Known switching classes of APN functions over GF(2^n) for n = 5,6,7,8: Difference between revisions
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<td rowspan="19"><math>7</math></td> | <td rowspan="19"><math>7</math></td> | ||
<td>1.1</td> | <td>1.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>3</sup></span> | ||
</ | </td> | ||
<td>3610</td> | <td>3610</td> | ||
<td>198</td> | <td>198</td> | ||
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<tr> | <tr> | ||
<td>1.2</td> | <td>1.2</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>3</sup> + Tr(x<sup>9</sup>)</span> | ||
</ | </td> | ||
<td>4026</td> | <td>4026</td> | ||
<td>212</td> | <td>212</td> | ||
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<tr> | <tr> | ||
<td>2.1</td> | <td>2.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>34</sup> + x<sup>18</sup> + x<sup>5</sup></span> | ||
</ | </td> | ||
<td>4034</td> | <td>4034</td> | ||
<td>210</td> | <td>210</td> | ||
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<tr> | <tr> | ||
<td>2.2</td> | <td>2.2</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>3</sup> + x<sup>17</sup> + x<sup>33</sup> + x<sup>34</sup></span> | ||
</ | </td> | ||
<td>4040</td> | <td>4040</td> | ||
<td>212</td> | <td>212</td> | ||
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<tr> | <tr> | ||
<td>3.1</td> | <td>3.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>5</sup></span> | ||
</ | </td> | ||
<td>3708</td> | <td>3708</td> | ||
<td>198</td> | <td>198</td> | ||
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<tr> | <tr> | ||
<td>4.1</td> | <td>4.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>9</sup></span> | ||
</ | </td> | ||
<td>3610</td> | <td>3610</td> | ||
<td>198</td> | <td>198</td> | ||
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<tr> | <tr> | ||
<td>5.1</td> | <td>5.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>13</sup></span> | ||
</ | </td> | ||
<td>4270</td> | <td>4270</td> | ||
<td>338</td> | <td>338</td> | ||
Line 225: | Line 225: | ||
<tr> | <tr> | ||
<td>6.1</td> | <td>6.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>57</sup></span> | ||
</ | </td> | ||
<td>4704</td> | <td>4704</td> | ||
<td>436</td> | <td>436</td> | ||
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<tr> | <tr> | ||
<td>7.1</td> | <td>7.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>-1</sup></span> | ||
</ | </td> | ||
<td>8128</td> | <td>8128</td> | ||
<td>4928</td> | <td>4928</td> | ||
Line 245: | Line 245: | ||
<tr> | <tr> | ||
<td>8.1</td> | <td>8.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>65</sup> + x<sup>10</sup> + x<sup>3</sup></span> | ||
</ | </td> | ||
<td>4038</td> | <td>4038</td> | ||
<td>212</td> | <td>212</td> | ||
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<tr> | <tr> | ||
<td>9.1</td> | <td>9.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>3</sup> + x<sup>9</sup> + x<sup>18</sup> + x<sup>66</sup></span> | ||
</ | </td> | ||
<td>4044</td> | <td>4044</td> | ||
<td>212</td> | <td>212</td> | ||
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<tr> | <tr> | ||
