Parameters:
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import re
import pandas as pd # Pandas for tables
from IPython.display import Latex
from IPython.display import HTML
def read_f(file_name):
"""Reading weight enumerators."""
with open(file_name, 'r') as fp:
wd = fp.read().split("]\n")[:-1] # "\n"
wd = np.array([list(map(int, re.findall(r"\d+", elem))) for elem in wd])
return wd
wd = read_f("weight_distrib_n2k8.txt") # Weight distribution
# print(wd.shape) # 256 entries: 255 for IPM codes and one for BKLC codes
alpha_all = ['$\\alpha^{%d}$'%i for i in np.arange(len(wd))]
d_all = np.zeros(len(wd))
B_all = np.zeros(len(wd))
for i in range(len(wd)):
d_all[i] = wd[i][2]
B_all[i] = wd[i][3]
See more setting of dataframe from https://mode.com/example-gallery/python_dataframe_styling/
# Set properties for th, td and caption elements in dataframe
th_props = [('font-size', '14px'), ('text-align', 'left'), ('font-weight', 'bold'), ('background-color', '#E0E0E0')]
td_props = [('font-size', '13px'), ('text-align', 'left'), ('min-width', '80px')]
cp_props = [('font-size', '16px'), ('text-align', 'center')]
# Set table styles
styles = [dict(selector="th", props=th_props), dict(selector="td", props=td_props), dict(selector="caption", props=cp_props)]
cm_1 = sns.light_palette("red", as_cmap=True)
cm_2 = sns.light_palette("purple", as_cmap=True, reverse=True)
df = pd.DataFrame({'$L_2$': alpha_all[:-1], '$d_{\mathcal{D}}^\perp$': d_all[:-1], '$B_{d_{\mathcal{D}}^\perp}$': B_all[:-1],
'Weight Enumerators': wd[:-1]})
pd.set_option('display.max_colwidth', 1000)
pd.set_option('display.width', 800)
(df.style
.background_gradient(cmap=cm_1, subset=['$d_{\mathcal{D}}^\perp$','$B_{d_{\mathcal{D}}^\perp}$' ])
.background_gradient(cmap=cm_2, subset=['$B_{d_{\mathcal{D}}^\perp}$' ])
.set_caption('Tab. I All linear codes for IPM with $n=2$ shares over $\mathbb{F}_{2^8}$.')
.set_table_styles(styles))
$L_2$ | $d_{\mathcal{D}}^\perp$ | $B_{d_{\mathcal{D}}^\perp}$ | Weight Enumerators | |
---|---|---|---|---|
0 | $\alpha^{0}$ | 2 | 8 | [0, 1, 2, 8, 4, 28, 6, 56, 8, 70, 10, 56, 12, 28, 14, 8, 16, 1] |
1 | $\alpha^{1}$ | 2 | 7 | [0, 1, 2, 7, 4, 21, 5, 8, 6, 35, 7, 32, 8, 35, 9, 48, 10, 21, 11, 32, 12, 7, 13, 8, 14, 1] |
2 | $\alpha^{2}$ | 2 | 6 | [0, 1, 2, 6, 4, 15, 5, 16, 6, 24, 7, 48, 8, 31, 9, 48, 10, 30, 11, 16, 12, 17, 14, 4] |
3 | $\alpha^{3}$ | 2 | 5 | [0, 1, 2, 5, 4, 10, 5, 20, 6, 24, 7, 48, 8, 41, 9, 40, 10, 33, 11, 16, 12, 12, 13, 4, 14, 2] |
4 | $\alpha^{4}$ | 2 | 4 | [0, 1, 2, 4, 4, 6, 5, 24, 6, 24, 7, 44, 8, 53, 9, 36, 10, 28, 11, 20, 12, 12, 13, 4] |
5 | $\alpha^{5}$ | 2 | 3 | [0, 1, 2, 3, 4, 3, 5, 24, 6, 29, 7, 38, 8, 57, 9, 46, 10, 23, 11, 18, 12, 11, 13, 2, 14, 1] |
6 | $\alpha^{6}$ | 2 | 2 | [0, 1, 2, 2, 4, 1, 5, 23, 6, 32, 7, 40, 8, 55, 9, 46, 10, 30, 11, 16, 12, 7, 13, 3] |
7 | $\alpha^{7}$ | 2 | 1 | [0, 1, 2, 1, 4, 1, 5, 23, 6, 36, 7, 40, 8, 51, 9, 46, 10, 33, 11, 16, 12, 3, 13, 3, 14, 2] |
8 | $\alpha^{8}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 34, 7, 36, 8, 55, 9, 46, 10, 28, 11, 20, 12, 5, 13, 1, 14, 2] |
9 | $\alpha^{9}$ | 4 | 5 | [0, 1, 4, 5, 5, 23, 6, 29, 7, 40, 8, 59, 9, 46, 10, 26, 11, 16, 12, 7, 13, 3, 14, 1] |
10 | $\alpha^{10}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 27, 7, 42, 8, 59, 9, 46, 10, 28, 11, 14, 12, 6, 13, 4, 14, 1] |
11 | $\alpha^{11}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 42, 8, 51, 9, 46, 10, 34, 11, 14, 12, 6, 13, 4] |
12 | $\alpha^{12}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
13 | $\alpha^{13}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 43, 8, 51, 9, 42, 10, 34, 11, 20, 12, 6, 15, 1] |
14 | $\alpha^{14}$ | 4 | 7 | [0, 1, 4, 7, 5, 22, 6, 28, 7, 44, 8, 51, 9, 40, 10, 36, 11, 20, 12, 5, 13, 2] |
15 | $\alpha^{15}$ | 4 | 7 | [0, 1, 4, 7, 5, 20, 6, 28, 7, 48, 8, 51, 9, 40, 10, 36, 11, 16, 12, 5, 13, 4] |
16 | $\alpha^{16}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
17 | $\alpha^{17}$ | 4 | 6 | [0, 1, 4, 6, 5, 19, 6, 32, 7, 46, 8, 51, 9, 44, 10, 30, 11, 18, 12, 6, 13, 1, 14, 2] |
18 | $\alpha^{18}$ | 3 | 1 | [0, 1, 3, 1, 4, 5, 5, 16, 6, 32, 7, 54, 8, 51, 9, 36, 10, 32, 11, 17, 12, 7, 13, 4] |
19 | $\alpha^{19}$ | 3 | 2 | [0, 1, 3, 2, 4, 6, 5, 12, 6, 31, 7, 54, 8, 55, 9, 42, 10, 24, 11, 16, 12, 10, 13, 2, 14, 1] |
20 | $\alpha^{20}$ | 3 | 3 | [0, 1, 3, 3, 4, 7, 5, 10, 6, 32, 7, 54, 8, 47, 9, 44, 10, 32, 11, 15, 12, 9, 13, 2] |
21 | $\alpha^{21}$ | 3 | 4 | [0, 1, 3, 4, 4, 7, 5, 9, 6, 32, 7, 52, 8, 47, 9, 46, 10, 32, 11, 16, 12, 9, 13, 1] |
22 | $\alpha^{22}$ | 3 | 5 | [0, 1, 3, 5, 4, 7, 5, 10, 6, 28, 7, 50, 8, 51, 9, 44, 10, 36, 11, 17, 12, 5, 13, 2] |
23 | $\alpha^{23}$ | 3 | 6 | [0, 1, 3, 6, 4, 7, 5, 10, 6, 28, 7, 44, 8, 51, 9, 52, 10, 36, 11, 14, 12, 5, 13, 2] |
24 | $\alpha^{24}$ | 3 | 7 | [0, 1, 3, 7, 4, 7, 5, 9, 6, 28, 7, 42, 8, 51, 9, 54, 10, 36, 11, 15, 12, 5, 13, 1] |
25 | $\alpha^{25}$ | 3 | 7 | [0, 1, 3, 7, 4, 6, 5, 8, 6, 30, 7, 44, 8, 51, 9, 54, 10, 34, 11, 13, 12, 6, 13, 2] |
26 | $\alpha^{26}$ | 3 | 6 | [0, 1, 3, 6, 4, 6, 5, 9, 6, 30, 7, 46, 8, 51, 9, 52, 10, 34, 11, 12, 12, 6, 13, 3] |
