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Article Contents

# Classification of $\mathbf{(3 \!\mod 5)}$ arcs in $\mathbf{ \operatorname{PG}(3,5)}$

• The proof of the non-existence of Griesmer $[104, 4, 82]_5$-codes is just one of many examples where extendability results are used. In a series of papers Landjev and Rousseva have introduced the concept of $(t\mod q)$-arcs as a general framework for extendability results for codes and arcs. Here we complete the known partial classification of $(3 \mod 5)$-arcs in $\operatorname{PG}(3,5)$ and uncover two missing, rather exceptional, examples disproving a conjecture of Landjev and Rousseva. As also the original non-existence proof of Griesmer $[104, 4, 82]_5$-codes is affected, we present an extended proof to fill this gap.

Mathematics Subject Classification: Primary: 51E22; Secondary: 51E21, 94B05.

 Citation:

• Table 1.  Number of isomorphism types of strong $(3\mod 5)$-arcs in $\operatorname{PG}(2,5)$ and their corresponding minihypers

 $\# \mathcal{K}$ $m$ $\# \mathcal{B}$ line mult. weights # isomorphism types 18 3 3 $0,1,2,3$ $0,1,2,3$ 4 23 4 9 $1,2,3,4$ $5,6,7,8$ 1 28 5 15 $2,3,4,5$ $10,11,12,13$ 1 33 6 21 $3,4,5,6$ $15,16,17,18$ 10 38 7 27 $4,5,6,7$ $20,21,22,23$ 23 43 8 33 $5,6,7,8$ $25,26,27,28$ 53 48 9 39 $6,7,8,9$ $30,31,32,33$ 49 53 10 45 $7,8,9,10$ $35,36,37,38$ 17 58 11 51 $8,9,10,11$ $40,41,42,43$ 11 63 12 57 $9,10,11,12$ $45,46,47,48$ 9 68 13 63 $10,11,12,13$ $50,51,52,53$ 6 73 14 69 $11,12,13,14$ $55,56,57,58$ 0 78 15 75 $12,13,14,15$ $60,61,62,63$ 0 83 16 81 $13,14,15,16$ $65,66,67,68$ 0 88 17 87 $14,15,16,17$ $70,71,72,73$ 0 93 18 93 $15,16,17,18$ $75,76,77,78$ 1

Table 2.  Different line types of strong $(3\mod 5)$-arcs in $\operatorname{PG}(2,5)$

 $\mathcal{K}(L)$ type of $L$ name 3 $(3,0,0,0,0,0)$ $A_1$ $(2,1,0,0,0,0)$ $A_2$ $(1,1,1,0,0,0)$ $A_3$ 8 $(3,3,2,0,0,0)$ $B_1$ $(3,3,1,1,0,0)$ $B_2$ $(3,2,2,1,0,0)$ $B_3$ $(3,2,1,1,1,0)$ $B_4$ $(3,1,1,1,1,1)$ $B_5$ $(2,2,2,2,0,0)$ $B_6$ $(2,2,2,1,1,0)$ $B_7$ $(2,2,1,1,1,1)$ $B_8$ 13 $(3,3,3,3,1,0)$ $C_1$ $(3,3,3,2,2,0)$ $C_2$ $(3,3,3,2,1,1)$ $C_3$ $(3,3,2,2,2,1)$ $C_4$ $(3,2,2,2,2,2)$ $C_5$ 18 $(3,3,3,3,3,3)$ $D_1$

Table 3.  Details for the computations for $178\le \# \mathcal{K}\le 273$

 $n$ 178 183 188 193 198 203 208 213 218 223 $\#\mathcal{E}_n$ 31 36 46 75 180 174 176 179 177 179 time in h 3078 351 998 972 1434 1787 2368 2661 3214 3110 $n$ 228 233 238 243 248 253 258 263 268 273 $\#\mathcal{E}_n$ 176 180 177 170 176 170 161 173 148 111 time in h 3477 3448 3396 3150 2848 2042 1752 855 911 683

Table 4.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $18$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 0 12 16 0 0 0 0 0 0 0 3 0 0 0 0 0 0 16 12 3 0 1 3 0 25 0 0 0 0 3 0 0 0 0 0 0 0 0 0 15 15 0 1 1 4 25 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 20 5 5 1 1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 25 0 0 6 1

Table 5.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $23$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 6 12 4 0 3 6 0 0 0 0 0 0 0 0 0 0 0 18 6 4 3 1