<td>10.1</td> | <td>10.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>3</sup> + x<sup>12</sup> + x<sup>17</sup> + x<sup>33</sup></span> | ||
</ | </td> | ||
<td>4048</td> | <td>4048</td> | ||
<td>210</td> | <td>210</td> | ||
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<tr> | <tr> | ||
<td>10.2</td> | <td>10.2</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>3</sup> + x<sup>17</sup> + x<sup>20</sup> + x<sup>34</sup> + x<sup>66</sup></span> | ||
</ | </td> | ||
<td>4040</td> | <td>4040</td> | ||
<td>210</td> | <td>210</td> | ||
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<tr> | <tr> | ||
<td>11.1</td> | <td>11.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>3</sup> + x<sup>20</sup> + x<sup>34</sup> + x<sup>66</sup></span> | ||
</ | </td> | ||
<td>4048</td> | <td>4048</td> | ||
<td>210</td> | <td>210</td> | ||
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<tr> | <tr> | ||
<td>12.1</td> | <td>12.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>3</sup> + x<sup>12</sup> + x<sup>40</sup> + x<sup>72</sup></span> | ||
</ | </td> | ||
<td>4048</td> | <td>4048</td> | ||
<td>210</td> | <td>210</td> | ||
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<tr> | <tr> | ||
<td>13.1</td> | <td>13.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>3</sup> + x<sup>5</sup> + x<sup>10</sup> + x<sup>33</sup> + x<sup>34</sup></span> | ||
</ | </td> | ||
<td>4040</td> | <td>4040</td> | ||
<td>212</td> | <td>212</td> | ||
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<tr> | <tr> | ||
<td>14.1</td> | <td>14.1</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>3</sup> + x<sup>6</sup> + x<sup>34</sup> + x<sup>40</sup> + x<sup>72</sup></span> | ||
</ | </td> | ||
<td>4048</td> | <td>4048</td> | ||
<td>212</td> | <td>212</td> | ||
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<tr> | <tr> | ||
<td>14.2</td> | <td>14.2</td> | ||
<td>< | <td> | ||
x | <span class="htmlMath">x<sup>3</sup> + x<sup>5</sup> + x<sup>6</sup> + x<sup>12</sup> + x<sup>33</sup> + x<sup>34</sup></span> | ||
</ | </td> | ||
<td>4050</td> | <td>4050</td> | ||
<td>210</td> | <td>210</td> | ||
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<tr> | <tr> | ||
<td>14.3</td> | <td>14.3</td> | ||
<td>< | <td> | ||
u | <span class="htmlMath">u<sup>2</sup>x<sup>96</sup> + u<sup>78</sup>x<sup>80</sup> + u<sup>121</sup>x<sup>72</sup> + u<sup>49</sup>x<sup>68</sup> + u<sup>77</sup>x<sup>66</sup> + u<sup>29</sup>x<sup>65</sup> + u<sup>119</sup>x<sup>48</sup> + u<sup>117</sup>x<sup>40</sup> + u<sup>28</sup>x<sup>36</sup> + u<sup>107</sup>x<sup>34</sup> +u<sup>62</sup>x<sup>33</sup> +u<sup>125</sup>x<sup>24</sup> +u<sup>76</sup>x<sup>20</sup> +u<sup>84</sup>x<sup>18</sup> +u<sup>110</sup>x<sup>17</sup> +u<sup>49</sup>x<sup>12</sup> +u<sup>102</sup>x<sup>10</sup> +u<sup>69</sup>x<sup>9</sup> + u<sup>14</sup>x<sup>6</sup> + x<sup>5</sup> + x<sup>3</sup></span> | ||
</ | </td> | ||
<td>4046</td> | <td>4046</td> | ||
<td>212</td> | <td>212</td> |
Revision as of 17:29, 11 July 2020
Known switching classes of APN functions over [math]\displaystyle{ \mathbb{F}_{2^5} }[/math], [math]\displaystyle{ \mathbb{F}_{2^6} }[/math], [math]\displaystyle{ \mathbb{F}_{2^7} }[/math] and [math]\displaystyle{ \mathbb{F}_{2^8} }[/math].
