New sizes of complete arcs in PG(2,q)
Alexander A. Davydov, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco
aa r X i v : . [ m a t h . C O ] A ug New sizes of complete arcs in
P G (2 , q ) Alexander A. Davydov [email protected]
Institute for Information Transmission Problems, Russian Academy of Sciences,Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 127994, Russia
Giorgio Faina [email protected]
Stefano Marcugini [email protected]
Fernanda Pambianco [email protected]
Dipartimento di Matematica e Informatica, Universit`a degli Studi di Perugia, ViaVanvitelli 1, Perugia, 06123, Italy
Abstract.
New upper bounds on the smallest size t (2 , q ) of a complete arc inthe projective plane P G (2 , q ) are obtained for 853 ≤ q ≤ q = 3511 , , , , , , q ≤ q = 2401 , , t (2 , q ) < . √ q holds. The bounds are obtained by finding of new small com-plete arcs with the help of computer search using randomized greedy algorithms.Also new sizes of complete arcs are presented. Let
P G (2 , q ) be the projective plane over the Galois field F q . An n -arc is aset of n points no 3 of which are collinear. An n -arc is called complete ifit is not contained in an ( n + 1)-arc of P G (2 , q ). Surveys of results on arcscan be found in [9, 10]. In [10] the close relationship between the theory ofcomplete n -arcs, coding theory and mathematical statistics is presented. Inparticular a complete arc in a plane P G (2 , q ) , points of which are treated as3-dimensional q -ary columns, defines a parity check matrix of a q -ary linearcode with codimension 3, Hamming distance 4 and covering radius 2. Arcs canbe interpreted as linear maximum distance separable (MDS) codes and theyare related to optimal coverings arrays [8] and to superregular matrices [11].One of the main problems in the study of projective planes, which is alsoof interest in Coding Theory, is the determination of the spectrum of possiblesizes of complete arcs. Especially the problem of determining t (2 , q ), the sizeof the smallest complete arc in P G (2 , q ), is interesting.In Section 2 we give upper bounds on t (2 , q ) for 853 ≤ q ≤ q =3511 , , , , , , , q . For q ≤ q = 2401 , , t (2 , q ) < . √ q holds. For smaller q slightly smaller bounds hold. The upper bounds have beenobtained by finding of new small complete arcs with the help of the randomizedgreedy algorithms described in [1, Sect. 2], [5, Sect. 2].In Section 3 we present new sizes of complete arcs in P G (2 , q ) with169 ≤ q ≤
349 and q = 1013 , K -ARCS IN P G (2 , Q ), 853 ≤ Q ≤ k -arcs in P G (2 , q ) , ≤ q ≤ In the plane
P G (2 , q ), we denote t (2 , q ) the smallest known size of completearcs. For q ≤ t (2 , q ) < √ q are collected in [2, Tab. 1].In Tables 1 and 2, the values of t (2 , q ) for 853 ≤ q ≤ q =3511 , , , , , , , A q = (cid:4) . √ q − t (2 , q ) (cid:5) , B q a superior approximation of t (2 , q ) / √ q . Also, C q = (cid:4) √ q − t (2 , q ) (cid:5) . For all q in Table 1 and q = 2401 , , t (2 , q ) < . √ q .In [7], complete k -arcs are obtained with k = 4( √ q − q = p odd, q ≤ q = 2401. For even q = 2 h , 10 ≤ h ≤
15, the smallest known sizes of complete k -arcs in P G (2 , q ) are obtained in [3], see also [2, p. 35]. They are as follows: k = 124 , , , , , , for q = 2 , , , , , , respectively.Also, 6( √ q − , q ) , q = 4 h +1 , are constructed in [4]; for h ≤ , ) . InTables 1 and 2, we use the results of [3, 7] for q = 31 , , , , , .The rest of sizes k for small complete arcs in Tables 1 and 2 is obtained in thiswork by computer search with the help of the randomized greedy algorithms.From Tables 1 and 2, we obtain Theorems 1 and 2. Theorem 1. In P G (2 , q ) ,t (2 , q ) < . √ q for q ≤ , q = 2401 , , t (2 , q ) < . √ q for q ≤ , q = 1181 , , , , t (2 , q ) < . √ q for q ≤ , q = 1459 , , , , , , , , t (2 , q ) < . √ q for q ≤ , q = 1867 , , , , , , , . Theorem 2. In P G (2 , q ) ,t (2 , q ) < . √ q for q ≤ , q = 2401 , , t (2 , q ) < . √ q − for q ≤ , q = 1181 , , , , , , , , t (2 , q ) < . √ q − for q ≤ , q = 1429 , , , , , , , , , , , t (2 , q ) < . √ q − for q ≤ , q = 1699 , , , , t (2 , q ) < . √ q − for q ≤ , q = 2017 , , . Our methods allow us to obtain small arcs in
P G (2 , q ) for q ≤ c be a constant independent of q . Let t ( P q ) be the size of the smallestcomplete arc in any projective plane P q of order q . In [12], for sufficiently large Table 1. The smallest known sizes t = t (2 , q ) < . √ q of complete arcs inplanes P G (2 , q ). A q = (cid:4) . √ q − t (2 , q ) (cid:5) , B q > t (2 , q ) / √ qq t A q B q q t A q B q q t A q B q
853 118 13 4 .