Table 6.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $28$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 6 0 10 0 15 0 0 0 0 0 0 0 0 0 0 0 0 15 10 0 6 1

Table 7.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $33$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 0 0 10 0 15 0 0 6 0 0 0 0 0 0 0 0 0 10 15 0 6 1 0 3 7 2 8 2 7 1 0 0 1 0 0 0 0 0 0 11 12 3 5 1 0 6 4 0 6 12 0 0 0 0 3 0 0 0 0 0 0 12 9 6 4 1 0 6 4 2 4 8 4 0 0 2 1 0 0 0 0 0 0 12 9 6 4 2 0 6 4 3 3 6 6 0 0 3 0 0 0 0 0 0 0 12 9 6 4 1 0 9 1 3 0 9 3 0 3 3 0 0 0 0 0 0 0 13 6 9 3 1 2 8 1 8 6 4 0 0 0 1 0 0 0 0 1 0 0 15 5 5 6 1 4 5 2 5 4 10 0 0 0 0 0 1 0 0 0 0 0 15 5 5 6 1 8 4 0 16 0 0 0 0 1 0 0 2 0 0 0 0 0 18 1 4 8 1

Table 8.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $38$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 0 0 4 0 0 0 0 0 3 18 6 0 0 0 0 0 0 6 12 13 0 1 0 0 5 0 0 6 12 2 0 2 3 1 0 0 0 0 0 7 14 6 4 1 0 1 4 0 0 10 4 1 0 8 2 0 0 1 0 0 0 8 11 9 3 1 0 1 4 0 0 9 6 0 1 6 3 0 0 1 0 0 0 8 11 9 3 1 0 2 3 0 0 6 9 0 1 7 2 0 1 0 0 0 0 8 11 9 3 1 0 2 4 0 12 0 8 3 0 0 0 1 1 0 0 0 0 9 13 2 7 1 0 2 4 4 5 4 8 0 0 2 0 0 0 2 0 0 0 10 10 5 6 2 0 3 2 0 0 8 2 0 4 10 1 0 0 0 1 0 0 9 8 12 2 1 0 3 3 2 6 6 8 0 0 0 1 1 0 0 1 0 0 10 10 5 6 1 0 4 2 4 2 10 3 0 1 3 0 0 0 1 1 0 0 11 7 8 5 1 0 5 0 0 0 5 0 0 10 10 0 0 0 0 0 1 0 10 5 15 1 1 0 5 1 2 4 12 0 1 0 4 0 0 1 0 1 0 0 11 7 8 5 1 0 5 1 3 3 9 4 0 1 3 0 0 1 0 1 0 0 11 7 8 5 1 0 6 0 4 0 12 0 0 6 0 1 0 0 0 2 0 0 12 4 11 4 1 1 1 4 2 4 7 9 0 0 1 0 1 0 1 0 0 0 10 10 5 6 1 1 2 3 3 1 13 2 0 1 3 0 0 0 2 0 0 0 11 7 8 5 1 1 3 2 2 1 13 4 0 1 2 0 1 0 0 1 0 0 11 7 8 5 1 1 4 1 0 4 14 0 1 0 4 0 0 2 0 0 0 0 11 7 8 5 1 1 4 1 1 3 11 4 0 1 3 0 0 2 0 0 0 0 11 7 8 5 1 2 5 0 10 2 7 0 0 2 0 0 0 2 1 0 0 0 14 3 7 7 2 3 0 4 3 15 0 3 0 0 0 0 3 0 0 0 0 0 12 9 1 9 1