Also available is Magma code generating representatives from the switching classes.
[math]\displaystyle{ n }[/math] | [math]\displaystyle{ N^\circ }[/math] | [math]\displaystyle{ F(x) }[/math] | Γ-rank | Δ-rank | Aut(dev(GF))/22n | Aut(dev(GF))/22n |
---|---|---|---|---|---|---|
[math]\displaystyle{ 5 }[/math] | 1.1 | x3 | 330 | 42 | 4960 | 4960 |
1.2 | x5 | 330 | 42 | 4960 | 158720 | |
2.1 | x-1 | 496 | 232 | 310 | 310 | |
[math]\displaystyle{ 6 }[/math] | 1.1 | x3 | 1102 | 94 | 24192 | 48384 |
1.2 | x3 + u11x6 + ux9 | 1146 | 94 | 4032 | 8064 | |
2.1 | ux5 + x9 + u4x17 + ux18 + u4x20 + ux24 + u4x34 + ux40 | 1158 | 96 | 320 | 320 | |
2.2 | u7x3 + x5 + u3x9 + u4x10 + x17 + u6x18 | 1166 | 94 | 448 | 896 | |
2.3 | x3 + ux24 + x10 | 1166 | 96 | 896 | 896 | |
2.4 | x3 + u17(x17 + x18 + x20 + x24) | 1168 | 96 | 64 | 64 | |
2.5 | x3 + u11x5 + u13x9 + x17 + u11x33 + x48 | 1170 | 96 | 320 | 320 | |
2.6 | u25x5 + x9 + u38x12 + u25x18 + u25x36 | 1170 | 96 | 64 | 64 | |
2.7 | u40x5 + u10x6 + u62x20 + u35x33 + u15x34 + u29x48 | 1170 | 96 | 64 | 64 | |
2.8 | u34x6 + u52x9 + u48x12 + u6x20 + u9x33 + u23x34 + u25x40 | 1170 | 96 | 64 | 64 | |
2.9 | x9 + u4(x10 + x18) + u9(x12 + x20 + x40) | 1172 | 96 | 64 | 64 | |
2.10 | u52x3 + u47x5 + ux6 + u9x9 + u44x12 + u47x33 + u10x34 + u33x40 | 1172 | 96 | 64 | 64 | |
2.11 | u(x6 + x10 + x24 + x33) + x9 + u4x17 | 1174 | 96 | 64 | 64 | |
2.12 | x3 + u17(x17 + x18 + x20 + x24) + u14((u52x3 + u6x5 + u19x7 + u28x11 + u2x13)+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)2 + (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)4+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)8+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)16+ (u52x3 + u6x5 + u19x7 + u28x11 + u2x13)32+ (u2x)9 +(u2x)19 +(u2x)36 + x21+x42 | 1300 | 152 | 8 | 8 | |
[math]\displaystyle{ 7 }[/math] | 1.1 |
x3 |
3610 | 198 | 113792 | 113792 |
1.2 |
x3 + Tr(x9) |
4026 | 212 | 896 | 896 | |
2.1 |
x34 + x18 + x5 |
4034 | 210 | 896 | 896 | |
2.2 |
x3 + x17 + x33 + x34 |
4040 | 212 | 896 | 896 | |
3.1 |
x5 |
3708 | 198 | 113792 | 113792 | |
4.1 |
x9 |
3610 | 198 | 113792 | 14565376 | |
5.1 |
x13 |
4270 | 338 | 889 | 889 | |
6.1 |
x57 |
4704 | 436 | 889 | 889 | |
7.1 |
x-1 |
8128 | 4928 | 1778 | 1778 | |
8.1 |
x65 + x10 + x3 |
4038 | 212 | 896 | 896 | |
9.1 |
x3 + x9 + x18 + x66 |
4044 | 212 | 896 | 896 | |
10.1 |
x3 + x12 + x17 + x33 |
4048 | 210 | 896 | 896 | |
10.2 |
x3 + x17 + x20 + x34 + x66 |
4040 | 210 | 896 | 896 | |
11.1 |
x3 + x20 + x34 + x66 |
4048 | 210 | 896 | 896 | |
12.1 |
x3 + x12 + x40 + x72 |
4048 | 210 | 896 | 896 | |
13.1 |
x3 + x5 + x10 + x33 + x34 |
4040 | 212 | 896 | 896 | |
14.1 |
x3 + x6 + x34 + x40 + x72 |
4048 | 212 | 896 | 896 | |
14.2 |
x3 + x5 + x6 + x12 + x33 + x34 |
4050 | 210 | 896 | 896 | |
14.3 |
u2x96 + u78x80 + u121x72 + u49x68 + u77x66 + u29x65 + u119x48 + u117x40 + u28x36 + u107x34 +u62x33 +u125x24 +u76x20 +u84x18 +u110x17 +u49x12 +u102x10 +u69x9 + u14x6 + x5 + x3 |
4046 | 212 | 128 | 128 | |
[math]\displaystyle{ 8 }[/math] | 1.1 | [math]\displaystyle{ x^{3} }[/math] | 11818 | 420 | ||
1.2 | [math]\displaystyle{ x^{9} }[/math] | 12370 | 420 | |||