27 | $\alpha^{27}$ | 3 | 5 | [0, 1, 3, 5, 4, 5, 5, 10, 6, 32, 7, 50, 8, 51, 9, 44, 10, 32, 11, 17, 12, 7, 13, 2] |
28 | $\alpha^{28}$ | 3 | 5 | [0, 1, 3, 5, 4, 4, 5, 9, 6, 34, 7, 52, 8, 51, 9, 44, 10, 30, 11, 15, 12, 8, 13, 3] |
29 | $\alpha^{29}$ | 3 | 5 | [0, 1, 3, 5, 4, 3, 5, 10, 6, 35, 7, 48, 8, 53, 9, 50, 10, 28, 11, 11, 12, 7, 13, 4, 14, 1] |
30 | $\alpha^{30}$ | 3 | 5 | [0, 1, 3, 5, 4, 3, 5, 11, 6, 36, 7, 46, 8, 51, 9, 50, 10, 28, 11, 13, 12, 9, 13, 3] |
31 | $\alpha^{31}$ | 3 | 5 | [0, 1, 3, 5, 4, 4, 5, 11, 6, 34, 7, 48, 8, 51, 9, 44, 10, 30, 11, 19, 12, 8, 13, 1] |
32 | $\alpha^{32}$ | 3 | 5 | [0, 1, 3, 5, 4, 5, 5, 12, 6, 32, 7, 46, 8, 51, 9, 44, 10, 32, 11, 21, 12, 7] |
33 | $\alpha^{33}$ | 3 | 5 | [0, 1, 3, 5, 4, 6, 5, 14, 6, 30, 7, 40, 8, 51, 9, 50, 10, 34, 11, 19, 12, 6] |
34 | $\alpha^{34}$ | 3 | 5 | [0, 1, 3, 5, 4, 7, 5, 15, 6, 26, 7, 36, 8, 57, 9, 56, 10, 30, 11, 15, 12, 7, 13, 1] |
35 | $\alpha^{35}$ | 3 | 5 | [0, 1, 3, 5, 4, 8, 5, 16, 6, 24, 7, 34, 8, 57, 9, 56, 10, 32, 11, 17, 12, 6] |
36 | $\alpha^{36}$ | 3 | 4 | [0, 1, 3, 4, 4, 8, 5, 17, 6, 24, 7, 36, 8, 57, 9, 54, 10, 32, 11, 16, 12, 6, 13, 1] |
37 | $\alpha^{37}$ | 3 | 3 | [0, 1, 3, 3, 4, 7, 5, 17, 6, 26, 7, 40, 8, 57, 9, 52, 10, 30, 11, 13, 12, 7, 13, 3] |
38 | $\alpha^{38}$ | 3 | 2 | [0, 1, 3, 2, 4, 7, 5, 19, 6, 28, 7, 40, 8, 51, 9, 50, 10, 36, 11, 14, 12, 5, 13, 3] |
39 | $\alpha^{39}$ | 3 | 1 | [0, 1, 3, 1, 4, 5, 5, 21, 6, 33, 7, 42, 8, 47, 9, 42, 10, 38, 11, 21, 12, 3, 13, 1, 14, 1] |
40 | $\alpha^{40}$ | 4 | 5 | [0, 1, 4, 5, 5, 22, 6, 34, 7, 43, 8, 45, 9, 42, 10, 38, 11, 20, 12, 5, 15, 1] |
41 | $\alpha^{41}$ | 4 | 4 | [0, 1, 4, 4, 5, 22, 6, 36, 7, 44, 8, 45, 9, 40, 10, 36, 11, 20, 12, 6, 13, 2] |
42 | $\alpha^{42}$ | 4 | 4 | [0, 1, 4, 4, 5, 21, 6, 36, 7, 44, 8, 45, 9, 46, 10, 36, 11, 12, 12, 6, 13, 5] |
43 | $\alpha^{43}$ | 3 | 1 | [0, 1, 3, 1, 4, 4, 5, 22, 6, 36, 7, 38, 8, 45, 9, 48, 10, 36, 11, 17, 12, 6, 13, 2] |
44 | $\alpha^{44}$ | 3 | 2 | [0, 1, 3, 2, 4, 4, 5, 23, 6, 40, 7, 36, 8, 41, 9, 46, 10, 32, 11, 18, 12, 10, 13, 3] |
45 | $\alpha^{45}$ | 3 | 3 | [0, 1, 3, 3, 4, 5, 5, 21, 6, 38, 7, 36, 8, 41, 9, 48, 10, 34, 11, 17, 12, 9, 13, 3] |
46 | $\alpha^{46}$ | 3 | 4 | [0, 1, 3, 4, 4, 6, 5, 19, 6, 34, 7, 38, 8, 47, 9, 44, 10, 30, 11, 22, 12, 10, 13, 1] |
47 | $\alpha^{47}$ | 3 | 5 | [0, 1, 3, 5, 4, 8, 5, 16, 6, 30, 7, 40, 8, 47, 9, 46, 10, 34, 11, 19, 12, 8, 13, 2] |
48 | $\alpha^{48}$ | 3 | 6 | [0, 1, 3, 6, 4, 8, 5, 15, 6, 30, 7, 38, 8, 47, 9, 48, 10, 34, 11, 20, 12, 8, 13, 1] |
49 | $\alpha^{49}$ | 3 | 6 | [0, 1, 3, 6, 4, 7, 5, 15, 6, 34, 7, 38, 8, 41, 9, 48, 10, 38, 11, 20, 12, 7, 13, 1] |
50 | $\alpha^{50}$ | 3 | 6 | [0, 1, 3, 6, 4, 6, 5, 12, 6, 32, 7, 40, 8, 45, 9, 52, 10, 40, 11, 18, 12, 4] |
51 | $\alpha^{51}$ | 3 | 5 | [0, 1, 3, 5, 4, 6, 5, 13, 6, 32, 7, 42, 8, 45, 9, 50, 10, 40, 11, 17, 12, 4, 13, 1] |
52 | $\alpha^{52}$ | 3 | 4 | [0, 1, 3, 4, 4, 7, 5, 13, 6, 28, 7, 48, 8, 51, 9, 42, 10, 36, 11, 20, 12, 5, 13, 1] |
53 | $\alpha^{53}$ | 3 | 3 | [0, 1, 3, 3, 4, 8, 5, 15, 6, 29, 7, 54, 8, 49, 9, 30, 10, 34, 11, 23, 12, 6, 13, 3, 14, 1] |
54 | $\alpha^{54}$ | 3 | 2 | [0, 1, 3, 2, 4, 7, 5, 14, 6, 28, 7, 56, 8, 51, 9, 32, 10, 36, 11, 22, 12, 5, 13, 2] |
55 | $\alpha^{55}$ | 3 | 1 | [0, 1, 3, 1, 4, 7, 5, 15, 6, 24, 7, 58, 8, 63, 9, 30, 10, 24, 11, 21, 12, 9, 13, 3] |
56 | $\alpha^{56}$ | 4 | 10 | [0, 1, 4, 10, 5, 12, 6, 25, 7, 60, 8, 57, 9, 36, 10, 22, 11, 20, 12, 12, 14, 1] |
57 | $\alpha^{57}$ | 4 | 13 | [0, 1, 4, 13, 5, 7, 6, 28, 7, 53, 8, 63, 9, 38, 10, 20, 11, 26, 12, 3, 13, 3, 15, 1] |
58 | $\alpha^{58}$ | 4 | 13 | [0, 1, 4, 13, 5, 10, 6, 24, 7, 58, 8, 49, 9, 46, 10, 30, 11, 14, 12, 9, 14, 2] |
59 | $\alpha^{59}$ | 4 | 15 | [0, 1, 4, 15, 5, 8, 6, 30, 7, 43, 8, 53, 9, 54, 10, 26, 11, 20, 12, 3, 13, 2, 15, 1] |
60 | $\alpha^{60}$ | 4 | 15 | [0, 1, 4, 15, 5, 8, 6, 35, 7, 38, 8, 45, 9, 62, 10, 28, 11, 18, 12, 3, 13, 2, 14, 1] |
61 | $\alpha^{61}$ | 4 | 14 | [0, 1, 4, 14, 5, 9, 6, 38, 7, 34, 8, 43, 9, 68, 10, 26, 11, 14, 12, 6, 13, 3] |
62 | $\alpha^{62}$ | 4 | 13 | [0, 1, 4, 13, 5, 11, 6, 36, 7, 36, 8, 47, 9, 58, 10, 28, 11, 20, 12, 3, 13, 3] |
63 | $\alpha^{63}$ | 3 | 1 | [0, 1, 3, 1, 4, 10, 5, 14, 6, 30, 7, 48, 8, 43, 9, 50, 10, 34, 11, 15, 12, 10] |
64 | $\alpha^{64}$ | 3 | 2 | [0, 1, 3, 2, 4, 7, 5, 16, 6, 28, 7, 48, 8, 51, 9, 44, 10, 36, 11, 14, 12, 5, 13, 4] |
65 | $\alpha^{65}$ | 3 | 2 | [0, 1, 3, 2, 4, 6, 5, 16, 6, 28, 7, 48, 8, 57, 9, 44, 10, 28, 11, 14, 12, 8, 13, 4] |
66 | $\alpha^{66}$ | 3 | 2 | [0, 1, 3, 2, 4, 6, 5, 13, 6, 31, 7, 50, 8, 55, 9, 48, 10, 24, 11, 12, 12, 10, 13, 3, 14, 1] |
67 | $\alpha^{67}$ | 3 | 2 | [0, 1, 3, 2, 4, 5, 5, 15, 6, 32, 7, 52, 8, 51, 9, 38, 10, 32, 11, 18, 12, 7, 13, 3] |
68 | $\alpha^{68}$ | 3 | 2 | [0, 1, 3, 2, 4, 3, 5, 16, 6, 35, 7, 50, 8, 53, 9, 38, 10, 28, 11, 20, 12, 7, 13, 2, 14, 1] |
69 | $\alpha^{69}$ | 3 | 2 | [0, 1, 3, 2, 4, 3, 5, 16, 6, 35, 7, 50, 8, 53, 9, 38, 10, 28, 11, 20, 12, 7, 13, 2, 14, 1] |
