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><math>
<td>
x^{3}
<span class="htmlMath">x<sup>3</sup></span>
</math></td>
</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><math>
<td>
x^{3} + {\rm Tr}(x^{9})
<span class="htmlMath">x<sup>3</sup> + Tr(x<sup>9</sup>)</span>
</math></td>
</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><math>
<td>
x^{34} + x^{18} + x^{5}
<span class="htmlMath">x<sup>34</sup> + x<sup>18</sup> + x<sup>5</sup></span>
</math></td>
</td>
<td>4034</td>
<td>4034</td>
<td>210</td>
<td>210</td>
Line 185: Line 185:
<tr>
<tr>
<td>2.2</td>
<td>2.2</td>
<td><math>
<td>
x^{3} + x^{17} + x^{33} + x^{34}
<span class="htmlMath">x<sup>3</sup> + x<sup>17</sup> + x<sup>33</sup> + x<sup>34</sup></span>
</math></td>
</td>
<td>4040</td>
<td>4040</td>
<td>212</td>
<td>212</td>
Line 195: Line 195:
<tr>
<tr>
<td>3.1</td>
<td>3.1</td>
<td><math>
<td>
x^{5}
<span class="htmlMath">x<sup>5</sup></span>
</math></td>
</td>
<td>3708</td>
<td>3708</td>
<td>198</td>
<td>198</td>
Line 205: Line 205:
<tr>
<tr>
<td>4.1</td>
<td>4.1</td>
<td><math>
<td>
x^{9}
<span class="htmlMath">x<sup>9</sup></span>
</math></td>
</td>
<td>3610</td>
<td>3610</td>
<td>198</td>
<td>198</td>
Line 215: Line 215:
<tr>
<tr>
<td>5.1</td>
<td>5.1</td>
<td><math>
<td>
x^{13}
<span class="htmlMath">x<sup>13</sup></span>
</math></td>
</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><math>
<td>
x^{57}
<span class="htmlMath">x<sup>57</sup></span>
</math></td>
</td>
<td>4704</td>
<td>4704</td>
<td>436</td>
<td>436</td>
Line 235: Line 235:
<tr>
<tr>
<td>7.1</td>
<td>7.1</td>
<td><math>
<td>
x^{-1}
<span class="htmlMath">x<sup>-1</sup></span>
</math></td>
</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><math>
<td>
x^{65} + x^{10} + x^{3}
<span class="htmlMath">x<sup>65</sup> + x<sup>10</sup> + x<sup>3</sup></span>
</math></td>
</td>
<td>4038</td>
<td>4038</td>
<td>212</td>
<td>212</td>
Line 255: Line 255:
<tr>
<tr>
<td>9.1</td>
<td>9.1</td>
<td><math>
<td>
x^{3} + x^{9} + x^{18} + x^{66}
<span class="htmlMath">x<sup>3</sup> + x<sup>9</sup> + x<sup>18</sup> + x<sup>66</sup></span>
</math></td>
</td>
<td>4044</td>
<td>4044</td>
<td>212</td>
<td>212</td>
Line 265: Line 265:
<tr>
<tr>
<td>10.1</td>
<td>10.1</td>
<td><math>
<td>
x^{3} + x^{12} + x^{17} + x^{33}
<span class="htmlMath">x<sup>3</sup> + x<sup>12</sup> + x<sup>17</sup> + x<sup>33</sup></span>
</math></td>
</td>
<td>4048</td>
<td>4048</td>
<td>210</td>
<td>210</td>
Line 275: Line 275:
<tr>
<tr>
<td>10.2</td>
<td>10.2</td>
<td><math>
<td>
x^{3} + x^{17} + x^{20} + x^{34} + x^{66}
<span class="htmlMath">x<sup>3</sup> + x<sup>17</sup> + x<sup>20</sup> + x<sup>34</sup> + x<sup>66</sup></span>
</math></td>
</td>
<td>4040</td>
<td>4040</td>
<td>210</td>
<td>210</td>
Line 285: Line 285:
<tr>
<tr>
<td>11.1</td>
<td>11.1</td>
<td><math>
<td>
x^{3} + x^{20} + x^{34} + x^{66}
<span class="htmlMath">x<sup>3</sup> + x<sup>20</sup> + x<sup>34</sup> + x<sup>66</sup></span>
</math></td>
</td>
<td>4048</td>
<td>4048</td>
<td>210</td>
<td>210</td>
Line 295: Line 295:
<tr>
<tr>
<td>12.1</td>
<td>12.1</td>
<td><math>
<td>
x^{3} + x^{12} + x^{40} + x^{72}
<span class="htmlMath">x<sup>3</sup> + x<sup>12</sup> + x<sup>40</sup> + x<sup>72</sup></span>
</math></td>
</td>
<td>4048</td>
<td>4048</td>
<td>210</td>
<td>210</td>
Line 305: Line 305:
<tr>
<tr>
<td>13.1</td>
<td>13.1</td>
<td><math>
<td>
x^{3} + x^{5} + x^{10} + x^{33} + x^{34}
<span class="htmlMath">x<sup>3</sup> + x<sup>5</sup> + x<sup>10</sup> + x<sup>33</sup> + x<sup>34</sup></span>
</math></td>
</td>
<td>4040</td>
<td>4040</td>
<td>212</td>
<td>212</td>
Line 315: Line 315:
<tr>
<tr>
<td>14.1</td>
<td>14.1</td>
<td><math>
<td>
x^{3} + x^{6} + x^{34} + x^{40} + x^{72}
<span class="htmlMath">x<sup>3</sup> + x<sup>6</sup> + x<sup>34</sup> + x<sup>40</sup> + x<sup>72</sup></span>
</math></td>
</td>
<td>4048</td>
<td>4048</td>
<td>212</td>
<td>212</td>
Line 325: Line 325:
<tr>
<tr>
<td>14.2</td>
<td>14.2</td>
<td><math>
<td>
x^{3} + x^{5} + x^{6} + x^{12} + x^{33} + x^{34}
<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>
</math></td>
</td>
<td>4050</td>
<td>4050</td>
<td>210</td>
<td>210</td>
Line 335: Line 335:
<tr>
<tr>
<td>14.3</td>
<td>14.3</td>
<td><math>
<td>
u^{2}x^{96} + </math> <math>u^{78}x^{80} + </math> <math>u^{121}x^{72} + </math> <math>u^{49}x^{68} + </math> <math>u^{77}x^{66} + </math> <math>u^{29}x^{65} + </math> <math>u^{119}x^{48} + </math> <math>u^{117}x^{40} + </math> <math>u^{28}x^{36} + </math> <math>u^{107}x^{34} +u^{62}x^{33} +u^{125}x^{24} +u^{76}x^{20} +u^{84}x^{18} +u^{110}x^{17} +u^{49}x^{12} +u^{102}x^{10} +u^{69}x^{9} + </math> <math>u^{14}x^{6} + </math> <math>x^{5} + </math> <math>x^{3}
<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>
</math></td>
</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