Tables, bounds and graphics of sizes of complete arcs in the plane PG(2,q)

111 Pages·2017·1.55 MB·English

Tables, bounds and graphics of sizes of
complete arcs in the plane PG(2, q) for all
8
1 q 321007 and sporadic q in [323761 . . . 430007]
0
≤
2 obtained by an algorithm with fixed order of

n
a points (FOP)
J ∗

3

] Daniele Bartoli
O
C Dipartimento di Matematica e Informatica, Universit`a degli Studi di Perugia,
.
h Via Vanvitelli 1, Perugia, 06123, Italy. E-mail: [email protected]
t
a
Alexander A. Davydov
m
[ Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences

3 Bol’shoi Karetnyi per. 19, Moscow, 127051, Russian Federation. E-mail: [email protected]
v
9 Giorgio Faina
6
4 Dipartimento di Matematica e Informatica, Universit`a degli Studi di Perugia,
0
. Via Vanvitelli 1, Perugia, 06123, Italy. E-mail: [email protected]
4
0
Alexey A. Kreshchuk
4
1
Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences
:
v
Bol’shoi Karetnyi per. 19, Moscow, 127051, Russian Federation. E-mail: [email protected]
i
X
Stefano Marcugini and Fernanda Pambianco
r
a
Dipartimento di Matematica e Informatica, Universit`a degli Studi di Perugia,
Via Vanvitelli 1, Perugia, 06123, Italy. E-mail: stefano.marcugini,fernanda.pambianco @unipg.it
{ }
TheresearchofD.Bartoli,G.Faina,S.Marcugini,andF.PambiancowassupportedinpartbyMinistry
∗
for Education, University and Research of Italy (MIUR) (Project “Geometrie di Galois e strutture di inci-
denza”) and by the Italian National Group for Algebraic and Geometric Structures and their Applications
(GNSAGA-INDAM).TheresearchofA.A.DavydovandA.A.KreshchukwascarriedoutattheIITPRAS
attheexpenseoftheRussianFoundationforSciences(project14-50-00150). Thisworkhasbeencarriedout
usingcomputingresourcesofthefederalcollectiveusagecenterComplexforSimulationandDataProcessing
for Mega-science Facilities at NRC Kurchatov Institute, http://ckp.nrcki.ru/.
1
Abstract. In the previous works of the authors, a step-by-step algorithm FOP which uses
any fixed order of points in the projective plane PG(2,q) is proposed to construct small
complete arcs. In each step, the algorithm adds to a current arc the first point in the fixed
order not lying on the bisecants of the arc. The algorithm is based on the intuitive postulate
that PG(2,q) contains a sufficient number of relatively small complete arcs. Also, in the
previous papers, it is shown that the type of order on the points of PG(2,q) is not relevant.
A complete lexiarc in PG(2,q) is a complete arc obtained by the algorithm FOP using
the lexicographical order of points. In this work, we collect and analyze the sizes of
complete lexiarcs in the following regions:
all q 321007, q prime power;
≤
15 sporadic q’s in the interval [323761...430007], see (1.10).
In the work [9], the smallest known sizes of complete arcs in PG(2,q) are collected for all
q 160001, q prime power. The sizes of complete arcs, collected in this work and in [9],
≤
provide the following upper bounds on the smallest size t (2,q) of a complete arc in the
2
projective plane PG(2,q):
(cid:112) (cid:112)
t (2,q) < 0.998 3qlnq < 1.729 qlnq for 7 q 160001;
2
≤ ≤
(cid:112) (cid:112)
t (2,q) < 1.05 3qlnq < 1.819 qlnq for 7 q 321007.
2
≤ ≤
Our investigations and results allow to conjecture that the bound
t (2,q) < 1.05√3qlnq < 1.819√qlnq holds for all q 7.
2
≥
It is noted that sizes of the random complete arcs and complete lexiarcs behave similarly.
This work can be considered as a continuation and development of the paper [11].
Mathematics Subject Classification (2010). 51E21, 51E22, 94B05.