70 | $\alpha^{70}$ | 3 | 2 | [0, 1, 3, 2, 4, 3, 5, 16, 6, 35, 7, 50, 8, 53, 9, 38, 10, 28, 11, 20, 12, 7, 13, 2, 14, 1] |
71 | $\alpha^{71}$ | 3 | 1 | [0, 1, 3, 1, 4, 4, 5, 18, 6, 33, 7, 50, 8, 53, 9, 36, 10, 30, 11, 21, 12, 6, 13, 2, 14, 1] |
72 | $\alpha^{72}$ | 4 | 6 | [0, 1, 4, 6, 5, 18, 6, 32, 7, 50, 8, 51, 9, 38, 10, 30, 11, 22, 12, 6, 14, 2] |
73 | $\alpha^{73}$ | 4 | 7 | [0, 1, 4, 7, 5, 19, 6, 30, 7, 46, 8, 53, 9, 44, 10, 26, 11, 18, 12, 11, 13, 1] |
74 | $\alpha^{74}$ | 4 | 8 | [0, 1, 4, 8, 5, 19, 6, 27, 7, 46, 8, 55, 9, 44, 10, 28, 11, 18, 12, 8, 13, 1, 14, 1] |
75 | $\alpha^{75}$ | 4 | 7 | [0, 1, 4, 7, 5, 20, 6, 29, 7, 44, 8, 55, 9, 44, 10, 26, 11, 20, 12, 9, 14, 1] |
76 | $\alpha^{76}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 28, 7, 44, 8, 57, 9, 40, 10, 28, 11, 20, 12, 8, 13, 2] |
77 | $\alpha^{77}$ | 4 | 5 | [0, 1, 4, 5, 5, 24, 6, 30, 7, 38, 8, 57, 9, 46, 10, 26, 11, 18, 12, 9, 13, 2] |
78 | $\alpha^{78}$ | 4 | 5 | [0, 1, 4, 5, 5, 23, 6, 29, 7, 40, 8, 59, 9, 46, 10, 26, 11, 16, 12, 7, 13, 3, 14, 1] |
79 | $\alpha^{79}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 42, 8, 51, 9, 46, 10, 34, 11, 14, 12, 6, 13, 4] |
80 | $\alpha^{80}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
81 | $\alpha^{81}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
82 | $\alpha^{82}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 29, 7, 44, 8, 53, 9, 46, 10, 34, 11, 12, 12, 4, 13, 5, 14, 1] |
83 | $\alpha^{83}$ | 4 | 8 | [0, 1, 4, 8, 5, 21, 6, 24, 7, 44, 8, 57, 9, 46, 10, 32, 11, 12, 12, 6, 13, 5] |
84 | $\alpha^{84}$ | 4 | 9 | [0, 1, 4, 9, 5, 20, 6, 20, 7, 48, 8, 63, 9, 40, 10, 28, 11, 16, 12, 7, 13, 4] |
85 | $\alpha^{85}$ | 4 | 10 | [0, 1, 4, 10, 5, 18, 6, 18, 7, 51, 8, 63, 9, 42, 10, 30, 11, 12, 12, 6, 13, 4, 15, 1] |
86 | $\alpha^{86}$ | 4 | 9 | [0, 1, 4, 9, 5, 20, 6, 20, 7, 48, 8, 63, 9, 40, 10, 28, 11, 16, 12, 7, 13, 4] |
87 | $\alpha^{87}$ | 4 | 8 | [0, 1, 4, 8, 5, 19, 6, 24, 7, 49, 8, 57, 9, 42, 10, 32, 11, 14, 12, 6, 13, 3, 15, 1] |
88 | $\alpha^{88}$ | 4 | 7 | [0, 1, 4, 7, 5, 18, 6, 28, 7, 53, 8, 51, 9, 36, 10, 36, 11, 18, 12, 5, 13, 2, 15, 1] |
89 | $\alpha^{89}$ | 4 | 7 | [0, 1, 4, 7, 5, 16, 6, 34, 7, 54, 8, 41, 9, 38, 10, 38, 11, 18, 12, 7, 13, 2] |
90 | $\alpha^{90}$ | 4 | 6 | [0, 1, 4, 6, 5, 15, 6, 36, 7, 56, 8, 41, 9, 38, 10, 36, 11, 16, 12, 8, 13, 3] |
91 | $\alpha^{91}$ | 4 | 6 | [0, 1, 4, 6, 5, 18, 6, 36, 7, 48, 8, 41, 9, 44, 10, 36, 11, 16, 12, 8, 13, 2] |
92 | $\alpha^{92}$ | 4 | 6 | [0, 1, 4, 6, 5, 17, 6, 35, 7, 50, 8, 43, 9, 44, 10, 36, 11, 14, 12, 6, 13, 3, 14, 1] |
93 | $\alpha^{93}$ | 3 | 1 | [0, 1, 3, 1, 4, 5, 5, 18, 6, 32, 7, 50, 8, 51, 9, 36, 10, 32, 11, 21, 12, 7, 13, 2] |
94 | $\alpha^{94}$ | 3 | 2 | [0, 1, 3, 2, 4, 5, 5, 16, 6, 37, 7, 44, 8, 43, 9, 48, 10, 34, 11, 18, 12, 7, 14, 1] |
95 | $\alpha^{95}$ | 3 | 3 | [0, 1, 3, 3, 4, 4, 5, 16, 6, 36, 7, 43, 8, 45, 9, 48, 10, 36, 11, 17, 12, 6, 15, 1] |
96 | $\alpha^{96}$ | 3 | 4 | [0, 1, 3, 4, 4, 3, 5, 15, 6, 38, 7, 42, 8, 45, 9, 48, 10, 34, 11, 18, 12, 7, 13, 1] |
97 | $\alpha^{97}$ | 3 | 4 | [0, 1, 3, 4, 4, 4, 5, 19, 6, 40, 7, 36, 8, 41, 9, 50, 10, 32, 11, 16, 12, 10, 13, 3] |
98 | $\alpha^{98}$ | 3 | 4 | [0, 1, 3, 4, 4, 4, 5, 21, 6, 42, 7, 30, 8, 35, 9, 56, 10, 38, 11, 14, 12, 8, 13, 3] |
99 | $\alpha^{99}$ | 3 | 4 | [0, 1, 3, 4, 4, 4, 5, 21, 6, 42, 7, 30, 8, 35, 9, 56, 10, 38, 11, 14, 12, 8, 13, 3] |
100 | $\alpha^{100}$ | 3 | 4 | [0, 1, 3, 4, 4, 4, 5, 21, 6, 42, 7, 30, 8, 35, 9, 56, 10, 38, 11, 14, 12, 8, 13, 3] |
101 | $\alpha^{101}$ | 3 | 3 | [0, 1, 3, 3, 4, 5, 5, 22, 6, 39, 7, 32, 8, 37, 9, 54, 10, 40, 11, 13, 12, 5, 13, 4, 14, 1] |
102 | $\alpha^{102}$ | 3 | 2 | [0, 1, 3, 2, 4, 6, 5, 21, 6, 36, 7, 40, 8, 41, 9, 46, 10, 36, 11, 14, 12, 8, 13, 5] |
103 | $\alpha^{103}$ | 3 | 1 | [0, 1, 3, 1, 4, 7, 5, 16, 6, 37, 7, 42, 8, 39, 9, 56, 10, 34, 11, 13, 12, 9, 14, 1] |
104 | $\alpha^{104}$ | 4 | 7 | [0, 1, 4, 7, 5, 12, 6, 46, 7, 38, 8, 45, 9, 54, 10, 26, 11, 18, 12, 3, 13, 6] |
105 | $\alpha^{105}$ | 3 | 1 | [0, 1, 3, 1, 4, 6, 5, 12, 6, 38, 7, 52, 8, 43, 9, 50, 10, 26, 11, 11, 12, 14, 13, 2] |
106 | $\alpha^{106}$ | 3 | 1 | [0, 1, 3, 1, 4, 6, 5, 13, 6, 34, 7, 56, 8, 47, 9, 40, 10, 30, 11, 15, 12, 10, 13, 3] |
107 | $\alpha^{107}$ | 3 | 1 | [0, 1, 3, 1, 4, 7, 5, 13, 6, 32, 7, 58, 8, 47, 9, 34, 10, 32, 11, 21, 12, 9, 13, 1] |
108 | $\alpha^{108}$ | 3 | 1 | [0, 1, 3, 1, 4, 8, 5, 13, 6, 29, 7, 58, 8, 49, 9, 34, 10, 34, 11, 21, 12, 6, 13, 1, 14, 1] |
109 | $\alpha^{109}$ | 3 | 1 | [0, 1, 3, 1, 4, 8, 5, 13, 6, 29, 7, 58, 8, 49, 9, 34, 10, 34, 11, 21, 12, 6, 13, 1, 14, 1] |
110 | $\alpha^{110}$ | 3 | 1 | [0, 1, 3, 1, 4, 7, 5, 14, 6, 30, 7, 54, 8, 51, 9, 40, 10, 32, 11, 17, 12, 5, 13, 2, 14, 2] |
111 | $\alpha^{111}$ | 3 | 1 | [0, 1, 3, 1, 4, 8, 5, 15, 6, 25, 7, 52, 8, 59, 9, 40, 10, 28, 11, 19, 12, 4, 13, 1, 14, 3] |
112 | $\alpha^{112}$ | 3 | 1 | [0, 1, 3, 1, 4, 9, 5, 15, 6, 20, 7, 58, 8, 63, 9, 30, 10, 28, 11, 21, 12, 7, 13, 3] |
113 | $\alpha^{113}$ | 4 | 11 | [0, 1, 4, 11, 5, 14, 6, 24, 7, 54, 8, 53, 9, 42, 10, 30, 11, 18, 12, 7, 14, 2] |
