The thesis entitled “On the role of the message dimension and the finite field characteristic in linear network coding” is a presentation of my original research work. Thus, the number of columns of the matrices that multiply the vectors generated by the sources is equal to the message dimension.
Background
Moreover, they show that ad-dimensional vector linear solution over a finite field implies a scalar linear solution over some ring (can be non-commutative). 6] showed that all multicast networks have a scalar linear solution over a finite field whose size is greater than or equal to the number of terminals.
Capacity, Information Inequalities, and Rank Inequalities
Inequalities of linear rank place bounds on the dimensions of these vector subspaces (discrete polymatoid). A list of twenty-four new linear rank inequalities in five variables that are not information inequalities is shown in [29].
Matroids and Linear Network Coding
Contributions and Organization of this Thesis
In Chapter 3 we generalize this result to show that for every positive integer m≥2 there exists a network that has a w-dimensional vector linear solution if and only if w is a multiple of m. We then show that for for every positive integer m ≥ 2 there exists a network that has a w-dimensional vector linear solution if and only if w is greater than or equal to m.
Vector Linear Network Coding
If a network has a linear scalar solution over Fpa, then it also has a one-dimensional linear vector solution over Fp [17]. But if a network has a one-dimensional linear vector solution over Fp, then it may not necessarily have a linear scalar solution over Fpa [14].
Fractional Linear Network Coding
Matroids, Polymatroids, and Networks
Matroids
Let N be a network, and let S be the set of sources inN, E the set of edges inN, and V the set of vertices inN. The network N is a matroidal network with respect to M if there exists a function f: {S, E} → G such that the following conditions are met: in [8] it has been proven that a network has only a scalar linear solution over Fq if it is a matroidal network with respect to a matroid representative of Fq. a scalar linear solution was translated into a matroid representation). in [38] showed the opposite: if a network is matroidal with respect to a matroid that is representative. Fq, then it has a scalar linear solution over Fq. a representation of a matroid was translated into a scalar linear solution).
Discrete Polymatroids
4102] A network has a (k, n) fractional linear network coding solution over Fq if and only if it is a (k, n)-discrete polymatroidal network with respect to a discrete polymatroid D that can be represented over Fq. A network has a k-dimensional vector linear solution over Fq if and only if it is a (k, k)-discrete polymatroidal network with respect to a discrete polymatroid D that can be represented over Fq.
Some Conventions and Lemmas on Co-dimension
In [32] and [33] H(A) has been used to denote denotedim(A), but we have refrained from this notation for our original content in this thesis. iii) For two vector subspaces A and B in a finite dimensional vector space, < A, B > denotes the vector space spanned by the vectors in A∪B; dim(A, B) specifies the dimension of < A, B >;. This may be an abuse of notation asf−1() is not a well-defined function unless f() is one-to-one.) If A is a subspace of V, then the co-dimension of A in V is codimV(A ) = dim(V)−dim(A).
Dimensions of Vector Spaces Obey Information Inequalities
To show that for any positive integer m ≥ 2 there exists a network that has an m dimensional vector linear solution but no vector. The Dim-k network has a w-dimensional vector linear solution if and only if is a positive integer multiple of k.
![Figure 3.1: The M-network reproduced from [3].](https://thumb-ap.123doks.com/thumbv2/azpdfnet/10551060.0/56.892.177.728.179.1131/figure-3-1-the-m-network-reproduced-from.webp)
Role of Message Dimension in Fractional Linear Network Coding
The n-factored network of Nm has a (w, wn) fractional linear network coding solution if and only if w is an integer multiple of d. Theorem 14 shows that Nn has a (w, wn) fractional linear network coding solution if and only if Nm has a (wn, wn) fractional linear network coding solution. N has a (a, kl) fractionally linear solution if and only if N||=k has a (a, l) fractionally linear solution.
It can be seen from Theorem 15 that Nn (w, wn) has a partial linear solution if and only if w is an integer multiple of dk.
MDim-m network
For every positive integer m ≥ 3, the MDim-m network has a linear solution of a w-dimensional vector if and only if w is greater than or equal to m−1. For any positive integerm≥3, we now show that MDim-mnetwork has an aw-dimensional routing vector solution for any w ≥m−1 ('if part'). Now, if terminals requiring exactly the same set of resources are treated as one (since they receive information from the same set of edges, this does not affect solvability), then the MDim-m network is a subnet of the Dim-m network [44], so m has -dimensional vector routing solution.
Furthermore, an (m−1)-dimensional vector routing solution and anm-dimensional vector routing solution guarantee a (2m−1)-dimensional and (2m−2)-dimensional vector routing solution.
Network Coding Solution but No Routing Solution
The Char-q-y network
In [12], Joseph Connelly and Kenneth Zeger presented a network called the Char-q network which has a linear vector solution for each message dimension if and only if the finite field characteristics share q. We show that if the middle edges of the network do not transmit any information generated by the source y, then Char-q-y has a linear scalar linear solution over any finite field. But if the middle edges transmit any symbol generated by y, then the Char-q-y network has a linear vector solution if and only if the finite field characteristic divides q.