Keywords. Projective planes, complete arcs, small complete arcs, upper bounds, algo-
rithm FOP, randomized greedy algorithms
1 Introduction. The main results
Let PG(2,q) be the projective plane over the Galois field F of q elements. An n-arc is a set
q
of n points no three of which are collinear. An n-arc is called complete if it is not contained
in an (n+1)-arc of PG(2,q). For an introduction to projective geometries over finite fields
see [56,58,59,80].
The relationship among the theory of n-arcs, coding theory, and mathematical statistics
is presented in [40,58,59,63,66,67]. In particular, a complete arc in the plane PG(2,q), the
points of which are treated as 3-dimensional q-ary columns, defines a parity check matrix of
a q-ary linear code with codimension 3, Hamming distance 4, and covering radius 2. Arcs
can be interpreted as linear maximum distance separable (MDS) codes [85, Sec. 7], [87] and
2
they are related to optimal coverings arrays [54], superregular matrices [61], and quantum
codes [26].
A point set S PG(2,q) is 1-saturating if any point of PG(2,q) S is collinear with two
⊂ \
points in S [19–21,24,37,38,49,73,88]. In the literature, 1-saturating sets are also called
“saturated sets”, “spanning sets”, and “dense sets”. The points of a 1-saturating set in
PG(2,q) form a parity check matrix of a linear covering code with codimension 3, Hamming
distance 3 or 4, and covering radius 2. An open problem is to find small 1-saturating
sets (respectively, short covering codes). A complete arc in PG(2,q) is, in particular, a
1-saturating set; often the smallest known complete arc is the smallest known 1-saturating
set [24,37,38,49,75]. Let (cid:96) (2,q) be the smallest size of a 1-saturating set in PG(2,q). In [21],
1
combining approaches of [73] and [8], the following upper bound is proved for arbitrary q:
(cid:114)
(cid:112) q
(cid:96) (2,q) q(3lnq +lnlnq)+ +3. (1.1)
1
≤ 3lnq
Lett (2,q)bethe smallest size of a complete arc in the projective plane PG(2,q).
2
One of the main problems in the study of projective planes, which is also of interest in
coding theory, is the determination of the spectrum of possible sizes of complete arcs. In
particular, the value of t (2,q) is interesting. Finding estimates of the minimum size t (2,q)
2 2
is a hard open problem, see e.g. [60, Sec.4.10].
This paper is devoted to upper bounds on t (2,q).
2
Surveys of results on the sizes of plane complete arcs, methods of their construction, and
the comprehension of the relating properties can be found in [5,6,8–11,13,15,18,23,33,34,
43,49,55,56,58–60,62,76,77,80–82,84–86].
Problems connected with small complete arcs in PG(2,q) are considered in [1–18,22,24,
25,27,31–38,41–53,55–60,62,64,68–71,74–77,79–86,89–92], see also the references therein.
The exact values of t (2,q) are known only for q 32; see [2,24,31,32,42,52,56,57,70,71].
2
≤
The following lower bounds hold (see [3,27,79] and the references therein):
(cid:26)
√2q +1 for any q,
t (2,q) > (1.2)
2 √3q + 1 for q = ph, p prime, h = 1,2,3.
2
Let t( ) be the size of the smallest complete arc in any (not necessarily Desarguesian)
q
P
projective plane of order q. In [62], for q large enough, the following result is proved by
q
P
probabilistic methods :
t( ) √qlnC q, C 300, (1.3)
q
P ≤ ≤
where C is a constant independent of q (so-called universal or absolute constant). Surveys
and results of random constructions for geometrical objects in PG(2,q) can be found in
[19–21,28,45,62,65,73].
In PG(2,q), complete arcs are obtained by algebraic constructions (see [59, p.209]) with
sizes approximately 1q [1,13,64,81–83,92], 1q [13,64,84], 2q0.9 for q > 710 [81], and
3 4
3
2.46q0.75lnq for big prime q [53]. It is noted in [45, Sec. 8] that the smallest size of a
complete arc in PG(2,q) obtained via algebraic constructions is
cq3/4 (1.4)
where c is a universal constant [84, Sec. 3], [85, Th. 6.8].