1.3 | [math]\displaystyle{ x^{3}+{\rm Tr}(x^{9}) }[/math] | 13800 | 432 | |||
1.4 | [math]\displaystyle{ x^{9}+{\rm Tr}(x^{3}) }[/math] | 13804 | 434 | |||
1.5 | [math]\displaystyle{ x^{3}+u^{245}x^{33}+u^{183}x^{66}+u^{21}x^{144} }[/math] | 13842 | 436 | |||
1.6 | [math]\displaystyle{ x^{3} + u^{65}x^{18}+u^{120}x^{66}+u^{135}x^{144} }[/math] | 13848 | 438 | |||
1.7 | [math]\displaystyle{ u^{188}x^{192} + }[/math] [math]\displaystyle{ u^{129}x^{144} + }[/math] [math]\displaystyle{ u^{172}x^{132} + }[/math] [math]\displaystyle{ u^{138}x^{129} + }[/math] [math]\displaystyle{ u^{74}x^{96} + }[/math] [math]\displaystyle{ u^{244}x^{72} + }[/math] [math]\displaystyle{ u^{22}x^{66} + }[/math] [math]\displaystyle{ u^{178}x^{48} + }[/math] [math]\displaystyle{ u^{150}x^{36} + }[/math] [math]\displaystyle{ u^{146}x^{33} + }[/math] [math]\displaystyle{ u^{6}x^{24} + }[/math] [math]\displaystyle{ u^{60}x^{18} + }[/math] [math]\displaystyle{ u^{80}x^{12} + }[/math] [math]\displaystyle{ u^{140}x^{9} + }[/math] [math]\displaystyle{ u^{221}x^{6} + }[/math] [math]\displaystyle{ u^{19}x^{3} }[/math] | 14034 | 438 | |||
1.8 | [math]\displaystyle{ u^{37}x^{192} + }[/math] [math]\displaystyle{ u^{110}x^{144} + }[/math] [math]\displaystyle{ u^{40}x^{132} + }[/math] [math]\displaystyle{ u^{53}x^{129} + }[/math] [math]\displaystyle{ u^{239}x^{96} + }[/math] [math]\displaystyle{ u^{235}x^{72} + }[/math] [math]\displaystyle{ u^{126}x^{66} + }[/math] [math]\displaystyle{ u^{215}x^{48} + }[/math] [math]\displaystyle{ u^{96}x^{36} + }[/math] [math]\displaystyle{ u^{29}x^{33} + }[/math] [math]\displaystyle{ u^{19}x^{24} + }[/math] [math]\displaystyle{ u^{14}x^{18} + }[/math] [math]\displaystyle{ u^{139}x^{12} + }[/math] [math]\displaystyle{ u^{230}x^{9} + }[/math] [math]\displaystyle{ u^{234}x^{6} + }[/math] [math]\displaystyle{ u^{228}x^{3} }[/math] | 14032 | 438 | |||
1.9 | [math]\displaystyle{ u^{242}x^{192} + }[/math] [math]\displaystyle{ u^{100}x^{144} + }[/math] [math]\displaystyle{ u^{66}x^{132} + }[/math] [math]\displaystyle{ u^{230}x^{129} + }[/math] [math]\displaystyle{ u^{202}x^{96} + }[/math] [math]\displaystyle{ u^{156}x^{72} + }[/math] [math]\displaystyle{ u^{254}x^{66} + }[/math] [math]\displaystyle{ u^{18}x^{48} + }[/math] [math]\displaystyle{ u^{44}x^{36} + }[/math] [math]\displaystyle{ u^{95}x^{33} + }[/math] [math]\displaystyle{ u^{100}x^{24} + }[/math] [math]\displaystyle{ u^{245}x^{18} + }[/math] [math]\displaystyle{ u^{174}x^{12} + }[/math] [math]\displaystyle{ u^{175}x^{9} + }[/math] [math]\displaystyle{ u^{247}x^{6} + }[/math] [math]\displaystyle{ u^{166}x^{3} }[/math] | 14036 | 438 | |||
1.10 | [math]\displaystyle{ u^{100}x^{192} + }[/math] [math]\displaystyle{ u^{83}x^{144} + }[/math] [math]\displaystyle{ u^{153}x^{132} + }[/math] [math]\displaystyle{ u^{65}x^{129} + }[/math] [math]\displaystyle{ u^{174}x^{96} + }[/math] [math]\displaystyle{ u^{136}x^{72} + }[/math] [math]\displaystyle{ u^{46}x^{66}+ u^{55}x^{48}+ u^{224}x^{36}+ u^{180}x^{33}+ u^{179}x^{24}+u^{226}x^{18}+ u^{54}x^{12}+ u^{168}x^{9}+ u^{89}x^{6}+ u^{56}x^{3} }[/math] | 14036 | 438 | |||