Table 9.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $43$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 0 0 0 0 0 0 0 30 0 0 0 0 0 0 0 0 1 0 25 0 6 1 0 0 0 0 0 0 0 4 0 0 25 0 0 0 0 2 0 0 20 10 1 1 0 0 1 0 0 0 9 3 0 6 9 0 0 0 3 0 0 3 16 9 3 1 0 0 2 0 2 7 8 1 0 4 3 0 0 2 2 0 0 6 12 8 5 2 0 0 2 0 3 1 13 4 0 2 2 0 1 3 0 0 0 5 15 5 6 1 0 0 3 2 8 5 6 1 0 0 1 1 0 4 0 0 0 8 11 4 8 1 0 0 3 4 6 0 12 0 1 0 0 1 0 4 0 0 0 8 11 4 8 1 0 1 0 0 0 0 8 0 0 12 7 0 0 0 1 2 0 4 13 12 2 1 0 1 1 0 2 3 13 0 1 3 3 0 1 1 2 0 0 6 12 8 5 1 0 1 1 0 2 4 11 1 0 5 2 0 1 1 2 0 0 6 12 8 5 2 0 1 1 0 2 8 4 0 2 7 2 0 0 0 4 0 0 7 9 11 4 2 0 1 2 1 9 4 7 1 0 0 1 1 1 3 0 0 0 8 11 4 8 1 0 1 2 6 0 12 0 0 3 2 0 0 0 2 3 0 0 10 5 10 6 1 0 2 0 0 0 12 0 0 4 8 1 0 0 1 0 3 0 8 6 14 3 1 0 2 0 0 1 7 7 0 1 8 1 0 1 0 2 1 0 7 9 11 4 1 0 2 0 0 1 8 5 1 0 10 0 0 1 0 2 1 0 7 9 11 4 1 0 2 1 0 10 3 8 1 0 0 1 1 2 2 0 0 0 8 11 4 8 1 0 2 1 1 8 2 11 1 0 0 0 2 1 1 1 0 0 8 11 4 8 1 0 2 1 2 4 11 5 0 0 0 1 2 0 0 3 0 0 9 8 7 7 1 0 2 1 2 5 10 4 0 0 1 1 1 1 1 2 0 0 9 8 7 7 1 0 2 1 2 6 9 3 0 0 2 1 0 2 2 1 0 0 9 8 7 7 2 0 2 1 3 4 8 6 0 0 2 0 1 1 1 2 0 0 9 8 7 7 1 0 2 1 3 5 7 5 0 0 3 0 0 2 2 1 0 0 9 8 7 7 1 0 3 0 0 7 11 3 0 0 0 2 1 2 0 2 0 0 9 8 7 7 1 0 3 0 2 6 6 6 0 0 3 0 0 3 1 1 0 0 9 8 7 7 1 0 3 0 4 1 12 2 0 2 2 0 0 2 1 1 1 0 10 5 10 6 1 0 3 0 4 2 10 2 0 3 2 0 0 2 0 3 0 0 10 5 10 6 1 0 3 1 12 3 6 0 0 0 0 0 3 0 0 3 0 0 12 4 6 9 1 1 0 0 0 0 0 0 2 0 25 0 0 0 0 0 3 0 5 10 15 1 1 1 0 0 0 0 0 25 3 0 0 0 0 0 0 0 1 1 5 15 5 6 1 1 0 1 0 0 6 8 0 2 9 0 0 1 0 3 0 0 7 9 11 4 1 1 0 2 0 7 5 10 1 0 0 0 2 1 2 0 0 0 8 11 4 8 1 1 0 2 2 4 10 4 0 0 3 0 0 2 3 0 0 0 9 8 7 7 2 1 1 1 1 4 10 6 0 0 2 0 1 2 1 1 0 0 9 8 7 7 2 1 2 0 2 1 14 2 0 2 2 0 0 3 1 0 1 0 10 5 10 6 2 1 2 0 2 2 12 2 0 3 2 0 0 3 0 2 0 0 10 5 10 6 1 1 3 0 9 5 6 0 0 0 1 0 2 3 0 1 0 0 12 4 6 9 1 2 0 0 0 0 0 0 0 25 0 0 0 0 0 0 4 0 10 0 20 1 1 2 0 0 0 0 25 0 1 0 0 0 0 0 0 0 2 1 10 5 10 6 2 2 0 0 0 25 0 0 2 0 0 0 0 0 0 0 0 2 10 10 0 11 2 2 1 1 8 3 10 0 0 0 0 0 3 2 0 1 0 0 12 4 6 9 1 2 2 0 7 5 8 0 0 0 1 0 2 4 0 0 0 0 12 4 6 9 1 3 0 0 25 0 0 0 0 0 0 0 0 0 0 0 1 2 15 0 5 11 1 3 0 0 6 6 12 0 0 0 0 0 0 3 0 0 0 1 12 4 6 9 1