There is a substantial gap between the known upper bounds and the lower bounds on
t (2,q),see(1.2)–(1.4). Thegapisreducedifoneconsidersthelowerbound(1.2)forcomplete
2
arcs and the upper bound (1.1) for 1-saturating sets. However, though complete arcs are
1-saturating sets, they represent a narrower class of objects. Therefore, for complete arcs,
one may not use the bound (1.1) directly. Nevertheless, the common nature of complete arcs
and 1-saturating sets allows to hope for upper bounds on t (2,q) similar to (1.1). The hope
2
is supported by numerous experimental data and some probabilistic conjectures, see below.
In [62, p. 313], it is noted (with reference to the work [27]) that in a preliminary report of
1989, J.C. Fisher obtained by computer search complete arcs in many planes of small orders
and conjectured that the average size of a complete arc in PG(2,q) is about √3qlogq.
In [8], see also [7], an attempt to obtain a theoretical upper bound on t (2,q) with
2
the main term of the form c√qlnq, where c is a small universal constant, is done. The
reasonings of [8] are based on the explanation of the working mechanism of a step-by-step
greedy algorithm for constructing complete arcs in PG(2,q) and on quantitative estimations
of the algorithm. For more than half of the steps of the iterative process, these estimations
are proved rigorously. The natural (and well-founded) conjecture that they hold for the rest
of steps is done, see [8, Conject. 2]. As a result, in [8], the following conjectural upper bounds
are formulated.
Conjecture 1.1. [8] Let t (2,q) be the smallest size of a complete arc in the projective plane
2
PG(2,q). Under conjecture given in [8, Conject. 2], the following upper bounds hold:
(cid:114)
(cid:112) q
t (2,q) < q(3lnq +lnlnq +ln3)+ +3, (1.5)
2
3lnq
(cid:112) (cid:112)
t (2,q) < 1.87 qlnq < 1.08 3qlnq. (1.6)
2
Moreover, in [8] it is conjectured that the upper bounds (1.5), (1.6) hold for all q without
any extra conditions.
Denote by t (2,q) the smallest known size of a complete arc in the projective
2
plane PG(2,q). Clearly,
t (2,q) t (2,q).
2 2
≤
For even q = 2h, small complete arcs in planes are a base for inductive infinite families
of small complete caps in the projective spaces PG(N,q), see [36]. For h 17, the smallest
≤
known sizes of complete arcs in PG(2,2h) are collected in [6,8], see also [13,15,34,36,41].
4
Also, (6√q 6)-arcs in PG(2,42a+1), are constructed in [35]; for a 4 it is proved that they
− ≤
are complete. This gives a complete 3066-arc in PG(2,218). In particular, it holds that
t (2,29) = 85, t (2,210) = 124, t (2,211) = 199, t (2,212) = 300, t (2,213) = 449,
2 2 2 2 2
t (2,214) = 665, t (2,215) = 987, t (2,216) = 1453, t (2,217) = 2141, t (2,218) = 3066.
2 2 2 2 2
For prime powers q 13627, the values of t (2,q) (up to January 2013) are collected
2
≤
in [13,15]. For prime powers q 49727 and prime q 150001, the values of t (2,q) (up
2
≤ ≤
to August 2014) are collected in [6]. For q 151, a number of improvements of t (2,q), in
2
≤
comparison with [13,15], are given in [75] and cited in [6].
From the results of [6,13,15], see also [36,41,46,51], it follows that
t (2,q) < 4√q for q 841, q = 857,312,210,372,412,74;
2
≤
t (2,q) < 4.5√q for q 2647, q = 2659,2663,2683,2693,2753,2801;
2
≤
t (2,q) < 5√q for q 9497, q = 9539,9587,9613,9623,9649,9689,9923,9973;
2
≤
t (2,q) < 5.5√q for q 38557, q = 36481,37537,37963,38039,38197.
