In this entry I discuss the definition of (digital) higher order nets and sequences and some possible simplifications of the notation.

Digital higher order nets and sequences have been introduced in

- [1] J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal., 46, 1519–1553, 2008. doi: 10.1137/060666639 For a Matlab program generating higher order digital nets, numerical examples and plots of higher order digital nets see here.

whereas higher order nets have been introduced in

- [2] J. Dick and J. Baldeaux, Equidistribution properties of generalized nets and sequences. In: Proceedings of the MCQMC’08 conference, Montreal, Canada, P. L’Ecuyer and A. Owen (eds.), pp. 305–323, 2009. doi: 10.1007/978-3-642-04107-5_19 An earlier version can be found here.

There are several parameters occurring in the definition of higher order nets, namely and for higher order sequences we have the parameters

(Digital) higher order nets and sequences are point sets and sequences such that

where is the number of quadrature points and is the smoothness of the integrand

**Definitions**

For convenience we repeat the definitions of those (digital) higher order nets and sequences here. A discussion of the parameters occurring in these definitions follows in the subsequent section.

Defintion[Digital -net]

Let let be a real number and let be an integer. Let be the finite field of order , where is a prime power. Let where is the th row vector of for and If for all withthe vectors

are linearly independent over then the digital net with generating matrices is called a digital -net over

For and one obtains a digital -net over

Definition[Digital -sequence]

Let and and let be a real number. Let be the finite field of order , where is a prime power. Let where is the th row vector of the matrix for and Further let denote the left upper submatrix of for If for all the matrices generate a digital -net over then the digital sequence with generating matrices is a digital -sequence over

For one obtains a digital -sequence over

Before we can define the analogue geometrical concepts, we need the notion of a higher order elementary interval (which is actually not an interval anymore, but a union of intervals). We need some notation. Let let let let and let where the components and do not appear in and if By a higher order elementary interval we mean a subset of of the form

where is an integer and where for we have in case and if

An example of a higher order (or generalized) elementary interval can be found in [2].

Definition[-net]

Let be natural numbers, let be a real number, and let be an integer. Let be an integer and let The point set is called a -net in base if for all integers where withwhere for we set the empty sum the higher order elementary interval contains exactly point of for each

For and one obtains a -net in base

Definition[-sequence]

Let and be integers and be a real number. Let be a sequence of numbers in Then is a -sequence in base if for all and all we have that is a -net in base

For one obtains a -sequence in base

**Meaning of the parameters**

Some parameters have the same meaning as for classical -nets and -sequences, and are therefore not discussed further. These are

- the base or the finite field over which the net or sequence is constructed,
- the dimension,
- is equal to where is the number of points and is the logarithm in base

Now to the remaining parameters of (digital) higher order nets and (digital) higher order sequences. There are a few guiding principles which we follow in defining (digital) higher order nets and sequences:

- each parameter in the definition of higher order nets should have an analogue in the definition for digital higher order nets and vice versa,
- each parameter in the definition of higher order sequences should have an analogue in the definition for digital higher order sequences and vice versa,
- (digital) higher order sequences should be a direct extension of (digital) higher order nets,
- (digital) higher order nets should be a finite version of (digital) higher order sequences, and
- (digital) -nets should be a special case of (digital) higher order nets and analogously for sequences.

A consequence of these guiding principles is that all definitions are related to each other and hence one should not consider a definition just on its own. In other words, each definition should only be considered in conjunction with all other definitions.

One could of course ignore one or more of the above principles and arrive at a different definition of some object (which might appear more natural). However, it makes the objects more difficult to relate to each other. For instance, a digital -net over is a -net in base and analogously for (digital) higher order sequences. This is a natural result and holds since in the definitions the parameters of each object have direct analogues in the related object.

In the error bound above we said that denotes the smoothness of the integrand We make one assumption which makes the definition a bit more complicated, namely, we assume that is not a known fixed number, that is, we simply do not know the smoothness of the integrand. Since we assume that we do not know we cannot talk about an order net in this context. So we think of the remaining parameters as a function of and, for a given point set or sequence, we would like to know the value of the remaining parameters for each value of In this sense, it would also be possible to write for instance digital -net instead of digital -net (and analogously for the other three cases).

Using the guiding principles we can explain one more parameter immediately. Namely, is the number of rows of the generating matrices By principle (2) it follows that we also need to have the parameter in the definition of higher order nets (though it does not have such an immediate interpretation as for digital higher order nets).

The parameter is introduced because of principle (4). Namely, when one considers the first points of a digital -sequence, then the left upper submatrices of the generating matrices of the digital higher order sequence generate a digital -net. Hence, by principle (4) Notice that we cannot choose in this case, since is not a fixed number (i.e. the smoothness is not known).