Table 10.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $48$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 0 0 0 0 12 3 6 3 0 0 0 0 0 6 0 0 1 6 12 3 10 1 0 0 0 0 2 11 0 0 2 6 2 0 0 1 5 2 0 6 7 13 5 1 0 0 0 0 2 4 12 1 0 0 4 0 0 6 1 1 0 4 13 7 7 1 0 0 0 0 2 8 6 0 0 4 3 0 0 3 4 1 0 5 10 10 6 1 0 0 0 0 3 3 10 2 0 2 3 0 0 5 3 0 0 4 13 7 7 1 0 0 0 0 3 6 6 0 1 4 3 0 0 2 6 0 0 5 10 10 6 1 0 0 0 1 0 11 2 0 1 7 1 0 0 2 3 3 0 6 7 13 5 1 0 0 0 1 2 1 12 2 0 3 2 0 0 5 3 0 0 4 13 7 7 1 0 0 0 1 2 4 8 0 1 5 2 0 0 2 6 0 0 5 10 10 6 1 0 0 0 1 2 5 6 1 0 7 1 0 0 2 6 0 0 5 10 10 6 1 0 0 0 1 2 7 2 0 3 7 1 0 0 0 7 1 0 6 7 13 5 1 0 0 0 2 0 4 10 0 0 6 1 0 0 3 4 1 0 5 10 10 6 1 0 0 0 2 0 7 4 0 2 8 0 0 0 1 5 2 0 6 7 13 5 1 0 0 0 2 0 8 0 0 7 6 0 0 0 0 4 4 0 7 4 16 4 1 0 0 0 3 6 6 9 0 0 0 0 0 0 3 3 0 1 7 9 6 9 1 0 0 0 4 2 14 4 0 0 0 0 0 0 1 4 1 1 8 6 9 8 1 0 0 1 0 9 3 6 3 0 0 0 3 0 6 0 0 0 6 12 3 10 1 0 0 1 1 8 7 2 0 0 1 2 0 3 6 0 0 0 7 9 6 9 1 0 0 1 2 5 7 6 0 0 0 1 2 1 4 2 0 0 7 9 6 9 1 0 0 1 3 4 5 8 0 0 1 0 2 1 4 2 0 0 7 9 6 9 1 0 0 1 3 5 4 7 0 0 2 0 1 2 5 1 0 0 7 9 6 9 2 0 0 1 3 6 3 6 0 0 3 0 0 3 6 0 0 0 7 9 6 9 2 0 0 1 4 2 10 2 0 1 2 0 1 2 2 4 0 0 8 6 9 8 2 0 0 1 4 3 9 1 0 1 3 0 0 3 3 3 0 0 8 6 9 8 2 0 0 1 6 0 9 0 0 6 0 0 0 3 0 6 0 0 9 3 12 7 1 0 0 2 6 12 0 0 0 0 0 1 6 0 4 0 0 0 9 8 2 12 1 0 1 0 1 6 6 7 0 0 0 1 2 2 3 2 0 0 7 9 6 9 1 0 1 0 1 7 5 6 0 0 1 1 1 3 4 1 0 0 7 9 6 9 1 0 1 0 2 3 13 0 1 0 2 0 2 2 0 5 0 0 8 6 9 8 1 0 1 0 2 5 4 9 0 0 1 0 2 2 3 2 0 0 7 9 6 9 1 0 1 0 3 1 12 4 0 0 1 0 2 2 1 3 1 0 8 6 9 8 1 0 1 0 3 2 11 3 0 0 2 0 1 3 2 2 1 0 8 6 9 8 1 0 1 0 3 3 10 2 0 0 3 0 0 4 3 1 1 0 8 6 9 8 1 0 1 0 3 3 9 3 0 1 2 0 1 3 1 4 0 0 8 6 9 8 2 0 1 0 3 4 8 2 0 1 3 0 0 4 2 3 0 0 8 6 9 8 2 0 1 0 5 0 10 1 0 5 0 0 0 4 0 4 1 0 9 3 12 7 1 0 2 0 12 0 6 0 0 1 0 0 2 6 1 0 1 0 11 2 8 10 1 1 0 0 0 6 5 8 0 0 2 0 1 4 4 0 0 0 7 9 6 9 1 1 0 0 1 4 10 2 0 1 3 0 0 5 2 2 0 0 8 6 9 8 1 1 0 0 4 14 0 4 0 0 0 0 4 1 2 0 0 1 9 8 2 12 1 1 0 1 4 11 0 4 0 0 0 0 7 1 2 0 0 0 9 8 2 12 1 2 0 0 8 1 8 0 0 2 0 0 2 8 0 0 0 0 11 2 8 10 2