2
≤ (cid:54)
Let Q be a set of 34 sporadic q’s in the interval [160801...430007], see [9, Tab.3]. Let
1
Q be a set of values of q. We have
Q = 160801,161009,162007,163003,164011,165001,166013,167009,168013, (1.7)
1
{
169003,170503,178169,180001,185021,190027,200003,210011,250007,260003,
262144,270001,280001,290011,300007,330017,350003,360007,370003,380041,
390001,400009,410009,420001,430007 ;
}
Q = 2 q 160001,q prime power Q . (1.8)
1
{ ≤ ≤ } ∪
For q Q, the values of t (2,q) are obtained in the works [4–6,8,9,11,13,15,22], see also
2
∈
the references therein.
All the values of t (2,q) (up to June 2015), q Q, are collected in the work [9].
2
∈
Note that, for q Q, in the works [4–6,8,9,11–13,15], the most of the values t (2,q)
2
∈
have been obtained by computer search using randomized greedy algorithms described in
[11,13,15,18,33,34,39]. In each step, a step-by-step greedy algorithm adds to an incomplete
current arc a point providing the maximum possible (for the given step) number of new
covered points.
We denote by tG(2,q) the size of a complete arc in PG(2,q) collected in [9].
2
An important way to obtain small plane complete arcs is a step-by-step algorithm with
fixed order of points (FOP), see [10–12,16,18,22,25]. The algorithm FOP fixes a particular
order on points of PG(2,q). In each step, the algorithm FOP adds to an incomplete current
arc the next point in this order not lying on bisecants of this arc. For both prime and
non-prime q, a lexicographical order of points can be used, see Section 2.
5
Definition 1.2. We call a lexiarc an arc obtained by the algorithm FOP using the
lexicographical order of points.
Let tL(2,q) be the size of a complete lexiarc in the projective plane PG(2,q).
2
Note that the sizes of complete arcs obtained by the algorithm FOP vary insignificantly
with respect to the order of points, see [10,18,22,25]. In particular, for the lexicographical
order of points described in Section 2, in the case of prime q, the size tL(2,q) of a complete
2
lexiarc and its set of points depend on q only. No other factors affect size and structure of a
complete lexiarc.
Let L , L , L , and L be the following sets of values of q:
1 2 3
L = q 301813, q prime power ; (1.9)
1
{ ≤ }
L = 323761,326041,330017,332929,340007,344569,350003,360007, (1.10)
2
{
370003,380041,390001,400009,410009,420001,430007 ;
}
L = 301813 < q 321007, q prime power ; (1.11)
3
{ ≤ }
L = L L L = q 321007, q prime power L . (1.12)
1 2 3 2
∪ ∪ { ≤ } ∪
For q L L , the values of tL(2,q) are obtained in [10–12,16,18,25] and collected in the
∈ 1 ∪ 2 2
work [10]. For q L , the values of tL(2,q) are obtained in this work.
∈ 3 2
All the values of tL(2,q), q L, are collected in this work.
2 ∈
It should be emphasized that in this work, to obtain upper bounds, we use the computer
search results for all prime powers q in the regions q 160001 with greedy algorithms
≤
and q 321007 with the algorithm FOP. In this sense, we say that computer search in
≤
the noted regions is complete.
Complete arcs obtained by greedy algorithms have smaller sizes than complete lexiarcs,
however the greedy algorithms take essentially greater computer time than the algorithm
FOP. This is why the complete computer search for all prime powers q with the help of
greedy algorithms is done for q 160001 [9] whereas the complete search by algorithm FOP
≤
is executed for q 321007. So, we have, see Figure 1 in Section 3,
≤
(cid:26) tG(2,q) if q Q = 2 q 160001 Q
t (2,q) = 2 ∈ { ≤ ≤ }∪ 1 . (1.13)
2 tL(2,q) if q L Q = ( 160001 < q 321007 L ) Q
2 ∈ \ { ≤ }∪ 2 \ 1
In the works of the authors [4,15,18,22,25], non-standard types of upper bounds on
t (2,q) are proposed. In particular, in the paper [11], the function h(q) is defined so that
2
(cid:112)
t (2,q) = h(q) 3qlnq. (1.14)
2
Using the function h(q) instead of t (2,q) allows to do estimates and graphics more expres-
2
sive, see Figures 1, 6 in Section 3.