114 | $\alpha^{114}$ | 4 | 11 | [0, 1, 4, 11, 5, 14, 6, 25, 7, 52, 8, 51, 9, 48, 10, 30, 11, 12, 12, 9, 13, 2, 14, 1] |
115 | $\alpha^{115}$ | 3 | 1 | [0, 1, 3, 1, 4, 9, 5, 16, 6, 26, 7, 48, 8, 53, 9, 46, 10, 30, 11, 15, 12, 9, 13, 2] |
116 | $\alpha^{116}$ | 3 | 2 | [0, 1, 3, 2, 4, 6, 5, 18, 6, 28, 7, 44, 8, 57, 9, 44, 10, 28, 11, 18, 12, 8, 13, 2] |
117 | $\alpha^{117}$ | 3 | 3 | [0, 1, 3, 3, 4, 6, 5, 18, 6, 28, 7, 38, 8, 57, 9, 52, 10, 28, 11, 15, 12, 8, 13, 2] |
118 | $\alpha^{118}$ | 3 | 3 | [0, 1, 3, 3, 4, 6, 5, 18, 6, 30, 7, 36, 8, 51, 9, 58, 10, 34, 11, 9, 12, 6, 13, 4] |
119 | $\alpha^{119}$ | 3 | 3 | [0, 1, 3, 3, 4, 4, 5, 18, 6, 36, 7, 38, 8, 45, 9, 52, 10, 36, 11, 15, 12, 6, 13, 2] |
120 | $\alpha^{120}$ | 3 | 3 | [0, 1, 3, 3, 4, 3, 5, 17, 6, 37, 7, 42, 8, 47, 9, 46, 10, 34, 11, 19, 12, 5, 13, 1, 14, 1] |
121 | $\alpha^{121}$ | 3 | 3 | [0, 1, 3, 3, 4, 4, 5, 16, 6, 34, 7, 44, 8, 51, 9, 46, 10, 30, 11, 17, 12, 8, 13, 2] |
122 | $\alpha^{122}$ | 3 | 3 | [0, 1, 3, 3, 4, 4, 5, 17, 6, 32, 7, 42, 8, 57, 9, 46, 10, 24, 11, 19, 12, 10, 13, 1] |
123 | $\alpha^{123}$ | 3 | 2 | [0, 1, 3, 2, 4, 5, 5, 18, 6, 29, 7, 44, 8, 59, 9, 44, 10, 26, 11, 18, 12, 7, 13, 2, 14, 1] |
124 | $\alpha^{124}$ | 3 | 1 | [0, 1, 3, 1, 4, 5, 5, 20, 6, 27, 7, 44, 8, 65, 9, 42, 10, 20, 11, 19, 12, 9, 13, 2, 14, 1] |
125 | $\alpha^{125}$ | 4 | 4 | [0, 1, 4, 4, 5, 23, 6, 31, 7, 42, 8, 59, 9, 40, 10, 24, 11, 22, 12, 8, 13, 1, 14, 1] |
126 | $\alpha^{126}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 33, 7, 36, 8, 59, 9, 46, 10, 22, 11, 20, 12, 9, 13, 1, 14, 1] |
127 | $\alpha^{127}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 34, 7, 36, 8, 55, 9, 46, 10, 28, 11, 20, 12, 5, 13, 1, 14, 2] |
128 | $\alpha^{128}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 34, 7, 36, 8, 55, 9, 46, 10, 28, 11, 20, 12, 5, 13, 1, 14, 2] |
129 | $\alpha^{129}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 33, 7, 36, 8, 59, 9, 46, 10, 22, 11, 20, 12, 9, 13, 1, 14, 1] |
130 | $\alpha^{130}$ | 4 | 4 | [0, 1, 4, 4, 5, 23, 6, 31, 7, 42, 8, 59, 9, 40, 10, 24, 11, 22, 12, 8, 13, 1, 14, 1] |
131 | $\alpha^{131}$ | 3 | 1 | [0, 1, 3, 1, 4, 5, 5, 20, 6, 27, 7, 44, 8, 65, 9, 42, 10, 20, 11, 19, 12, 9, 13, 2, 14, 1] |
132 | $\alpha^{132}$ | 3 | 2 | [0, 1, 3, 2, 4, 5, 5, 18, 6, 29, 7, 44, 8, 59, 9, 44, 10, 26, 11, 18, 12, 7, 13, 2, 14, 1] |
133 | $\alpha^{133}$ | 3 | 3 | [0, 1, 3, 3, 4, 4, 5, 17, 6, 32, 7, 42, 8, 57, 9, 46, 10, 24, 11, 19, 12, 10, 13, 1] |
134 | $\alpha^{134}$ | 3 | 3 | [0, 1, 3, 3, 4, 4, 5, 16, 6, 34, 7, 44, 8, 51, 9, 46, 10, 30, 11, 17, 12, 8, 13, 2] |
135 | $\alpha^{135}$ | 3 | 3 | [0, 1, 3, 3, 4, 3, 5, 17, 6, 37, 7, 42, 8, 47, 9, 46, 10, 34, 11, 19, 12, 5, 13, 1, 14, 1] |
136 | $\alpha^{136}$ | 3 | 3 | [0, 1, 3, 3, 4, 4, 5, 18, 6, 36, 7, 38, 8, 45, 9, 52, 10, 36, 11, 15, 12, 6, 13, 2] |
137 | $\alpha^{137}$ | 3 | 3 | [0, 1, 3, 3, 4, 6, 5, 18, 6, 30, 7, 36, 8, 51, 9, 58, 10, 34, 11, 9, 12, 6, 13, 4] |
138 | $\alpha^{138}$ | 3 | 3 | [0, 1, 3, 3, 4, 6, 5, 18, 6, 28, 7, 38, 8, 57, 9, 52, 10, 28, 11, 15, 12, 8, 13, 2] |
139 | $\alpha^{139}$ | 3 | 2 | [0, 1, 3, 2, 4, 6, 5, 18, 6, 28, 7, 44, 8, 57, 9, 44, 10, 28, 11, 18, 12, 8, 13, 2] |
140 | $\alpha^{140}$ | 3 | 1 | [0, 1, 3, 1, 4, 9, 5, 16, 6, 26, 7, 48, 8, 53, 9, 46, 10, 30, 11, 15, 12, 9, 13, 2] |
141 | $\alpha^{141}$ | 4 | 11 | [0, 1, 4, 11, 5, 14, 6, 25, 7, 52, 8, 51, 9, 48, 10, 30, 11, 12, 12, 9, 13, 2, 14, 1] |
142 | $\alpha^{142}$ | 4 | 11 | [0, 1, 4, 11, 5, 14, 6, 24, 7, 54, 8, 53, 9, 42, 10, 30, 11, 18, 12, 7, 14, 2] |
143 | $\alpha^{143}$ | 3 | 1 | [0, 1, 3, 1, 4, 9, 5, 15, 6, 20, 7, 58, 8, 63, 9, 30, 10, 28, 11, 21, 12, 7, 13, 3] |
144 | $\alpha^{144}$ | 3 | 1 | [0, 1, 3, 1, 4, 8, 5, 15, 6, 25, 7, 52, 8, 59, 9, 40, 10, 28, 11, 19, 12, 4, 13, 1, 14, 3] |
145 | $\alpha^{145}$ | 3 | 1 | [0, 1, 3, 1, 4, 7, 5, 14, 6, 30, 7, 54, 8, 51, 9, 40, 10, 32, 11, 17, 12, 5, 13, 2, 14, 2] |
146 | $\alpha^{146}$ | 3 | 1 | [0, 1, 3, 1, 4, 8, 5, 13, 6, 29, 7, 58, 8, 49, 9, 34, 10, 34, 11, 21, 12, 6, 13, 1, 14, 1] |
147 | $\alpha^{147}$ | 3 | 1 | [0, 1, 3, 1, 4, 8, 5, 13, 6, 29, 7, 58, 8, 49, 9, 34, 10, 34, 11, 21, 12, 6, 13, 1, 14, 1] |
148 | $\alpha^{148}$ | 3 | 1 | [0, 1, 3, 1, 4, 7, 5, 13, 6, 32, 7, 58, 8, 47, 9, 34, 10, 32, 11, 21, 12, 9, 13, 1] |
149 | $\alpha^{149}$ | 3 | 1 | [0, 1, 3, 1, 4, 6, 5, 13, 6, 34, 7, 56, 8, 47, 9, 40, 10, 30, 11, 15, 12, 10, 13, 3] |
150 | $\alpha^{150}$ | 3 | 1 | [0, 1, 3, 1, 4, 6, 5, 12, 6, 38, 7, 52, 8, 43, 9, 50, 10, 26, 11, 11, 12, 14, 13, 2] |
151 | $\alpha^{151}$ | 4 | 7 | [0, 1, 4, 7, 5, 12, 6, 46, 7, 38, 8, 45, 9, 54, 10, 26, 11, 18, 12, 3, 13, 6] |
152 | $\alpha^{152}$ | 3 | 1 | [0, 1, 3, 1, 4, 7, 5, 16, 6, 37, 7, 42, 8, 39, 9, 56, 10, 34, 11, 13, 12, 9, 14, 1] |
153 | $\alpha^{153}$ | 3 | 2 | [0, 1, 3, 2, 4, 6, 5, 21, 6, 36, 7, 40, 8, 41, 9, 46, 10, 36, 11, 14, 12, 8, 13, 5] |
154 | $\alpha^{154}$ | 3 | 3 | [0, 1, 3, 3, 4, 5, 5, 22, 6, 39, 7, 32, 8, 37, 9, 54, 10, 40, 11, 13, 12, 5, 13, 4, 14, 1] |
155 | $\alpha^{155}$ | 3 | 4 | [0, 1, 3, 4, 4, 4, 5, 21, 6, 42, 7, 30, 8, 35, 9, 56, 10, 38, 11, 14, 12, 8, 13, 3] |