Table 11.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $53$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 0 0 0 0 5 10 0 0 0 0 2 2 4 4 4 0 0 6 7 8 10 1 0 0 0 0 6 4 5 1 0 0 1 3 2 8 1 0 0 5 10 5 11 1 0 0 0 1 3 9 3 0 0 0 1 3 3 3 5 0 0 6 7 8 10 1 0 0 0 1 4 3 8 1 0 0 0 4 1 7 2 0 0 5 10 5 11 1 0 0 0 1 6 0 8 0 1 0 1 2 3 9 0 0 0 5 10 5 11 1 0 0 0 2 2 7 5 0 0 1 0 3 3 3 5 0 0 6 7 8 10 1 0 0 0 2 3 6 4 0 0 2 0 2 4 4 4 0 0 6 7 8 10 2 0 0 0 3 0 11 1 0 1 1 0 1 6 2 3 2 0 7 4 11 9 1 0 0 0 3 2 8 0 0 2 2 0 0 7 2 4 1 0 7 4 11 9 2 0 0 0 9 3 6 0 0 0 0 0 3 6 0 3 0 1 9 3 7 12 1 0 0 1 9 0 6 0 0 0 0 0 6 6 0 3 0 0 9 3 7 12 1 0 1 0 1 12 0 0 2 0 0 0 11 0 4 0 0 0 7 9 1 14 1 0 1 0 8 2 4 0 0 0 1 0 5 8 0 2 0 0 9 3 7 12 1 1 0 0 10 0 0 0 0 5 0 0 0 15 0 0 0 0 10 0 10 11 1 1 0 0 6 2 6 0 0 0 1 0 5 9 0 1 0 0 9 3 7 12 1

Table 12.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $58$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 0 0 0 0 0 10 0 1 0 0 0 5 5 0 10 0 0 5 5 10 11 1 0 0 0 0 3 3 3 0 0 1 1 3 5 9 3 0 0 4 8 7 12 1 0 0 0 1 1 2 6 0 0 1 0 4 4 8 4 0 0 4 8 7 12 1 0 0 0 1 1 5 2 0 1 1 0 3 7 2 8 0 0 5 5 10 11 1 0 0 0 1 1 6 1 0 0 2 0 2 8 4 5 1 0 5 5 10 11 1 0 0 0 1 2 4 1 0 1 2 0 2 8 3 7 0 0 5 5 10 11 1 0 0 0 1 3 0 4 0 0 3 0 2 6 10 2 0 0 4 8 7 12 1 0 0 0 1 3 3 0 0 1 3 0 1 9 4 6 0 0 5 5 10 11 1 0 0 0 2 1 4 0 0 4 0 0 0 12 0 6 2 0 6 2 13 10 1 0 0 0 3 6 0 3 0 0 0 0 9 3 6 0 0 1 6 7 3 15 1 0 0 1 3 3 0 3 0 0 0 0 12 3 6 0 0 0 6 7 3 15 1

Table 13.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $63$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 0 0 0 0 0 0 0 0 0 0 5 0 0 15 10 1 0 0 10 10 11 1 0 0 0 0 0 0 0 0 0 3 2 1 2 6 15 2 0 1 7 13 10 1 0 0 0 0 0 0 0 0 1 4 0 0 6 2 12 6 0 2 4 16 9 1 0 0 0 0 0 1 4 1 0 0 0 4 2 14 4 0 1 2 9 6 14 1 0 0 0 0 0 3 3 0 0 0 0 3 6 6 9 0 1 3 6 9 13 1 0 0 0 0 0 6 0 0 0 0 0 0 12 3 6 3 1 4 3 12 12 1 0 0 0 4 1 2 0 0 0 0 0 4 14 0 4 0 2 6 2 8 15 1 0 0 1 0 0 0 0 0 0 3 0 3 9 9 6 0 0 3 6 9 13 1 0 0 1 0 0 0 0 0 3 0 0 3 12 0 12 0 0 4 3 12 12 1

Table 14.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $68$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 1 0 0 25 6 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 25 3 2 0 5 15 11 1 0 0 0 0 0 0 0 2 0 0 0 0 0 25 0 1 3 0 10 5 16 1 0 0 0 0 3 0 0 0 0 0 0 6 6 12 0 0 4 3 6 4 18 1 1 0 0 0 0 0 0 0 0 0 0 0 25 0 0 2 3 5 0 10 16 1 1 0 0 0 0 0 0 1 0 0 0 25 0 0 0 0 4 5 5 0 21 1

Table 15.  Strong $(3\mod 5)$-arcs in ${\rm{PG}}(2, 5)$ of cardinality $93$

 $A_1$ $A_2$ $A_3$ $B_1$ $B_2$ $B_3$ $B_4$ $B_5$ $B_6$ $B_7$ $B_8$ $C_1$ $C_2$ $C_3$ $C_4$ $C_5$ $D_1$ $\lambda_0$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\#$ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 31 0 0 0 31 1
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