1.11 | [math]\displaystyle{ u^{77}x^{192} + }[/math] [math]\displaystyle{ u^{133}x^{144} + }[/math] [math]\displaystyle{ u^{47}x^{132} + }[/math] [math]\displaystyle{ u^{229}x^{129} + }[/math] [math]\displaystyle{ u^{23}x^{96} + }[/math] [math]\displaystyle{ u^{242}x^{72} + }[/math] [math]\displaystyle{ u^{242}x^{66} + }[/math] [math]\displaystyle{ u^{245}x^{48} + }[/math] [math]\displaystyle{ u^{212}x^{36} + }[/math] [math]\displaystyle{ u^{231}x^{33} + }[/math] [math]\displaystyle{ u^{174}x^{24} + }[/math] [math]\displaystyle{ u^{216}x^{18} + }[/math] [math]\displaystyle{ u^{96}x^{12} + }[/math] [math]\displaystyle{ u^{253}x^{9} + }[/math] [math]\displaystyle{ u^{154}x^{6} + }[/math] [math]\displaystyle{ u^{71}x^{3} }[/math] | 14032 | 438 | |||
1.12 | [math]\displaystyle{ u^{220}x^{192} + }[/math] [math]\displaystyle{ u^{94}x^{144} + }[/math] [math]\displaystyle{ u^{70}x^{132} + }[/math] [math]\displaystyle{ u^{159}x^{129} + }[/math] [math]\displaystyle{ u^{145}x^{96} + }[/math] [math]\displaystyle{ u^{160}x^{72} + }[/math] [math]\displaystyle{ u^{74}x^{66} + }[/math] [math]\displaystyle{ u^{184}x^{48} + }[/math] [math]\displaystyle{ u^{119}x^{36} + }[/math] [math]\displaystyle{ u^{106}x^{33} + }[/math] [math]\displaystyle{ u^{253}x^{24} + }[/math] [math]\displaystyle{ wx^{18} + }[/math] [math]\displaystyle{ u^{90}x^{12} + }[/math] [math]\displaystyle{ u^{169}x^{9} + }[/math] [math]\displaystyle{ u^{118}x^{6} + }[/math] [math]\displaystyle{ u^{187}x^{3} }[/math] | 14034 | 438 | |||
1.13 | [math]\displaystyle{ u^{98}x^{192} + }[/math] [math]\displaystyle{ u^{225}x^{144} + }[/math] [math]\displaystyle{ u^{111}x^{132} + }[/math] [math]\displaystyle{ u^{238}x^{129} + }[/math] [math]\displaystyle{ u^{182}x^{96} + }[/math] [math]\displaystyle{ u^{125}x^{72} + }[/math] [math]\displaystyle{ u^{196}x^{66} + }[/math] [math]\displaystyle{ u^{219}x^{48} + }[/math] [math]\displaystyle{ u^{189}x^{36} + }[/math] [math]\displaystyle{ u^{199}x^{33} + }[/math] [math]\displaystyle{ u^{181}x^{24} + }[/math] [math]\displaystyle{ u^{110}x^{18} + }[/math] [math]\displaystyle{ u^{19}x^{12} + }[/math] [math]\displaystyle{ u^{175}x^{9} + }[/math] [math]\displaystyle{ u^{133}x^{6} + }[/math] [math]\displaystyle{ u^{47}x^{3} }[/math] | 14030 | 438 | |||
1.14 | [math]\displaystyle{ u^{236}x^{192} + }[/math] [math]\displaystyle{ u^{212}x^{160} + }[/math] [math]\displaystyle{ u^{153}x^{144} + }[/math] [math]\displaystyle{ u^{185}x^{136} + }[/math] [math]\displaystyle{ u^{3}x^{132} + }[/math] [math]\displaystyle{ u^{89}x^{130} + }[/math] [math]\displaystyle{ u^{189}x^{129} + }[/math] [math]\displaystyle{ u^{182}x^{96} + }[/math] [math]\displaystyle{ u^{105}x^{80} + }[/math] [math]\displaystyle{ u^{232}x^{72} + }[/math] [math]\displaystyle{ u^{219}x^{68} + }[/math] [math]\displaystyle{ u^{145}x^{66} + }[/math] [math]\displaystyle{ u^{171}x^{65} + }[/math] [math]\displaystyle{ u^{107}x^{48} + }[/math] [math]\displaystyle{ u^{179}x^{40} + }[/math] [math]\displaystyle{ u^{227}x^{36} + }[/math] [math]\displaystyle{ u^{236}x^{34} + }[/math] [math]\displaystyle{ u^{189}x^{33} + }[/math] [math]\displaystyle{ u^{162}x^{24} + }[/math] [math]\displaystyle{ u^{216}x^{20} + }[/math] [math]\displaystyle{ u^{162}x^{18} + }[/math] [math]\displaystyle{ u^{117}x^{17} + }[/math] [math]\displaystyle{ u^{56}x^{12} + }[/math] [math]\displaystyle{ u^{107}x^{10} + }[/math] [math]\displaystyle{ u^{236}x^{9} + }[/math] [math]\displaystyle{ u^{253}x^{6} + }[/math] [math]\displaystyle{ u^{180}x^{5} + }[/math] [math]\displaystyle{ u^{18}x^{3} }[/math] | 14046 | 454 | |||