05 1087 137 11 4 .
16 1327 155 8 4 . .
07 1091 138 10 4 .
18 1331 155 9 4 . .
07 1093 138 10 4 .
18 1361 157 9 4 . .
06 1097 138 11 4 .
17 1367 158 8 4 . .
06 1103 138 11 4 .
16 1369 144 22 3 . .
08 1109 138 11 4 .
15 1373 158 8 4 . .
08 1117 140 10 4 .
19 1381 159 8 4 . .
07 1123 139 11 4 .
15 1399 160 8 4 . .
09 1129 140 11 4 .
17 1409 160 8 4 . .
08 1151 142 10 4 .
19 1423 161 8 4 . .
10 1153 142 10 4 .
19 1427 162 7 4 . .
11 1163 143 10 4 .
20 1429 161 9 4 . .
12 1171 144 9 4 .
21 1433 161 9 4 . .
11 1181 144 10 4 .
20 1439 161 9 4 . .
13 1187 145 10 4 .
21 1447 162 9 4 . .
12 1193 145 10 4 .
20 1451 163 8 4 . .
88 1201 146 9 4 .
22 1453 164 7 4 . .
12 1213 147 9 4 .
23 1459 164 7 4 . .
11 1217 147 9 4 .
22 1471 164 8 4 . .
13 1223 147 10 4 .
21 1481 164 9 4 . .
12 1229 148 9 4 .
23 1483 165 8 4 . .
13 1231 148 9 4 .
22 1487 166 7 4 . .
12 1237 148 10 4 .
21 1489 166 7 4 . .
16 1249 149 10 4 .
22 1493 166 7 4 . .
12 1259 150 9 4 .
23 1499 166 8 4 . .
14 1277 151 9 4 .
23 1511 166 8 4 . .
14 1279 151 9 4 .
23 1523 168 7 4 . .
88 1283 152 9 4 .
25 1531 169 7 4 . .
12 1289 152 9 4 .
24 1543 169 7 4 . .
14 1291 152 9 4 .
24 1549 170 7 4 . .
16 1297 153 9 4 .
25 1553 170 7 4 . .
14 1301 153 9 4 .
25 1559 170 7 4 . .
17 1303 153 9 4 .
24 1567 171 7 4 . .
15 1307 153 9 4 .
24 1571 171 7 4 . .
18 1319 154 9 4 .
25 1579 172 6 4 . .
16 1321 154 9 4 .
24 1583 172 7 4 . K -ARCS IN P G (2 , Q ), 853 ≤ Q ≤ Table 1 (continue). The smallest known sizes t = t (2 , q ) < . √ q ofcomplete arcs in planes P G (2 , q ). A q = (cid:4) . √ q − t (2 , q ) (cid:5) , B q > t (2 , q ) / √ qq t A q B q q t A q B q q t A q B q .
33 1867 190 4 4 .
40 2129 206 1 4 . .
33 1871 191 3 4 .
42 2131 206 1 4 . .
35 1873 191 3 4 .
42 2137 206 2 4 . .
34 1877 191 3 4 .
41 2141 206 2 4 . .
34 1879 191 4 4 .
41 2143 207 1 4 . .
33 1889 191 4 4 .
40 2153 207 1 4 . .
33 1901 191 5 4 .
39 2161 207 2 4 . .
34 1907 192 4 4 .
40 2179 209 1 4 . .
35 1913 192 4 4 .
39 2187 209 1 4 . .
35 1931 194 3 4 .
42 2197 208 2 4 . .
35 1933 194 3 4 .
42 2203 209 2 4 . .
34 1949 194 4 4 .
40 2207 210 1 4 . .
34 1951 195 3 4 .
42 2209 210 1 4 . .
91 1973 196 3 4 .
42 2213 210 1 4 . .
36 1979 196 4 4 .
41 2221 210 2 4 . .
37 1987 197 3 4 .
42 2237 211 1 4 . .
35 1993 196 4 4 .
40 2239 211 1 4 . .
36 1997 198 3 4 .
44 2243 211 2 4 . .
