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

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  • 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:

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  • 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
     | Show Table
    DownLoad: CSV

    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 $
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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