Due to the demands of the terminalt1, from equations (4.1) and (4.6), we have the following equations.
Network G 1
This network is a combination of the M network and a Char-2-¯y network (with sources ¯a and ¯y common to both networks). For any odd number, the network G1 has a d-dimensional linear vector solution if and only if the finite field characteristic divides q. We show that if the characteristic of the finite field does not divide, then G1 does not have an odd-dimensional linear vector solution.
We show that G1 has a linear scalar solution (thus having a linear vector solution for each message dimension) if the finite field characteristic divides q.
Network G 2
The networkG2 has a two-dimensional vector linear solution if and only if the characteristic of the finite field divides q0. Assume that G2 has a two-dimensional vector linear solution even if the finite field characteristic does not divideq0. The networkG2 has a five-dimensional vector linear solution if and only if the characteristic of the finite field divides q0.
Assume that G2 has a 5-dimensional vector linear solution when the characteristic of the finite field does not divide q0.
Network G 3
G3 has a scalar linear solution if and only if it divides the characteristic of the finite field q1. We show that there is a scalar linear solution if the characteristic of the finite field divides q1. Then, similar to case I.2.1.2, it can be proven that N3 does not have a two-dimensional vector linear solution in this case.
Then, similar to Case I.2.1.2, it can be proved that N3 does not have a 2-dimensional vector linear solution under this case.
Main Results
But G1 has a 3-dimensional vector linear solution if and only if the characteristic of the finite field belongs to P1∪P2∪P3. This implies that G2 has a 2-dimensional vector linear solution if and only if the characteristic of the finite field belongs to {P1, P2}. So the network G12 has a 2-dimensional vector linear solution if and only if the characteristic of the finite field belongs to {P1, P2}.
Then network G12 has a 3-dimensional vector linear solution if and only if the characteristic of the finite field belongs to P2.
Discussion
First Set of Inequalities
- Application of Inequality in Equation (5.1)
Then the following linear rank inequality holds if V is a vector space over a finite field whose characteristic does not belong to {p1, p2, . pl}, but may not apply otherwise:. Before giving the proof, we show that this inequality may not hold asq = 0 over the finite field (noteq= 0 when the characteristic belongs to the given set of primes). Let ui be the 1 dimensional vector space spanned by the q+ 1-length vector whose ith element is 1 and all other elements are zero.
Then applying the feature-dependent linear rank inequality shown in equation (5.1), we have the following equation, (Fig. 5.1 shows the variables corresponding to the sources and the edges).
Second Set of Inequalities
- Application of Inequality in Equation (5.2)
We show here that this inequality may not hold if q has an inverse over the finite field (which is equivalent to saying that the characteristic of the finite field does not belong to {p1, p2, . . , pl}). Letui is the 1-dimensional vector space spanned by the q+ vector of 1 length whose element is 1 and all other elements are zero. Then, applying the characteristic-dependent linear rank inequality from equation (5.2), we get the following equation (Fig. 5.2 shows the variables corresponding to the sources and the edges).
Third Set of Inequalities
- Application of Inequality in Equation (5.2)
We derive here another feature-dependent linear rank inequality by defining the set S in equation (5.149) in a different way. How many characteristic-dependent linear rank inequalities are there if the number of variables is fixed? When the number of variables is six or less, there is no characteristic-dependent linear rank inequality.
It is not known whether the number of feature-dependent linear rank inequalities with a given finite number of variables is also finite. There also exists a network that has a vector linear solution if and only if the message dimension is greater than or equal to m and the characteristic of the finite field belongs to P. Zeger, “Feature-dependent linear rank inequalities with applications to network coding,” IEEE Transactions on Information Theory, vol.
A Note on the Proofs of the Inequalities (5.1), (), and (5.3)
Proof of Inequality Shown in 5.1
=ifBiBj+fBiY +fBiZ =I over a subspaceBi0 of Bi where. iii) fCA+fCY =I over a subspace CO of C where. j=1,j6. For 1≤i≤q−1, consider the following composite functions. Note that these equations hold for each different values of i in the given range). 5.76). Since from equation (5.56) fW A is invertible over fZWfAZ( ¯A); fW A is also invertible over fZWfBiZ(RBi).
Then from equation (5.40) it can be deduced that fZCfBiZ is one-to-one over SBi.
Proof of the Inequality Shown in Equation (5.2)
Proof of the Inequality Shown in Equation (5.3)
Discussion
It was already shown in [3] that a network can have a rate 1 linear solution, but it cannot have a (1,1) fractional linear solution. It is known that a network can have a d-dimensional vector-linear solution over Fq but have no scalar linear solution over any finite field whose magnitude is less than or equal to qd. We showed that for any prime p there exists a non-multicast network which has a scalar linear solution if and only if the size of the finite field is a power of p but has a 2-dimensional vector linear solution over all limited fields.
Another question that remains open is whether linear rank equalities can also capture the fact that a network can have a vector linear solution but no scalar linear solution.