37 1999 198 3 4 .
43 2251 212 1 4 . .
37 2003 198 3 4 .
43 2267 213 1 4 . .
38 2011 199 2 4 .
44 2269 213 1 4 . .
37 2017 199 3 4 .
44 2273 214 0 4 . .
36 2027 199 3 4 .
43 2281 214 0 4 . .
38 2029 200 2 4 .
45 2287 215 0 4 . .
37 2039 201 2 4 .
46 2293 215 0 4 . .
37 2048 201 2 4 .
45 2297 215 0 4 . .
34 2053 201 2 4 .
44 2309 215 1 4 . .
38 2063 202 2 4 .
45 2311 216 0 4 . .
38 2069 202 2 4 .
45 2333 217 0 4 . .
39 2081 203 2 4 .
45 2339 217 0 4 . .
40 2083 203 2 4 .
45 2341 217 0 4 . .
38 2087 203 2 4 .
45 2347 218 0 4 . .
40 2089 203 2 4 .
45 2351 218 0 4 . .
40 2099 204 2 4 .
46 2357 218 0 4 . .
40 2111 205 1 4 .
47 2371 218 1 4 . .
41 2113 205 1 4 .
46 2377 219 0 4 . Table 2. The smallest known sizes t = t (2 , q ) < √ q of complete arcs in planes P G (2 , q ). A q = (cid:4) . √ q − t (2 , q ) (cid:5) , B q > t (2 , q ) / √ q , C q = (cid:4) √ q − t (2 , q ) (cid:5) q t A q C q B q q t C q B q q t C q B q .
51 2551 229 23 4 .
54 2713 237 23 4 . .
51 2557 229 23 4 .
53 2719 238 22 4 . .
51 2579 230 23 4 .
53 2729 238 23 4 . .
52 2591 231 23 4 .
54 2731 238 23 4 . .
52 2593 231 23 4 .
54 2741 239 22 4 . .
92 2609 232 23 4 .
55 2749 239 23 4 . .
51 2617 233 22 4 .
56 2753 239 23 4 . .
50 2621 233 22 4 .
56 2767 241 22 4 . .
51 2633 232 24 4 .
53 2777 241 22 4 . .
50 2647 234 23 4 .
55 2789 241 23 4 . .
52 2657 233 24 4 .
53 2791 242 22 4 . .
51 2659 233 24 4 .
52 2797 241 23 4 . .
52 2663 235 23 4 .
56 2801 242 22 4 . .
51 2671 236 22 4 .
57 2803 242 22 4 . .
53 2677 236 22 4 .
57 2809 242 23 4 . .
53 2683 236 22 4 .
56 2819 242 23 4 . .
54 2687 236 23 4 .
56 2833 243 23 4 . .
53 2689 236 23 4 .
56 2837 244 22 4 . .
52 2693 237 22 4 .
57 2843 244 22 4 . .
53 2699 237 22 4 .
57 2851 244 22 4 . .
53 2707 237 23 4 .
56 2857 245 22 4 . .
54 2711 237 23 4 .
56 2861 245 22 4 . . q , it is proved that t ( P q ) ≤ √ q log c q , c = 300. The logarithm basis is not notedas the estimate is asymptotic. For definiteness, we use the binary logarithms.We introduce D q ( c ) and D q ( c ) as follows: t (2 , q ) = D q ( c ) √ q log c q, t (2 , q ) = D q ( c ) √ q log c q. From Tables 1, 2 and [2, Tab. 1], we obtain Observation 1.
Observation 1.
Let 173 ≤ q ≤ q = 5 , , , , , , , .Then (i) . > D q (1) > . . > D q (1) if 467 ≤ q ; 0 . > D q (1) if1013 ≤ q ; 0 . > D q (1) if 1399 ≤ q ; 0 . > D q (1) if 1889 ≤ q . So, D q (1) hasa tendency to decreasing. (ii) . < D q ( ) < . D q ( ) < .
27 if q ≤ D q ( ) < .
32 if q ≤ D q ( ) < .
325 if q ≤ D q ( ) < .
335 if q ≤ D q ( ) has K -ARCS IN P G (2 , Q ), 853 ≤ Q ≤ a tendency to increasing. (iii) . < D q (0 . < . D q (0 .
75) oscillate about theaverage value 0 . . < D q (0 . < .
743 if 173 ≤ q ≤ , . < D q (0 . < .
741 if 1009 ≤ q ≤ , . < D q (0 . < .
738 if 2003 ≤ q ≤ . (1)Moreover, let b t (2 , q ) = 0 . √ q log . q, ∆ q = t (2 , q ) − b t (2 , q ) , P q = 100∆ q t (2 , q ) % . It holds that − . ≤ ∆ q ≤ . . (2) − . < P q < .