156 | $\alpha^{156}$ | 3 | 4 | [0, 1, 3, 4, 4, 4, 5, 21, 6, 42, 7, 30, 8, 35, 9, 56, 10, 38, 11, 14, 12, 8, 13, 3] |
157 | $\alpha^{157}$ | 3 | 4 | [0, 1, 3, 4, 4, 4, 5, 21, 6, 42, 7, 30, 8, 35, 9, 56, 10, 38, 11, 14, 12, 8, 13, 3] |
158 | $\alpha^{158}$ | 3 | 4 | [0, 1, 3, 4, 4, 4, 5, 19, 6, 40, 7, 36, 8, 41, 9, 50, 10, 32, 11, 16, 12, 10, 13, 3] |
159 | $\alpha^{159}$ | 3 | 4 | [0, 1, 3, 4, 4, 3, 5, 15, 6, 38, 7, 42, 8, 45, 9, 48, 10, 34, 11, 18, 12, 7, 13, 1] |
160 | $\alpha^{160}$ | 3 | 3 | [0, 1, 3, 3, 4, 4, 5, 16, 6, 36, 7, 43, 8, 45, 9, 48, 10, 36, 11, 17, 12, 6, 15, 1] |
161 | $\alpha^{161}$ | 3 | 2 | [0, 1, 3, 2, 4, 5, 5, 16, 6, 37, 7, 44, 8, 43, 9, 48, 10, 34, 11, 18, 12, 7, 14, 1] |
162 | $\alpha^{162}$ | 3 | 1 | [0, 1, 3, 1, 4, 5, 5, 18, 6, 32, 7, 50, 8, 51, 9, 36, 10, 32, 11, 21, 12, 7, 13, 2] |
163 | $\alpha^{163}$ | 4 | 6 | [0, 1, 4, 6, 5, 17, 6, 35, 7, 50, 8, 43, 9, 44, 10, 36, 11, 14, 12, 6, 13, 3, 14, 1] |
164 | $\alpha^{164}$ | 4 | 6 | [0, 1, 4, 6, 5, 18, 6, 36, 7, 48, 8, 41, 9, 44, 10, 36, 11, 16, 12, 8, 13, 2] |
165 | $\alpha^{165}$ | 4 | 6 | [0, 1, 4, 6, 5, 15, 6, 36, 7, 56, 8, 41, 9, 38, 10, 36, 11, 16, 12, 8, 13, 3] |
166 | $\alpha^{166}$ | 4 | 7 | [0, 1, 4, 7, 5, 16, 6, 34, 7, 54, 8, 41, 9, 38, 10, 38, 11, 18, 12, 7, 13, 2] |
167 | $\alpha^{167}$ | 4 | 7 | [0, 1, 4, 7, 5, 18, 6, 28, 7, 53, 8, 51, 9, 36, 10, 36, 11, 18, 12, 5, 13, 2, 15, 1] |
168 | $\alpha^{168}$ | 4 | 8 | [0, 1, 4, 8, 5, 19, 6, 24, 7, 49, 8, 57, 9, 42, 10, 32, 11, 14, 12, 6, 13, 3, 15, 1] |
169 | $\alpha^{169}$ | 4 | 9 | [0, 1, 4, 9, 5, 20, 6, 20, 7, 48, 8, 63, 9, 40, 10, 28, 11, 16, 12, 7, 13, 4] |
170 | $\alpha^{170}$ | 4 | 10 | [0, 1, 4, 10, 5, 18, 6, 18, 7, 51, 8, 63, 9, 42, 10, 30, 11, 12, 12, 6, 13, 4, 15, 1] |
171 | $\alpha^{171}$ | 4 | 9 | [0, 1, 4, 9, 5, 20, 6, 20, 7, 48, 8, 63, 9, 40, 10, 28, 11, 16, 12, 7, 13, 4] |
172 | $\alpha^{172}$ | 4 | 8 | [0, 1, 4, 8, 5, 21, 6, 24, 7, 44, 8, 57, 9, 46, 10, 32, 11, 12, 12, 6, 13, 5] |
173 | $\alpha^{173}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 29, 7, 44, 8, 53, 9, 46, 10, 34, 11, 12, 12, 4, 13, 5, 14, 1] |
174 | $\alpha^{174}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
175 | $\alpha^{175}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
176 | $\alpha^{176}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 42, 8, 51, 9, 46, 10, 34, 11, 14, 12, 6, 13, 4] |
177 | $\alpha^{177}$ | 4 | 5 | [0, 1, 4, 5, 5, 23, 6, 29, 7, 40, 8, 59, 9, 46, 10, 26, 11, 16, 12, 7, 13, 3, 14, 1] |
178 | $\alpha^{178}$ | 4 | 5 | [0, 1, 4, 5, 5, 24, 6, 30, 7, 38, 8, 57, 9, 46, 10, 26, 11, 18, 12, 9, 13, 2] |
179 | $\alpha^{179}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 28, 7, 44, 8, 57, 9, 40, 10, 28, 11, 20, 12, 8, 13, 2] |
180 | $\alpha^{180}$ | 4 | 7 | [0, 1, 4, 7, 5, 20, 6, 29, 7, 44, 8, 55, 9, 44, 10, 26, 11, 20, 12, 9, 14, 1] |
181 | $\alpha^{181}$ | 4 | 8 | [0, 1, 4, 8, 5, 19, 6, 27, 7, 46, 8, 55, 9, 44, 10, 28, 11, 18, 12, 8, 13, 1, 14, 1] |
182 | $\alpha^{182}$ | 4 | 7 | [0, 1, 4, 7, 5, 19, 6, 30, 7, 46, 8, 53, 9, 44, 10, 26, 11, 18, 12, 11, 13, 1] |
183 | $\alpha^{183}$ | 4 | 6 | [0, 1, 4, 6, 5, 18, 6, 32, 7, 50, 8, 51, 9, 38, 10, 30, 11, 22, 12, 6, 14, 2] |
184 | $\alpha^{184}$ | 3 | 1 | [0, 1, 3, 1, 4, 4, 5, 18, 6, 33, 7, 50, 8, 53, 9, 36, 10, 30, 11, 21, 12, 6, 13, 2, 14, 1] |
185 | $\alpha^{185}$ | 3 | 2 | [0, 1, 3, 2, 4, 3, 5, 16, 6, 35, 7, 50, 8, 53, 9, 38, 10, 28, 11, 20, 12, 7, 13, 2, 14, 1] |
186 | $\alpha^{186}$ | 3 | 2 | [0, 1, 3, 2, 4, 3, 5, 16, 6, 35, 7, 50, 8, 53, 9, 38, 10, 28, 11, 20, 12, 7, 13, 2, 14, 1] |
187 | $\alpha^{187}$ | 3 | 2 | [0, 1, 3, 2, 4, 3, 5, 16, 6, 35, 7, 50, 8, 53, 9, 38, 10, 28, 11, 20, 12, 7, 13, 2, 14, 1] |
188 | $\alpha^{188}$ | 3 | 2 | [0, 1, 3, 2, 4, 5, 5, 15, 6, 32, 7, 52, 8, 51, 9, 38, 10, 32, 11, 18, 12, 7, 13, 3] |
189 | $\alpha^{189}$ | 3 | 2 | [0, 1, 3, 2, 4, 6, 5, 13, 6, 31, 7, 50, 8, 55, 9, 48, 10, 24, 11, 12, 12, 10, 13, 3, 14, 1] |
190 | $\alpha^{190}$ | 3 | 2 | [0, 1, 3, 2, 4, 6, 5, 16, 6, 28, 7, 48, 8, 57, 9, 44, 10, 28, 11, 14, 12, 8, 13, 4] |
191 | $\alpha^{191}$ | 3 | 2 | [0, 1, 3, 2, 4, 7, 5, 16, 6, 28, 7, 48, 8, 51, 9, 44, 10, 36, 11, 14, 12, 5, 13, 4] |
192 | $\alpha^{192}$ | 3 | 1 | [0, 1, 3, 1, 4, 10, 5, 14, 6, 30, 7, 48, 8, 43, 9, 50, 10, 34, 11, 15, 12, 10] |
193 | $\alpha^{193}$ | 4 | 13 | [0, 1, 4, 13, 5, 11, 6, 36, 7, 36, 8, 47, 9, 58, 10, 28, 11, 20, 12, 3, 13, 3] |
194 | $\alpha^{194}$ | 4 | 14 | [0, 1, 4, 14, 5, 9, 6, 38, 7, 34, 8, 43, 9, 68, 10, 26, 11, 14, 12, 6, 13, 3] |
195 | $\alpha^{195}$ | 4 | 15 | [0, 1, 4, 15, 5, 8, 6, 35, 7, 38, 8, 45, 9, 62, 10, 28, 11, 18, 12, 3, 13, 2, 14, 1] |
196 | $\alpha^{196}$ | 4 | 15 | [0, 1, 4, 15, 5, 8, 6, 30, 7, 43, 8, 53, 9, 54, 10, 26, 11, 20, 12, 3, 13, 2, 15, 1] |
197 | $\alpha^{197}$ | 4 | 13 | [0, 1, 4, 13, 5, 10, 6, 24, 7, 58, 8, 49, 9, 46, 10, 30, 11, 14, 12, 9, 14, 2] |
198 | $\alpha^{198}$ | 4 | 13 | [0, 1, 4, 13, 5, 7, 6, 28, 7, 53, 8, 63, 9, 38, 10, 20, 11, 26, 12, 3, 13, 3, 15, 1] |
199 | $\alpha^{199}$ | 4 | 10 | [0, 1, 4, 10, 5, 12, 6, 25, 7, 60, 8, 57, 9, 36, 10, 22, 11, 20, 12, 12, 14, 1] |