The following theorem summarizes the main results of this work. The theorem is based
on the complete computer search, the results of which are collected in [5,6,9–11,13,15,16],
(see also the references therein) and in Tables 1 – 6 of this paper.
6
Theorem 1.3. Let t (2,q) be the smallest size of a complete arc in the projective plane
2
PG(2,q). Let t (2,q) be the smallest known size of a complete arc in PG(2,q). Let L be
2 2
given by (1.10). The following upper bounds hold:
(cid:112) (cid:112)
t (2,q) < t (2,q) < 0.998 3qlnq < 1.729 qlnq for 7 q 160001; (1.15)
2 2
≤ ≤
(cid:112) (cid:112)
t (2,q) < t (2,q) < 1.05 3qlnq < 1.819 qlnq for 7 q 321007 and q L . (1.16)
2 2 2
≤ ≤ ∈
For q 160001, complete arcs in PG(2,q) satisfying the upper bounds (1.15) can be
≤
constructed with the help of the step-by-step greedy algorithm which adds to an incomplete
current arc, in each step, a point providing the maximum possible (for the given step) number
of new covered points.
For 160001 < q 321007 and q L , complete arcs in PG(2,q) satisfying the upper
2
≤ ∈
bounds (1.16) can be constructed as lexiarcs with the help of the algorithm FOP using the
lexicographical order of points. The only exception is q = 178169 for which a complete arc
satisfying (1.16) can be obtained by the greedy algorithm.
Calculations executed for sporadic q 430007 (q L of (1.10)) strengthen the confi-
2
≤ ∈
dence for validity of the bounds of Theorem 1.3, see also Figures 1 – 3 in Section 3. Also, it
is important that the bounds (1.15), (1.16) are close to the conjectural (but well-founded)
bounds of [8], see (1.5), (1.6) in Conjecture 1.1 and Figures 4, 5 in Section 3. On the whole,
our investigations and results (again see figures in Section 3) allow to conjecture that the
bound (1.16) holds for all q.
Conjecture 1.4. Let t (2,q) be the smallest size of a complete arc in the projective plane
2
PG(2,q). The following upper bound holds:
(cid:112) (cid:112)
t (2,q) < 1.05 3qlnq < 1.819 qlnq for all q 7.
2
≥
The paper is organized as follows. In Section 2, the algorithm with fixed order of points
(FOP)isconsidered. Thelexicographicalorderofpointsisdescribedandlexiarcsaredefined.
In Section 3, upper bounds on t (2,q), based on sizes of complete lexiarcs in PG(2,q) and
2
on sizes of complete arcs collected in [9], are proposed. In Section 4, the common nature
lexiarcs and random arcs is considered. In Section 5, the list of tables with sizes of complete
lexiarcs in the projective plane PG(2,q) is given. In Conclusion the results of this paper are
briefly analyzed. In Appendix, tables of sizes of complete lexiarcs in the projective plane
PG(2,q), obtained in this paper and in the previous works of the authors, are collected.
2 Algorithm with fixed order of points (FOP). Lexiarcs
2.1 Step-by-step algorithm FOP
We use results and approaches of the works [10,12,16,18,22,25]. Consider the projective
plane PG(2,q) and fix a particular order on its points. The algorithm FOP builds a complete
7
arc iteratively, step-by-step.
Let K(j) be the arc obtained on the j-th step. On the next step, the first point in the
fixed order not lying on the bisecants of K(j) is added to K(j).
Algorithm FOP.SupposethatthepointsofPG(2,q)areorderedasA ,A ,...,A .
1 2 q2+q+1
Consider the empty set as root of the search and let K(j) be the partial solution obtained in
the j-th step, as extension of the root. We put
K(0) = , K(1) = A , K(2) = A ,A , m(1) = 2, (2.1)
1 1 2
∅ { } { }
K(j+1) = K(j) A ,
m(j)
∪{ }
m(j) = min i [m(j 1)+1,q2 +q +1] (cid:64) P,Q K(j) : A ,P,Q are collinear ,
i
{ ∈ − | ∈ }
i.e. m(j) is the minimum subscript i such that the corresponding point A is not saturated
i
by K(j). The process ends when a complete arc is obtained.
