1 Introduction
1.1 Covering codes. The length function. Saturating sets in projective spaces
Let be the Galois field with elements. Let be the
dimensional vector space over
Denote by a ary linear code of length and codimension (redundancy) , that is, a subspace of of dimension The sphere of radius with center in is the set where is the Hamming distance between the vectors and .Definition 1.1.
 (i)

The covering radius of a linear code is the least integer such that the space is covered by spheres of radius centered at the codewords.
 (ii)

A linear code has covering radius if every column of is equal to a linear combination of at most columns of a parity check matrix of the code, and is the smallest value with such property.
Definitions 1.1(i) and 1.1(ii) are equivalent. Let an code be an code of covering radius . Let an code be an code of minimum distance . For an introduction to coverings of vector Hamming spaces over finite fields, see [7, 8].
The covering density of an code is defined as the ratio of the total volume of all spheres of radius centered at the codewords to the volume of the space . By Definition 1.1(i), we have . In the other words,
(1.1) 
The covering quality of a code is better if its covering density is smaller. For fixed , and , the covering density of an code decreases with decreasing .
Codes investigated from the point view of the covering quality are usually called covering codes [8]; see an online bibliography [23], works [7, 9, 10, 11, 13, 14, 15, 16, 21, 22], and the references therein.
This work is devoted to nonbinary covering codes with radius . Note that for relatively small many results are given in [11, 13, 14] and the references therein.
Definition 1.2.
From (1.1), see also Definition 1.1(ii), one can get an approximate lower bound on . In particular, if is considerable larger than (this is the natural situation in covering codes investigations) and if is large enough, we have
and, in a more general form,
(1.2) 
where is independent of but it is possible that is dependent of and . In [11], see also the references therein including [9, 13], the bound (1.2) is given in another (asymptotic) form and infinite families of covering codes, achieving the bound, are obtained for the following situations:
Here are integers, is a prime power.
In the general case, for arbitrary , the problem to achieve the bound (1.2) is open.
In the last decades, upper bounds on have been intensively investigated, see [8, 7, 9, 10, 11, 13, 14, 15, 16, 21, 22, 23] and the references therein.
The goal of this work is to obtain new upper bounds on the length functions and with and arbitrary , in particular with where is a prime power. It is an open problems.
Let be the dimensional projective space over the field ; see [17, 18, 19] for an introduction to the projective spaces over finite fields, see also [15, 18, 21, 22] for connections between coding theory and Galois geometries.
Effective methods to obtain upper bounds on are connected with saturating sets in .
Definition 1.3.
A point set is saturating if for any point of there exist points in generating a subspace of containing , and is the smallest value with such property.
By Definition 1.3, every point from can be written as a linear combination of at most points of a saturating set, cf. Definition 1.1(ii).
Saturating sets are considered, for instance, in [2, 3, 4, 7, 9, 10, 11, 12, 13, 15, 20, 16, 21, 22, 26]. In the literature, saturating sets are also called “saturated sets”, “spanning sets”, “dense sets”.
Let be the smallest size of a saturating set in .
If ary positions of a column of an parity check matrix of an code are treated as the homogeneous coordinates of a point in then this parity check matrix defines an saturating set of size in [9, 10, 16, 11, 15, 20, 21, 22]. So, there is a onetoone correspondence between codes and saturating sets in . Therefore,
in particular, , .
Complete arcs in are an important class of saturating sets. An arc in with is a set of points such that no
points belong to the same hyperplane of
. An arc of is complete if it is not contained in an arc of . A complete arc in is an saturating set. Points (in the homogeneous coordinates) of a complete arc in , treated as columns, form a parity check matrix of an maximum distance separable (MDS) code. If these codes are quasiperfect.Let be the smallest size of a complete arc in . By above,
1.2 Covering codes with radius 3
For arbitrary , covering codes of length close to the lower bound (1.2) are known only for [11, 13]. In particular, the following bounds are obtained by algebraic constructions [11, Sect. 5, eq. (5.2)], [13, Th. 12]:
If or , covering codes of length close to the lower bound (1.2) are known only when , where is a prime power [9, 10, 11, 16]. In particular, the following bounds are obtained by algebraic constructions, see [9, 10], [11, Sect. 5, eqs. (5.3),(5.4)]:
For arbitrary , in the literature, computer results are given for codes with [14, Tab. 1] and codes with [10, Tab. 1], [14, Tab. 2].