Next we discuss the parameter If one considers only (digital) higher order nets, then one could argue that the parameter is redundant. But to understand one needs to consider (digital) higher order sequences rather than (digital) higher order nets. The parameter then appears in the definition of (digital) higher order nets because of principle (3), we want (digital) higher order sequences to be a direct extension of (digital) higher order nets. If one ignores principle (3) then one could define (digital) higher order nets without using

For sequences, it has been shown that in the definition of (digital) higher order sequences cannot grow faster than as The value of must then satisfy that this sum diverges as fast or faster than as

**Simplification 1: Assume that is known**

This is not really a simplification of the notation, since it does not have the same level of generality anymore as the above concepts. Nonetheless, this case is valuable since it is much simpler to understand.

So assume now that is a given number. For digital higher order nets we can set the parameter to always if is known. The same can be done for higher order nets and hence the parameter is not needed in this case. Similarly we can just assume that and hence this parameter is also not needed.

Now consider the parameter In the general case the situation is the following: First choose a sequence , then choose a value of and and then find out what are the parameters such that is a -sequence. But now the situation is different. We can now first fix and then choose a sequence which is good for the given value of . The important point is that, using for instance the construction from [1], we know that there exists a sequence for which we can choose such that Sequences for which we cannot choose are not really of interest since they do not yield the optimal rate of convergence (notice that is not possible, see [Theorem 4, 2]. Hence we can restrict ourselves to sequences for which In fact, we can simply choose and With this understanding we do not need to include the parameters (in the general case this is not possible, since currently there is no sequence for which for all ). Since we can always choose for sequences, we can do the same for nets. Thus we can set and for known

Thus, in this case we could simply speak of -nets and -sequences or (digital) order -nets and (digital) order -sequences, as suggested by Art.

With our current knowledge, this information is usually sufficient, since known propagation rules provide enough information about the remaining parameters. That is, for arbitrary , we can obtain useful values for for (digital) order nets or (digital) order sequences for the case when And this information is competitive with respect to the parameters obtained from the current constructions.

However, this somewhat hides the question whether there are constructions which are better than what can be obtained from the current propagation rules for the cases when In other words, it would be interesting to know whether a sequence exists for which for all Currently this is only possible for in a chosen range.

In the notion of extensibility we can describe this in the following way:

- fixed case: is known and we can define (digital) order nets and (digital) order sequences;
- finitely extensible case: unknown and we know higher order nets and sequences which are optimal for all where is an arbitrarily chosen finite number;
- infinitely extensible case: unknown and we know higher order nets and sequences which are optimal for all

Because of the known propagation rules, for some papers which only deal with the first or second case, the notation of (digital) order nets and (digital) order sequences can be sufficient. A solution to the third case is currently not known.

**Simplification 2: Removing and **

We now go back to the general case where is not known and consider all the parameters in this setting.

We now provide some definitions which do not include the parameters and Though has a natural meaning for digital higher order nets (the number of rows of the generating matrices), one could do without specifying it. For sequences on the other hand, one could simply set in the definition and not mention it as a parameter. In this case the definition could be modified to the following (we only consider the digital case, the other case can be modified analogously).

Defintion[Digital -net]

Let let be a real number and let be an integer. Let be the finite field of order , where is a prime power. Let (for some integer ) where is the th row vector of for and If for all withthe vectors

are linearly independent over then the digital net with generating matrices is called a digital -net over

Definition[Digital -sequence]

Let and and let be a real number. Let be the finite field of order , where is a prime power. Let where is the th row vector of the matrix for and Further let denote the left upper submatrix of for If for all the matrices generate a digital -net over then the digital sequence with generating matrices is a digital -sequence over

This is also a possible definition. It does however hide the dependence of the convergence rate on the number For instance consider the following result, first stated with the usual definition and then with the simplified definition:

- the convergence rate of the integration error using a digital -net cannot, in general, be asymptotically smaller than
- the convergence rate of the integration error using a digital -net with generating matrices cannot, in general, be asymptotically smaller than

I believe the first version is somewhat simpler, since is included as a parameter. This is to illustrate the motivation for including as a parameter. Similarly, some propagation rules also benefit from including the parameter To summarize, yes one can do without specifying but in some instances it is easier if it is included in the notation.

The motivation for including is principle 3, but could reasonably be left out. The reason for writing -nets for higher order nets is guiding principle 1. The case for higher order sequences also follows from the guiding principles.

**Simplification 3: Removing **

Now we turn to This parameter is important for higher order sequences and is included in higher order nets because of guiding principle 3. So let us discuss the meaning of for higher order sequences. The important higher order sequences are those for which see [page 15, 2] for more information. Now depends on and is a parameter which can be chosen independent of Currently there is no higher order sequence known for which for all For the constructions currently known we only have for all for some finite and arbitrarily chosen Currently there is no sequence known for which for all Currently the best known sequences have parameters of the form

for some arbitrary but fixed If there would be explicit constructions of higher order sequences for which for all then the parameter would be redundant. As long as this is not the case, the parameter is important when is not known.

For (digital) higher order nets, the parameter is included because of guiding principle 4.

**Conclusion**

To conclude, yes, some simplifications of the notation for (digital) higher order nets and sequences are possible and make sense in certain situations. Especially the assumption that is known simplifies the notation considerably. If is not known then one could do without and to some extend Still I believe that it is useful to adhere to the guiding principles to keep the notation consistent with each other. Comments are welcome.