200 | $\alpha^{200}$ | 3 | 1 | [0, 1, 3, 1, 4, 7, 5, 15, 6, 24, 7, 58, 8, 63, 9, 30, 10, 24, 11, 21, 12, 9, 13, 3] |
201 | $\alpha^{201}$ | 3 | 2 | [0, 1, 3, 2, 4, 7, 5, 14, 6, 28, 7, 56, 8, 51, 9, 32, 10, 36, 11, 22, 12, 5, 13, 2] |
202 | $\alpha^{202}$ | 3 | 3 | [0, 1, 3, 3, 4, 8, 5, 15, 6, 29, 7, 54, 8, 49, 9, 30, 10, 34, 11, 23, 12, 6, 13, 3, 14, 1] |
203 | $\alpha^{203}$ | 3 | 4 | [0, 1, 3, 4, 4, 7, 5, 13, 6, 28, 7, 48, 8, 51, 9, 42, 10, 36, 11, 20, 12, 5, 13, 1] |
204 | $\alpha^{204}$ | 3 | 5 | [0, 1, 3, 5, 4, 6, 5, 13, 6, 32, 7, 42, 8, 45, 9, 50, 10, 40, 11, 17, 12, 4, 13, 1] |
205 | $\alpha^{205}$ | 3 | 6 | [0, 1, 3, 6, 4, 6, 5, 12, 6, 32, 7, 40, 8, 45, 9, 52, 10, 40, 11, 18, 12, 4] |
206 | $\alpha^{206}$ | 3 | 6 | [0, 1, 3, 6, 4, 7, 5, 15, 6, 34, 7, 38, 8, 41, 9, 48, 10, 38, 11, 20, 12, 7, 13, 1] |
207 | $\alpha^{207}$ | 3 | 6 | [0, 1, 3, 6, 4, 8, 5, 15, 6, 30, 7, 38, 8, 47, 9, 48, 10, 34, 11, 20, 12, 8, 13, 1] |
208 | $\alpha^{208}$ | 3 | 5 | [0, 1, 3, 5, 4, 8, 5, 16, 6, 30, 7, 40, 8, 47, 9, 46, 10, 34, 11, 19, 12, 8, 13, 2] |
209 | $\alpha^{209}$ | 3 | 4 | [0, 1, 3, 4, 4, 6, 5, 19, 6, 34, 7, 38, 8, 47, 9, 44, 10, 30, 11, 22, 12, 10, 13, 1] |
210 | $\alpha^{210}$ | 3 | 3 | [0, 1, 3, 3, 4, 5, 5, 21, 6, 38, 7, 36, 8, 41, 9, 48, 10, 34, 11, 17, 12, 9, 13, 3] |
211 | $\alpha^{211}$ | 3 | 2 | [0, 1, 3, 2, 4, 4, 5, 23, 6, 40, 7, 36, 8, 41, 9, 46, 10, 32, 11, 18, 12, 10, 13, 3] |
212 | $\alpha^{212}$ | 3 | 1 | [0, 1, 3, 1, 4, 4, 5, 22, 6, 36, 7, 38, 8, 45, 9, 48, 10, 36, 11, 17, 12, 6, 13, 2] |
213 | $\alpha^{213}$ | 4 | 4 | [0, 1, 4, 4, 5, 21, 6, 36, 7, 44, 8, 45, 9, 46, 10, 36, 11, 12, 12, 6, 13, 5] |
214 | $\alpha^{214}$ | 4 | 4 | [0, 1, 4, 4, 5, 22, 6, 36, 7, 44, 8, 45, 9, 40, 10, 36, 11, 20, 12, 6, 13, 2] |
215 | $\alpha^{215}$ | 4 | 5 | [0, 1, 4, 5, 5, 22, 6, 34, 7, 43, 8, 45, 9, 42, 10, 38, 11, 20, 12, 5, 15, 1] |
216 | $\alpha^{216}$ | 3 | 1 | [0, 1, 3, 1, 4, 5, 5, 21, 6, 33, 7, 42, 8, 47, 9, 42, 10, 38, 11, 21, 12, 3, 13, 1, 14, 1] |
217 | $\alpha^{217}$ | 3 | 2 | [0, 1, 3, 2, 4, 7, 5, 19, 6, 28, 7, 40, 8, 51, 9, 50, 10, 36, 11, 14, 12, 5, 13, 3] |
218 | $\alpha^{218}$ | 3 | 3 | [0, 1, 3, 3, 4, 7, 5, 17, 6, 26, 7, 40, 8, 57, 9, 52, 10, 30, 11, 13, 12, 7, 13, 3] |
219 | $\alpha^{219}$ | 3 | 4 | [0, 1, 3, 4, 4, 8, 5, 17, 6, 24, 7, 36, 8, 57, 9, 54, 10, 32, 11, 16, 12, 6, 13, 1] |
220 | $\alpha^{220}$ | 3 | 5 | [0, 1, 3, 5, 4, 8, 5, 16, 6, 24, 7, 34, 8, 57, 9, 56, 10, 32, 11, 17, 12, 6] |
221 | $\alpha^{221}$ | 3 | 5 | [0, 1, 3, 5, 4, 7, 5, 15, 6, 26, 7, 36, 8, 57, 9, 56, 10, 30, 11, 15, 12, 7, 13, 1] |
222 | $\alpha^{222}$ | 3 | 5 | [0, 1, 3, 5, 4, 6, 5, 14, 6, 30, 7, 40, 8, 51, 9, 50, 10, 34, 11, 19, 12, 6] |
223 | $\alpha^{223}$ | 3 | 5 | [0, 1, 3, 5, 4, 5, 5, 12, 6, 32, 7, 46, 8, 51, 9, 44, 10, 32, 11, 21, 12, 7] |
224 | $\alpha^{224}$ | 3 | 5 | [0, 1, 3, 5, 4, 4, 5, 11, 6, 34, 7, 48, 8, 51, 9, 44, 10, 30, 11, 19, 12, 8, 13, 1] |
225 | $\alpha^{225}$ | 3 | 5 | [0, 1, 3, 5, 4, 3, 5, 11, 6, 36, 7, 46, 8, 51, 9, 50, 10, 28, 11, 13, 12, 9, 13, 3] |
226 | $\alpha^{226}$ | 3 | 5 | [0, 1, 3, 5, 4, 3, 5, 10, 6, 35, 7, 48, 8, 53, 9, 50, 10, 28, 11, 11, 12, 7, 13, 4, 14, 1] |
227 | $\alpha^{227}$ | 3 | 5 | [0, 1, 3, 5, 4, 4, 5, 9, 6, 34, 7, 52, 8, 51, 9, 44, 10, 30, 11, 15, 12, 8, 13, 3] |
228 | $\alpha^{228}$ | 3 | 5 | [0, 1, 3, 5, 4, 5, 5, 10, 6, 32, 7, 50, 8, 51, 9, 44, 10, 32, 11, 17, 12, 7, 13, 2] |
229 | $\alpha^{229}$ | 3 | 6 | [0, 1, 3, 6, 4, 6, 5, 9, 6, 30, 7, 46, 8, 51, 9, 52, 10, 34, 11, 12, 12, 6, 13, 3] |
230 | $\alpha^{230}$ | 3 | 7 | [0, 1, 3, 7, 4, 6, 5, 8, 6, 30, 7, 44, 8, 51, 9, 54, 10, 34, 11, 13, 12, 6, 13, 2] |
231 | $\alpha^{231}$ | 3 | 7 | [0, 1, 3, 7, 4, 7, 5, 9, 6, 28, 7, 42, 8, 51, 9, 54, 10, 36, 11, 15, 12, 5, 13, 1] |
232 | $\alpha^{232}$ | 3 | 6 | [0, 1, 3, 6, 4, 7, 5, 10, 6, 28, 7, 44, 8, 51, 9, 52, 10, 36, 11, 14, 12, 5, 13, 2] |
233 | $\alpha^{233}$ | 3 | 5 | [0, 1, 3, 5, 4, 7, 5, 10, 6, 28, 7, 50, 8, 51, 9, 44, 10, 36, 11, 17, 12, 5, 13, 2] |
234 | $\alpha^{234}$ | 3 | 4 | [0, 1, 3, 4, 4, 7, 5, 9, 6, 32, 7, 52, 8, 47, 9, 46, 10, 32, 11, 16, 12, 9, 13, 1] |
235 | $\alpha^{235}$ | 3 | 3 | [0, 1, 3, 3, 4, 7, 5, 10, 6, 32, 7, 54, 8, 47, 9, 44, 10, 32, 11, 15, 12, 9, 13, 2] |
236 | $\alpha^{236}$ | 3 | 2 | [0, 1, 3, 2, 4, 6, 5, 12, 6, 31, 7, 54, 8, 55, 9, 42, 10, 24, 11, 16, 12, 10, 13, 2, 14, 1] |
237 | $\alpha^{237}$ | 3 | 1 | [0, 1, 3, 1, 4, 5, 5, 16, 6, 32, 7, 54, 8, 51, 9, 36, 10, 32, 11, 17, 12, 7, 13, 4] |
238 | $\alpha^{238}$ | 4 | 6 | [0, 1, 4, 6, 5, 19, 6, 32, 7, 46, 8, 51, 9, 44, 10, 30, 11, 18, 12, 6, 13, 1, 14, 2] |
239 | $\alpha^{239}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
240 | $\alpha^{240}$ | 4 | 7 | [0, 1, 4, 7, 5, 20, 6, 28, 7, 48, 8, 51, 9, 40, 10, 36, 11, 16, 12, 5, 13, 4] |
241 | $\alpha^{241}$ | 4 | 7 | [0, 1, 4, 7, 5, 22, 6, 28, 7, 44, 8, 51, 9, 40, 10, 36, 11, 20, 12, 5, 13, 2] |
242 | $\alpha^{242}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 43, 8, 51, 9, 42, 10, 34, 11, 20, 12, 6, 15, 1] |
