
n
} finite fields with characteristic in

- is called a

- is called an

By definition of linear random variables, we note any information inequality is an inequality which is also satisfied by dimensions of spans of vector spaces. The following inequality is the first linear rank inequality which is not information inequality.

Example 2. (_{1}, A_{2}, A_{3}, A_{4} sub-spaces of a finite dimensional vector space,

We can think a characteristic-dependent linear rank inequality like a linear rank inequality that is true over some fields.

We say that _{1} , _{
n
} are mutually complementary subspaces in _{1}, A_{n}. In this case, _{
I
} denotes the

The inequalities of the following lemmas, that we will use later, are valid for linear random variables that hold some additional conditions.

We remark that we use the following notation of intervals: [j, k] := _{
j
} +… + _{
k
} is denoted by _{[tk]}, and A_{0}: = A_{Ø} := O.

Lemma 3. _{
1
}
_{
n
}
_{
1
}
_{
n
}
_{
1
}

_{[i]} = _{
i+1
} ∩ _{[i]}

The equality holds if and only if I (_{
1
} ; A_{2} )= I (_{1}; A’_{2}). In other words, _{
1
} ∩ _{
2
}
_{1} ∩ A’_{2} because A’_{1} ≤ A_{i}. Now, we suppose the case

The equality holds if and only if I _{
i+1
}
_{[i]}

_{
i
}
_{
i
} for all i, we have _{
i+1
} ∩ _{[i]}
_{
i+1
} ∩ _{[i]}

Lemma 4.

The equality holds if and only if

Since

Let

We calculate the rank of the matrix

For a column _{
i
} of B, the set {j ^{
j
}
_{
¡
}
_{i}; if there is no confusion, by abuse of notation, we identify _{
i
}

where

The theorem of this paper shows three characteristic-dependent linear rank inequalities. The proof is guided by the matrix

Theorem 5. _{
1
}
_{
2
}
_{
3
}
_{
4
}
_{
5
}
_{
6
}
_{
7
}
_{
8
}
_{
9
}
_{
10
}
_{
1
}
_{
2
}
_{
3
}
_{
4
}
_{
5
}
_{
6
}
_{
7
}
_{
8
}
_{
9
}
_{
10
}

We remark that these inequalities do not in general hold over other fields whose characteristic is different to the described characteristic. A counterexample would be in ^{
10
}
_{
i
}
_{
i
}
_{
i
}
_{
i
}

The proof is given at the end of the section. Before, we introduce some propositions and lemmas that will help its development.

Let _{
1
} ⊕ _{
n
}
_{
1
}
_{
i-1
} + _{
i+1
}
_{
n
} is a direct sum for all i. We say that _{
1
}
_{
n
}

Proposition 6. _{
i
}
_{
1
}

Hence, _{i}. By property of the tuple of complementary vector subspaces, _{
I
}

As a consequence a non-zero element of _{
1
} , _{
n
}

Consider any _{i}. Let _{
bi
} be the _{
i
} -projection of

Where

Take

where _{
j
} ) _{j} is the canonical basis in ^{
j
}
_{
It
}
_{
k
} for some _{
k
} = 1, and 0 in otherwise. If

We have the following proposition.

Proposition 7. _{
1
} ⊕ _{
n
}
_{
i
}
^{
'
}

Now, let

The other implication is obtained from

Example 8. _{
1
}
_{
2
}
_{
3
} and _{
I3
}
_{
I2
}
_{
I3
} .

As a consequence of Proposition 6, if each _{
i
} and _{
b1
}
_{
Ii
}

Corollary 9. _{
1
}
_{
n
}
^{
'
}
_{
i
}
_{
i
} ∈_{
I
}

If we take

Corollary 10. _{
1
}
_{10},C)

In the rest of the paper, we only work with the matrix

Proposition 11. _{
x
}
_{
2
}
_{
5
}
_{
e
}
_{
7
}
_{
8
}
_{
9
}
_{10}, B_{1}, B_{2}, B_{3}
_{
4
}
_{
5
}
_{
6
}
_{
r
}
_{
8
}
_{
9
}
_{
10
}
_{
1
}
_{10},C)

Therefore, using conditions (i) and (ii), we have _{
bi
}
_{
bt
}
_{
i
}

Define _{1}, and for

Let ^{(0)} = ^{(i)},a subspace of ^{(i-1)} which is a complementary subspace to

in

Let Ċ denote the subspace ^{(10)}. The tuple

Lemma 12. ^{
'
}

One can use all (hese inequalities along with the definitions of _{
B
}
_{
B
} (A[_{10}]), ∇_{
B
} (A[_{10}]) (these definitions were previously given) to obtain the described inequalities.

Lemma 13.

Then, we apply the inequalities from Lemma 12.

Lemma 14.

Hence,

Then, we apply the inequalities from Lemma 12 and the inequality presented at the beginning of the proof.

To prove the inequality 1, from Lemma 13, the vector sub-spaces

One can note H _{
10
}

Using all these inequalities we obtain the desired inequality.

The inequality 2 is obtained in a similar way; and from the inequality 1, it is easy to note that the inequality 2 also holds over fields whose characteristic is 2.

To prove the inequality 3, from Lemma 14, the vector subspaces

One can note

Using all these inequalities we obtain the desired inequality.

In this paper, we produce three characteristic-dependent linear rank inequalities in 21 variables using as guide a suitable binary matrix whose rank is different over at least three different fields (specifically, the rank depends on the characteristic of the field). The first inequality is true over fields whose characteristic is 2; the second inequality is true over fields whose characteristic is 2 or 3; and the third inequality is true over field whose characteristic is neither 2 nor 3. We hope that the technique presented here can be used to produce other inequalities, choosing other suitable matrices. In future work, the independence or dependence of these inequalities and their possible applications to Network Coding must be studied.

The first author thanks the support provided by COLCIEN-CIAS and the second author thanks the support provided by Universidad Nacional de Colombia.

Jorge Cossio

The first author created the central idea of the manuscript and both authors contributed to its development and writing.

The authors declare that they have no conflict of interest.