Remark 2.1. A point of PG(2,q) can be treated as a 3-vector column of a code parity check
matrix. Then, formally, the algorithm FOP can be considered as a version of the recursive
g-paritycheckalgorithmforgreedycodesofcodimensionr = 3andminimumdistanced = 4,
see e.g. [30, p. 25], [72], [78, Section 7].
However, in coding theory, for given r,d, the aim is to get a long code while our goal is
to obtain a short complete arc. Moreover, our estimates and computer search show that for
r = 3, d = 4, the algorithm FOP gives “bad” codes, essentially shorter than “good” codes
corresponding to ovals and hyperovals.
Finally, note that we do not use a word “greedy”in the name of the algorithm FOP, as in
projective geometry the terms “greedy algorithm” and “randomized greedy algorithm” are
traditionally connected with other approaches, see [11,13,15,18,33,34,39].
2.2 Lexicographical order of points
In the beginning, we consider q prime. Let the elements of the field F = 0,1,...,q 1 be
q
{ − }
treated as integers modulo q. Let the points A of PG(2,q) be represented in homogenous
i
coordinates so that
A = (x(i),x(i),x(i)), x(i) F , (2.2)
i 0 1 2 j ∈ q
where the leftmost non-zero element is 1. For A , we put
i
i = x(i)q2 +x(i)q +x(i). (2.3)
0 1 2
So, the homogenous coordinates of a point A are treated as its number i written in the q-ary
i
scale of notation. Recall that the points of PG(2,q) are ordered as A ,A ,...,A .
1 2 q2+q+1
It is important that for such lexicographical order for prime q, the size tL(2,q)
2
of a complete lexiarc and its set of points depend on q only. No other factors affect
size and structure of a complete lexiarc.
8
Now let q = pm, p prime, m 2. Let F (x) be a primitive polynomial of F and let α
q pm
be a root of F (x). Elements of F≥ are represented by integers as follows
q pm
F = F = 0,1 = α0,2 = α1,...,q 1 = αq−2 .
pm q
{ − }
Then we again use (2.2) and (2.3). However for non-prime q the size tL(2,q) of a complete
2
lexiarc depends on q and on the form of the polynomial F (x). In this work we use primitive
q
polynomials that are created by the program system MAGMA [29] by default, see Table A
where polynomials for q < 36000 are given. In any case, the choice of the polynomial changes
the size of complete lexiarc inessentially.
We have noted in Introduction that, in general, the sizes of complete arcs obtained by
the algorithm FOP vary insignificantly with respect to the order of points. In particular,
in [18,22,25] the so-called Singer order of points (based on the Singer group of collineations)
is considered and it is shown that in the region 5000 < q 40009, q prime, the difference
≤
between the sizes of complete arcs obtained using the lexicographical order and the Singer
order is less then 2% and none of the two orders gives the smallest size for all q.
2.3 Starting points of lexiarcs in PG(2,q), q prime
Proposition 2.2. Let q be a prime. Then the j-th point of a lexiarc in PG(2,q) is the same
for all q q (j) where q (j) is large enough.
0 0
≥
Proof. Suppose that, in (2.1), at a certain step j we have K(j) K(j−1) = P , with P = A .
s
\ { }
A point Q = A K(j) will be the next chosen point in the extension process if and only if
r
(cid:54)∈
all the points A with i [s+1,r 1] are covered by K(j). That is, for any i [s+1,r 1] at
i
∈ − ∈ −
least one of the determinants of the coordinates of the points P ,P ,A , with P ,P K(j),
1 2 i 1 2
∈
is equal to zero modulo q. This can happen only for two reasons:
1. det(P ,P ,A ) = 0: we say that A is “absolutely” covered by K(j);
1 2 i i
2. det(P ,P ,A ) = m = 0, with m 0 mod q.
1 2 i
(cid:54) ≡
It is clear that, for q large enough, q does not divide any of the possible m = det(P ,P ,A )
1 2 i
and then, at least for j “small”, the points covered are just the absolutely covered points.