31% if 173 ≤ q ≤ , − . < P q < .
93% if 1009 ≤ q ≤ , − . < P q < .
54% if 2003 ≤ q ≤ . (3)In other words, b t (2 , q ) = 0 . √ q log . can be treated as a predicted value of t (2 , q ). Then ∆ q is the difference between the smallest known size t (2 , q ) of complete arcs and the predicted value. Finally, P q is this differencein percentage terms of the smallest known size.By (2),(3), the magnitude of the difference ∆ q is smaller than two. Themagnitude of the percentage value P q is smaller than two for q < q > P q is decreasing with growth of q .Also, by (1), the region of D q (0 .
75) is decreasing with growth of q .The graphs of values of D q (0 . q , and P q are shown on Figures 1-3.Examples for great q are given in Table 3.Table 3. The smallest known sizes t = t (2 , q ) < √ q of complete arcs inplanes P G (2 , q ) with great q . B q > t (2 , q ) / √ q , C q = (cid:4) √ q − t (2 , q ) (cid:5) q t C q B q D q (1) D q ( ) D q ( ) q t C q B q D q (1) D q ( ) D q ( )3511 278 18 4 .
70 0 .
398 1 .
367 0 . .
83 0 .
390 1 .
372 0 . .
72 0 .
393 1 .
362 0 . .
85 0 .
389 1 .
373 0 . .
79 0 .
394 1 .
374 0 . .
88 0 .
390 1 .
379 0 . .
83 0 .
392 1 .
375 0 . .
87 0 .
387 1 .
372 0 . Figure 1: The values of D q (0 .
75) = t (2 ,q ) √ q log . q , 173 ≤ q ≤ q = 5 , , , , , , , The examples confirm Observation 1. So, along with B q , the values D q ( c ), inparticular with c = 0 .
75, can be useful for estimates of complete arcs sizes.Note that a complete 302-arc of Table 3 improves the result of [3] for q = 2 .From Tables 1-3 and [2, Tab. 1], we obtain Theorem 3. Theorem 3.
Let ≤ q ≤ and q = 3511 , , , , , , , . Then t (2 , q ) < . √ q log . q. Taking into account (1) and Table 3, we assume that the following upperbound on the smallest size t (2 , q ) of complete arc in the plane P G (2 , q ) holds. Conjecture 1.
It holds that t (2 , q ) < . √ q log . q, ≤ q. P G (2 , Q ) Figure 2: The values of ∆ q = t (2 , q ) − . √ q log . q , 173 ≤ q ≤ q = 5 , , , , , , , P G (2 , q ) Let m (2 , q ) be the greatest size of complete arcs in P G (2 , q ). For odd q , m (2 , q ) = q + 1. For even q , m (2 , q ) = q + 2. For q = p there is the complete( q − √ q + 1)-arc [10]. For q odd there is a complete ( q + 5)-arc [13]. For q ≡ ≤ q ≤ q ≡ , q ≤
337 [6], there is acomplete ( q + 7)-arc. For even q ≥ ( q + 4)-arc [9].For even q , let M q = ( q + 4). For odd q , let M q = ( q + 7) if either q ≡ ≤ q ≤ q ≡ q ≤ M q = ( q + 5).Below we suppose that t (2 , q ) is given in [2, Tab. 1] for q ≤ q = 343,and in Tables 1 and 2 of this paper for 853 ≤ q ≤ Figure 3: The values of P q = q t (2 ,q ) %, 173 ≤ q ≤ q = 5 , , , , , , , have obtained the value t (2 , Theorem 4. In P G (2 , q ) with ≤ q ≤ , ≤ q ≤ , and q = 1013 , , there are complete k -arcs of all the sizes in the region t (2 , q ) ≤ k ≤ M q . In P G (2 , there are complete k -arcs of sizes k = 55 − , , , .Proof. For 25 ≤ q ≤
167 the assertion follows from [1, Tab. 2] and [2, Tab. 2].For 169 ≤ q ≤
349 and q = 1013 , Conjecture 2.
Let ≤ q ≤ be an odd prime. Then in P G (2 , q ) thereare complete k -arcs of all the sizes in the region t (2 , q ) ≤ k ≤ M q . Moreover,complete k -arcs with t (2 , q ) ≤ k ≤ ( q + 5) can be obtained by the randomizedgreedy algorithms of [1, 5] with a new approach to creation of starting data. Our methods are applicable using our present computers for q ≤ References [1] A. A. Davydov, G. Faina, S. Marcugini, and F. Pambianco, Computersearch in projective planes for the sizes of complete arcs,
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