243 | $\alpha^{243}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
244 | $\alpha^{244}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 42, 8, 51, 9, 46, 10, 34, 11, 14, 12, 6, 13, 4] |
245 | $\alpha^{245}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 27, 7, 42, 8, 59, 9, 46, 10, 28, 11, 14, 12, 6, 13, 4, 14, 1] |
246 | $\alpha^{246}$ | 4 | 5 | [0, 1, 4, 5, 5, 23, 6, 29, 7, 40, 8, 59, 9, 46, 10, 26, 11, 16, 12, 7, 13, 3, 14, 1] |
247 | $\alpha^{247}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 34, 7, 36, 8, 55, 9, 46, 10, 28, 11, 20, 12, 5, 13, 1, 14, 2] |
248 | $\alpha^{248}$ | 2 | 1 | [0, 1, 2, 1, 4, 1, 5, 23, 6, 36, 7, 40, 8, 51, 9, 46, 10, 33, 11, 16, 12, 3, 13, 3, 14, 2] |
249 | $\alpha^{249}$ | 2 | 2 | [0, 1, 2, 2, 4, 1, 5, 23, 6, 32, 7, 40, 8, 55, 9, 46, 10, 30, 11, 16, 12, 7, 13, 3] |
250 | $\alpha^{250}$ | 2 | 3 | [0, 1, 2, 3, 4, 3, 5, 24, 6, 29, 7, 38, 8, 57, 9, 46, 10, 23, 11, 18, 12, 11, 13, 2, 14, 1] |
251 | $\alpha^{251}$ | 2 | 4 | [0, 1, 2, 4, 4, 6, 5, 24, 6, 24, 7, 44, 8, 53, 9, 36, 10, 28, 11, 20, 12, 12, 13, 4] |
252 | $\alpha^{252}$ | 2 | 5 | [0, 1, 2, 5, 4, 10, 5, 20, 6, 24, 7, 48, 8, 41, 9, 40, 10, 33, 11, 16, 12, 12, 13, 4, 14, 2] |
253 | $\alpha^{253}$ | 2 | 6 | [0, 1, 2, 6, 4, 15, 5, 16, 6, 24, 7, 48, 8, 31, 9, 48, 10, 30, 11, 16, 12, 17, 14, 4] |
254 | $\alpha^{254}$ | 2 | 7 | [0, 1, 2, 7, 4, 21, 5, 8, 6, 35, 7, 32, 8, 35, 9, 48, 10, 21, 11, 32, 12, 7, 13, 8, 14, 1] |
We focus on the the linear codes with greater $d_{\mathcal{D}}^\perp$, which are better in the sense of side-channel resistance (from our paper).
# Finding the indices of d_C=4
d_index = []
d_C = 4
for i in range(len(wd)):
if wd[i][2] == d_C:
d_index.append(i)
#d_index = np.array(d_index)
print(len(d_index))
94
def highlight(s, threshold, column):
is_min = pd.Series(data=False, index=s.index)
is_min[column] = (s.loc[column] <= threshold)
return ['background-color: gold' if is_min.any() else '' for v in is_min]
df_4 = pd.DataFrame({'$L_2$': np.array(alpha_all)[d_index], '$d_{\mathcal{D}}^\perp$': d_all[d_index],
'$B_{d_{\mathcal{D}}^\perp}$': B_all[d_index], 'Weight Enumerators': wd[d_index]})
df_4 = df_4.sort_values(by=['$B_{d_{\mathcal{D}}^\perp}$'], ascending=True)
(df_4.style
.apply(highlight, threshold=3, column=['$B_{d_{\mathcal{D}}^\perp}$'], axis=1)
.background_gradient(cmap=cm_2, subset=['$B_{d_{\mathcal{D}}^\perp}$' ])
.set_caption('Tab. II Linear codes for IPM with $d_{\mathcal{D}}^\perp=4$.')
.set_table_styles(styles))
$L_2$ | $d_{\mathcal{D}}^\perp$ | $B_{d_{\mathcal{D}}^\perp}$ | Weight Enumerators | |
---|---|---|---|---|
0 | $\alpha^{8}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 34, 7, 36, 8, 55, 9, 46, 10, 28, 11, 20, 12, 5, 13, 1, 14, 2] |
48 | $\alpha^{129}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 33, 7, 36, 8, 59, 9, 46, 10, 22, 11, 20, 12, 9, 13, 1, 14, 1] |
47 | $\alpha^{128}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 34, 7, 36, 8, 55, 9, 46, 10, 28, 11, 20, 12, 5, 13, 1, 14, 2] |
45 | $\alpha^{126}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 33, 7, 36, 8, 59, 9, 46, 10, 22, 11, 20, 12, 9, 13, 1, 14, 1] |
46 | $\alpha^{127}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 34, 7, 36, 8, 55, 9, 46, 10, 28, 11, 20, 12, 5, 13, 1, 14, 2] |
93 | $\alpha^{247}$ | 4 | 3 | [0, 1, 4, 3, 5, 25, 6, 34, 7, 36, 8, 55, 9, 46, 10, 28, 11, 20, 12, 5, 13, 1, 14, 2] |
82 | $\alpha^{214}$ | 4 | 4 | [0, 1, 4, 4, 5, 22, 6, 36, 7, 44, 8, 45, 9, 40, 10, 36, 11, 20, 12, 6, 13, 2] |
81 | $\alpha^{213}$ | 4 | 4 | [0, 1, 4, 4, 5, 21, 6, 36, 7, 44, 8, 45, 9, 46, 10, 36, 11, 12, 12, 6, 13, 5] |
44 | $\alpha^{125}$ | 4 | 4 | [0, 1, 4, 4, 5, 23, 6, 31, 7, 42, 8, 59, 9, 40, 10, 24, 11, 22, 12, 8, 13, 1, 14, 1] |
12 | $\alpha^{42}$ | 4 | 4 | [0, 1, 4, 4, 5, 21, 6, 36, 7, 44, 8, 45, 9, 46, 10, 36, 11, 12, 12, 6, 13, 5] |
11 | $\alpha^{41}$ | 4 | 4 | [0, 1, 4, 4, 5, 22, 6, 36, 7, 44, 8, 45, 9, 40, 10, 36, 11, 20, 12, 6, 13, 2] |
49 | $\alpha^{130}$ | 4 | 4 | [0, 1, 4, 4, 5, 23, 6, 31, 7, 42, 8, 59, 9, 40, 10, 24, 11, 22, 12, 8, 13, 1, 14, 1] |
25 | $\alpha^{77}$ | 4 | 5 | [0, 1, 4, 5, 5, 24, 6, 30, 7, 38, 8, 57, 9, 46, 10, 26, 11, 18, 12, 9, 13, 2] |
92 | $\alpha^{246}$ | 4 | 5 | [0, 1, 4, 5, 5, 23, 6, 29, 7, 40, 8, 59, 9, 46, 10, 26, 11, 16, 12, 7, 13, 3, 14, 1] |
1 | $\alpha^{9}$ | 4 | 5 | [0, 1, 4, 5, 5, 23, 6, 29, 7, 40, 8, 59, 9, 46, 10, 26, 11, 16, 12, 7, 13, 3, 14, 1] |
10 | $\alpha^{40}$ | 4 | 5 | [0, 1, 4, 5, 5, 22, 6, 34, 7, 43, 8, 45, 9, 42, 10, 38, 11, 20, 12, 5, 15, 1] |
83 | $\alpha^{215}$ | 4 | 5 | [0, 1, 4, 5, 5, 22, 6, 34, 7, 43, 8, 45, 9, 42, 10, 38, 11, 20, 12, 5, 15, 1] |
68 | $\alpha^{178}$ | 4 | 5 | [0, 1, 4, 5, 5, 24, 6, 30, 7, 38, 8, 57, 9, 46, 10, 26, 11, 18, 12, 9, 13, 2] |
67 | $\alpha^{177}$ | 4 | 5 | [0, 1, 4, 5, 5, 23, 6, 29, 7, 40, 8, 59, 9, 46, 10, 26, 11, 16, 12, 7, 13, 3, 14, 1] |
26 | $\alpha^{78}$ | 4 | 5 | [0, 1, 4, 5, 5, 23, 6, 29, 7, 40, 8, 59, 9, 46, 10, 26, 11, 16, 12, 7, 13, 3, 14, 1] |
2 | $\alpha^{10}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 27, 7, 42, 8, 59, 9, 46, 10, 28, 11, 14, 12, 6, 13, 4, 14, 1] |