Therefore, when q is large enough the lexiarcs share a certain number of points.
The values of q (j) can be found with the help of calculations based on the proof of
0
Proposition 2.2. Also, we can directly consider lexiarcs constructed by the algorithm FOP
for a convenient region of q.
Example 2.3. Values of q (j), j 24, together with the homogenous coordinates (a ,a ,a )
0 0 1 2
≤
of the common points, are given in Table B. So, for all prime q 251 the first 24 points of
a lexiarc are as in Table B. Since tL(2,251) = 63 [16], we know≥24/63 ≈ 38% of complete
2
lexiarc points for q = 251. For growing q this percentage decreases (relatively slowly). For
instance, for q ≈ 270000 it is ≈ 20%.
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Table A. Primitive polynomials used for lexiarcs in PG(2,q) with non-prime q
q = pm primitive q = pm primitive q = pm primitive
polynomial polynomial polynomial
4 = 22 x2+x+1 8 = 23 x3+x+1 9 = 32 x2+2x+2
16 = 24 x4+x3+1 25 = 52 x2+x+2 27 = 33 x3+2x2+x+1
32 = 25 x5+x3+1 49 = 72 x2+x+3 64 = 26 x6+x4+x3+1
81 = 34 x4+x+2 121 = 112 x2+4x+2 125 = 53 x3+3x+2
128 = 27 x7+x+1 169 = 132 x2+x+2 243 = 35 x5+2x+1
256 = 28 x8+x4+x3+ 289 = 172 x2+x+3 343 = 73 x3+3x+2
x2+1
361 = 192 x2+x+2 512 = 29 x9+x4+1 529 = 232 x2+2x+5
625 = 54 x4+x2+2x+2 729 = 36 x6+x+2 841 = 292 x2+24x+2
961 = 312 x2+29x+3 1024 = 210 x10+x6+x5+ 1331 = 113 x3+2x+9
x3+x2+x+1
1369 = 372 x2+33x+2 1681 = 412 x2+38x+6 1849 = 432 x2+x+3
2048 = 211 x11+x2+1 2187 = 37 x7+x2+2x+1 2197 = 133 x3+x2+7
2209 = 472 x2+x+13 2401 = 74 x4+5x2+4x+3 2809 = 532 x2+49x+2
3125 = 55 x5+4x+2 3481 = 592 x2+58x+2 3721 = 612 x2+60x+2
4096 = 212 x12+x8+x2+ 4489 = 672 x2+63x+2 4913 = 173 x3+x+14
x+1
5041 = 712 x2+69x+7 5329 = 732 x2+70x+5 6241 = 792 x2+78x+3
6561 = 38 x8+2x5+x4+ 6859 = 193 x3+4x+17 6889 = 832 x2+82x+2
2x2+2x+2
7921 = 892 x2+82x+3 8192 = 213 x13+x4+x3+ 9409 = 972 x2+96x+5
x+1
10201 = 1012 x2+97x+2 10609 = 1032 x2+102x+5 11449 = 1072 x2+103x+2
11881 = 1092 x2+108x+6 12167 = 233 x3+2x+18 12769 = 1132 x2+101x+3
14641 = 114 x4+8x2+10x+2 15625 = 56 x6+x4+4x3+ 16129 = 1272 x2+126x+3
x2+2
16384 = 214 x14+x7+x5+ 16807 = 75 x5+x+4 17161 = 1312 x2+83x+127
x3+1
18769 = 1372 x2+95x+20 19321 = 1392 x2+72x+111 19683 = 39 x9+2x3+2x2+
x+1
22201 = 1492 x2+144x+34 22801 = 1512 x2+16x+51 24389 = 293 x3+2x+27
24649 = 1572 x2+153x+82 26569 = 1632 x2+153x+82 27889 = 1672 x2+156x+90
28561 = 134 x4+3x2+12x+2 29791 = 313 x3+x+28 29929 = 1732 x2+80x+104
32041 = 1792 x2+67x+154 32761 = 1812 x2+11x+78 32768 = 215 x15+x+1
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