In this work, by computer search, we obtain new results for quasiperfect MDS codes with , and quasiperfect Almost MDS codes with . This gives upper bounds on and for a set of values essentially greater than the one in [10, 14].
Theorem 1.4.
Let and . For the length function and for the smallest size of a saturating set in the projective space the following upper bounds hold:
(1.3)  
(1.4)  
(1.5) 
Remark 1.5.
We emphasize that, for and , the new bounds of Theorem 1.4 have the form
As far as it is known to the authors, such bounds have not been previously described in the literature.
Our results, in particular figures and observations in Sections 3 and 4, allow us to conjecture the following.
Conjecture 1.6.
Let and . For the length function and for the smallest size of a saturating set in the projective space the following upper bounds hold:
The paper is organized as follows. In Section 2, we describe a leximatrix algorithm to obtain parity check matrices of covering codes. In Sections 3 and 4, upper bounds on the length functions and are considered. In Conclusion, the results of this work are briefly analyzed; some tasks for investigation of the leximatrix algorithm are formulated. In Appendix, tables with sizes of codes obtained in this work are given.
2 Leximatrix algorithm to obtain parity check matrices of covering codes
The following is a version of the recursive gparity check algorithm for greedy codes, see e.g. [6, p. 25], [24], [25, Section 7].
Let be the Galois field with elements.
If is prime, the elements of are treated as integers modulo .
If with prime and , the elements of are represented by integers as follows: where is a root of a primitive polynomial of .
For a ary code of codimension , covering radius , and minimum distance , we construct a parity check matrix from nonzero columns of the form
where the first (leftmost) nonzero element is 1. The number of distinct columns is . For we put . We order the columns in the list as . The columns of the list are candidates to be included in the parity check matrix.
By the above arguments connected with the formula for and the order of columns, a column is treated as its number in our list written in the ary scale of notation. The considered order of columns is lexicographical.
The first column of the list should be included into the matrix. Then stepbystep, one takes the next column from the list which cannot be represented as a linear combination of at most columns already chosen. The process ends when no new column may be included into the matrix. The obtained matrix is a parity check matrix of an code.
We call a leximatrix the obtained parity check matrix. We call a leximatrix code the corresponding code.
It is important to note that for prime , length of a leximatrix code and the form of the leximatrix depend on and only. No other factors affect code length and structure. Actually, assume that after some step a current matrix is obtained. At the next step we should remove from our current list all columns that are linear combination of or less columns of the current matrix. For prime and the given , the result of removing is unequivocal; hence, the next column is taken uniquely.
For nonprime , the length of a leximatrix code depends on and on the form of the primitive polynomial of the field. In this work, we use primitive polynomials that are created by the program system MAGMA [5] by default, see Table A. In any case, the choice of the polynomial changes the leximatrix code length unessentially.
By the leximatrix algorithm, if , we obtain the ary Hamming code. If , we obtain a quasiperfect code; for such code is an MDS code and corresponds to a complete arc in . If , we obtain a quasiperfect code; for such code is an MDS code and corresponds to a complete arc in ; for it is an Almost MDS code.
Let be length of the ary leximatrix code of codimension and covering radius . It is assumed that for a nonprime field , one uses the primitive polynomial created by the program system MAGMA [5] by default; in particular, for nonprime , the polynomial from Table A should be taken.
Future, we represent length of an leximatrix code in the form
(2.1) 
where is a coefficient entirely given by (if is prime) or by , and the primitive polynomial of (if is nonprime).
3 Upper bounds on the length functions
The following properties of the leximatrix algorithm are useful for implementation.
Proposition 3.1.
Let be a prime. Then the th column of the leximatrix of an code is the same for all where is large enough.
Proof.
Let be the matrix obtained in the th step of the leximatrix algorithm. Here is a column of the matrix. A column from the list, not included in , is covered by if it can be represented as a linear combination of at most 3 columns of . Suppose that , where is the th column in the lexicographical list of candidates. A column is the next chosen column, if and only if all the columns with are covered by . This means that, for any , at least one of the determinants , with , is equal to zero modulo . This can happen only in two cases:

, we say that is “absolutely” covered by ;

, but .
For large enough, does not divide any of the possible values of and then, at least for relatively small, the columns covered are just the absolutely covered columns. Therefore, when is large enough the leximatrices share a certain number of columns. ∎
The values of can be found with the help of calculations based on the proof of Proposition 3.1. Also, we can directly consider leximatrices for a convenient region of .
Example 3.2.
Values of , , together with columns , are given in Table B. So, for all prime (resp. ) the first 14 (resp. 20) columns of a parity check matrix of an MDS leximatrix code are as in Table B.
Table B. The first 20 columns of parity check matrices of leximatrix MDS codes, prime
1  0  0  0  1  2  11  1  7  11  8  67 
2  0  0  1  0  2  12  1  8  6  13  109 
3  0  1  0  0  2  13  1  9  13  16  199 
4  1  0  0  0  2  14  1  10  12  22  233 
5  1  1  1  1  2  15  1  11  7  29  269 
6  1  2  3  4  5  16  1  12  22  15  769 
7  1  3  2  5  11  17  1  13  16  20  769 
8  1  4  5  3  29  18  1  14  17  7  1283 
9  1  5  4  2  41  19  1  15  21  10  1283 
10  1  6  8  9  41  20  1  16  9  38  1321 
Proposition 3.3.
 (i)

There exists a code, length of which satisfies .
 (ii)

There exist quasiperfect MDS leximatrix codes of length for and .
Proof.
Lengths of leximatrix quasiperfect MDS codes are collected in Table 1 (see Appendix) and presented in Figure 1 by the bottom solid black curve. The bound is shown in Figure 1 by the top dashed red curve.
We denote by the difference between the bound and length of the leximatrix code. Let be the corresponding percent difference. Thus,
The difference and the percent difference are presented in Figures 2 and 3, respectively.
By (2.1), we represent length of an leximatrix code in the form
(3.1) 
where is a coefficient entirely given by (if is prime) or by and the primitive polynomial of the field (if is nonprime). The coefficients are shown in Figure 4.
Observation 3.4.
Remark 3.5.
It is interesting that the oscillation of the coefficients around a horizontal line, in principle, is similar to the oscillation of the values around a horizontal line in [3, Fig. 12, Observation 3.7], [4, Fig. 5, Observation 3.7].
In the papers [3, 4], small complete arcs in the projective plane are constructed by computer search using algorithm with fixed order of points (FOP). These arcs correspond to quasiperfect MDS codes while the algorithm FOP is analogous to the leximatrix algorithm of Section 2. Moreover, the value is defined in [3, 4] as . So, see (3.1), the coefficients and the values have the similar nature. It is possible that the oscillations mentioned have similar reasons too. However, in the present time the enigma of the oscillations is incomprehensible,
4 Upper bounds on the length functions
Proposition 4.1.
 (i)

There exist codes with for .
 (ii)

There exist Almost MDS leximatrix codes with for .
Proof.
Lengths of leximatrix Almost MDS codes are collected in Table 2 (see Appendix) and presented in Figure 5 by the bottom solid black curve. The bound is shown in Figure 5 by the top dashed red curve.
We denote by the difference between the bound and length of the leximatrix code. Let be the corresponding percent difference. Thus,
By (2.1), we represent length of an leximatrix code in the form
(4.1) 
where is a coefficient entirely given by (if is prime) or by and the primitive polynomial of the field (if is nonprime). The coefficients are shown in Figure
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