1.15 | [math]\displaystyle{ u^{27}x^{192} + }[/math] [math]\displaystyle{ u^{167}x^{144} + }[/math] [math]\displaystyle{ u^{26}x^{132} + }[/math] [math]\displaystyle{ u^{231}x^{129} + }[/math] [math]\displaystyle{ u^{139}x^{96} + }[/math] [math]\displaystyle{ u^{30}x^{72} + }[/math] [math]\displaystyle{ u^{139}x^{66} + }[/math] [math]\displaystyle{ u^{203}x^{48} + }[/math] [math]\displaystyle{ u^{36}x^{36} + }[/math] [math]\displaystyle{ u^{210}x^{33} + }[/math] [math]\displaystyle{ u^{195}x^{24} + }[/math] [math]\displaystyle{ u^{12}x^{18} + }[/math] [math]\displaystyle{ u^{43}x^{12} + }[/math] [math]\displaystyle{ u^{97}x^{9} + }[/math] [math]\displaystyle{ u^{61}x^{6} + }[/math] [math]\displaystyle{ u^{39}x^{3} }[/math] | 14036 | 454 | |||
1.16 | [math]\displaystyle{ u^{6}x^{192} + }[/math] [math]\displaystyle{ u^{85}x^{144} + }[/math] [math]\displaystyle{ u^{251}x^{132} + }[/math] [math]\displaystyle{ u^{215}x^{129} + }[/math] [math]\displaystyle{ u^{229}x^{96} + }[/math] [math]\displaystyle{ u^{195}x^{72} + }[/math] [math]\displaystyle{ u^{152}x^{66} + }[/math] [math]\displaystyle{ u^{173}x^{48} + }[/math] [math]\displaystyle{ u^{209}x^{36} + }[/math] [math]\displaystyle{ u^{165}x^{33} + }[/math] [math]\displaystyle{ u^{213}x^{24} + }[/math] [math]\displaystyle{ u^{214}x^{18} + }[/math] [math]\displaystyle{ u^{158}x^{12} + }[/math] [math]\displaystyle{ u^{146}x^{9} + }[/math] [math]\displaystyle{ x^{6} + }[/math] [math]\displaystyle{ u^{50}x^{3} }[/math] | 14032 | 438 | |||
1.17 | [math]\displaystyle{ u^{164}x^{192} + }[/math] [math]\displaystyle{ u^{224}x^{144} + }[/math] [math]\displaystyle{ u^{59}x^{132} + }[/math] [math]\displaystyle{ u^{124}x^{129} + }[/math] [math]\displaystyle{ u^{207}x^{96} + }[/math] [math]\displaystyle{ u^{211}x^{72} + }[/math] [math]\displaystyle{ u^{5}x^{66} + }[/math] [math]\displaystyle{ u^{26}x^{48} + }[/math] [math]\displaystyle{ u^{20}x^{36} + }[/math] [math]\displaystyle{ u^{101}x^{33} + }[/math] [math]\displaystyle{ u^{175}x^{24} + }[/math] [math]\displaystyle{ u^{241}x^{18} + }[/math] [math]\displaystyle{ x^{12} + }[/math] [math]\displaystyle{ u^{15}x^{9} + }[/math] [math]\displaystyle{ u^{217}x^{6} + }[/math] [math]\displaystyle{ u^{212}x^{3} }[/math] | 14028 | 438 | |||
2.1 | [math]\displaystyle{ x^{3}+ x^{17}+u^{16}(x^{18}+x^{33})+u^{15}x^{48} }[/math] | 13200 | 414 | |||
3.1 | [math]\displaystyle{ x^{3}+ u^{24}x^{6}+u^{182}x^{132}+u^{67}x^{192} }[/math] | 14024 | 438 | |||
4.1 | [math]\displaystyle{ x^{3}+x^{6}+x^{68}+x^{80}+x^{132}+x^{160} }[/math] | 14040 | 454 | |||
5.1 | [math]\displaystyle{ x^{3}+x^{5}+x^{18}+x^{40}+x^{66} }[/math] | 14044 | 446 | |||
6.1 | [math]\displaystyle{ x^{3}+x^{12}+x^{40}+x^{66}+x^{130} }[/math] | 14046 | 438 | |||
7.1 | [math]\displaystyle{ x^{57} }[/math] | 15358 | 960 |