4 | $\alpha^{12}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
53 | $\alpha^{163}$ | 4 | 6 | [0, 1, 4, 6, 5, 17, 6, 35, 7, 50, 8, 43, 9, 44, 10, 36, 11, 14, 12, 6, 13, 3, 14, 1] |
54 | $\alpha^{164}$ | 4 | 6 | [0, 1, 4, 6, 5, 18, 6, 36, 7, 48, 8, 41, 9, 44, 10, 36, 11, 16, 12, 8, 13, 2] |
55 | $\alpha^{165}$ | 4 | 6 | [0, 1, 4, 6, 5, 15, 6, 36, 7, 56, 8, 41, 9, 38, 10, 36, 11, 16, 12, 8, 13, 3] |
63 | $\alpha^{173}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 29, 7, 44, 8, 53, 9, 46, 10, 34, 11, 12, 12, 4, 13, 5, 14, 1] |
64 | $\alpha^{174}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
9 | $\alpha^{17}$ | 4 | 6 | [0, 1, 4, 6, 5, 19, 6, 32, 7, 46, 8, 51, 9, 44, 10, 30, 11, 18, 12, 6, 13, 1, 14, 2] |
3 | $\alpha^{11}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 42, 8, 51, 9, 46, 10, 34, 11, 14, 12, 6, 13, 4] |
66 | $\alpha^{176}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 42, 8, 51, 9, 46, 10, 34, 11, 14, 12, 6, 13, 4] |
69 | $\alpha^{179}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 28, 7, 44, 8, 57, 9, 40, 10, 28, 11, 20, 12, 8, 13, 2] |
73 | $\alpha^{183}$ | 4 | 6 | [0, 1, 4, 6, 5, 18, 6, 32, 7, 50, 8, 51, 9, 38, 10, 30, 11, 22, 12, 6, 14, 2] |
84 | $\alpha^{238}$ | 4 | 6 | [0, 1, 4, 6, 5, 19, 6, 32, 7, 46, 8, 51, 9, 44, 10, 30, 11, 18, 12, 6, 13, 1, 14, 2] |
85 | $\alpha^{239}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
88 | $\alpha^{242}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 43, 8, 51, 9, 42, 10, 34, 11, 20, 12, 6, 15, 1] |
89 | $\alpha^{243}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
90 | $\alpha^{244}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 42, 8, 51, 9, 46, 10, 34, 11, 14, 12, 6, 13, 4] |
91 | $\alpha^{245}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 27, 7, 42, 8, 59, 9, 46, 10, 28, 11, 14, 12, 6, 13, 4, 14, 1] |
65 | $\alpha^{175}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
8 | $\alpha^{16}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
5 | $\alpha^{13}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 43, 8, 51, 9, 42, 10, 34, 11, 20, 12, 6, 15, 1] |
29 | $\alpha^{81}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
30 | $\alpha^{82}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 29, 7, 44, 8, 53, 9, 46, 10, 34, 11, 12, 12, 4, 13, 5, 14, 1] |
27 | $\alpha^{79}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 30, 7, 42, 8, 51, 9, 46, 10, 34, 11, 14, 12, 6, 13, 4] |
39 | $\alpha^{91}$ | 4 | 6 | [0, 1, 4, 6, 5, 18, 6, 36, 7, 48, 8, 41, 9, 44, 10, 36, 11, 16, 12, 8, 13, 2] |
38 | $\alpha^{90}$ | 4 | 6 | [0, 1, 4, 6, 5, 15, 6, 36, 7, 56, 8, 41, 9, 38, 10, 36, 11, 16, 12, 8, 13, 3] |
24 | $\alpha^{76}$ | 4 | 6 | [0, 1, 4, 6, 5, 22, 6, 28, 7, 44, 8, 57, 9, 40, 10, 28, 11, 20, 12, 8, 13, 2] |
40 | $\alpha^{92}$ | 4 | 6 | [0, 1, 4, 6, 5, 17, 6, 35, 7, 50, 8, 43, 9, 44, 10, 36, 11, 14, 12, 6, 13, 3, 14, 1] |
20 | $\alpha^{72}$ | 4 | 6 | [0, 1, 4, 6, 5, 18, 6, 32, 7, 50, 8, 51, 9, 38, 10, 30, 11, 22, 12, 6, 14, 2] |
28 | $\alpha^{80}$ | 4 | 6 | [0, 1, 4, 6, 5, 21, 6, 30, 7, 46, 8, 51, 9, 40, 10, 34, 11, 18, 12, 6, 13, 3] |
21 | $\alpha^{73}$ | 4 | 7 | [0, 1, 4, 7, 5, 19, 6, 30, 7, 46, 8, 53, 9, 44, 10, 26, 11, 18, 12, 11, 13, 1] |
41 | $\alpha^{104}$ | 4 | 7 | [0, 1, 4, 7, 5, 12, 6, 46, 7, 38, 8, 45, 9, 54, 10, 26, 11, 18, 12, 3, 13, 6] |
7 | $\alpha^{15}$ | 4 | 7 | [0, 1, 4, 7, 5, 20, 6, 28, 7, 48, 8, 51, 9, 40, 10, 36, 11, 16, 12, 5, 13, 4] |
6 | $\alpha^{14}$ | 4 | 7 | [0, 1, 4, 7, 5, 22, 6, 28, 7, 44, 8, 51, 9, 40, 10, 36, 11, 20, 12, 5, 13, 2] |
86 | $\alpha^{240}$ | 4 | 7 | [0, 1, 4, 7, 5, 20, 6, 28, 7, 48, 8, 51, 9, 40, 10, 36, 11, 16, 12, 5, 13, 4] |
87 | $\alpha^{241}$ | 4 | 7 | [0, 1, 4, 7, 5, 22, 6, 28, 7, 44, 8, 51, 9, 40, 10, 36, 11, 20, 12, 5, 13, 2] |
70 | $\alpha^{180}$ | 4 | 7 | [0, 1, 4, 7, 5, 20, 6, 29, 7, 44, 8, 55, 9, 44, 10, 26, 11, 20, 12, 9, 14, 1] |
72 | $\alpha^{182}$ | 4 | 7 | [0, 1, 4, 7, 5, 19, 6, 30, 7, 46, 8, 53, 9, 44, 10, 26, 11, 18, 12, 11, 13, 1] |
23 | $\alpha^{75}$ | 4 | 7 | [0, 1, 4, 7, 5, 20, 6, 29, 7, 44, 8, 55, 9, 44, 10, 26, 11, 20, 12, 9, 14, 1] |
52 | $\alpha^{151}$ | 4 | 7 | [0, 1, 4, 7, 5, 12, 6, 46, 7, 38, 8, 45, 9, 54, 10, 26, 11, 18, 12, 3, 13, 6] |
37 | $\alpha^{89}$ | 4 | 7 | [0, 1, 4, 7, 5, 16, 6, 34, 7, 54, 8, 41, 9, 38, 10, 38, 11, 18, 12, 7, 13, 2] |
36 | $\alpha^{88}$ | 4 | 7 | [0, 1, 4, 7, 5, 18, 6, 28, 7, 53, 8, 51, 9, 36, 10, 36, 11, 18, 12, 5, 13, 2, 15, 1] |
57 | $\alpha^{167}$ | 4 | 7 | [0, 1, 4, 7, 5, 18, 6, 28, 7, 53, 8, 51, 9, 36, 10, 36, 11, 18, 12, 5, 13, 2, 15, 1] |
56 | $\alpha^{166}$ | 4 | 7 | [0, 1, 4, 7, 5, 16, 6, 34, 7, 54, 8, 41, 9, 38, 10, 38, 11, 18, 12, 7, 13, 2] |
62 | $\alpha^{172}$ | 4 | 8 | [0, 1, 4, 8, 5, 21, 6, 24, 7, 44, 8, 57, 9, 46, 10, 32, 11, 12, 12, 6, 13, 5] |
31 | $\alpha^{83}$ | 4 | 8 | [0, 1, 4, 8, 5, 21, 6, 24, 7, 44, 8, 57, 9, 46, 10, 32, 11, 12, 12, 6, 13, 5] |
58 | $\alpha^{168}$ | 4 | 8 | [0, 1, 4, 8, 5, 19, 6, 24, 7, 49, 8, 57, 9, 42, 10, 32, 11, 14, 12, 6, 13, 3, 15, 1] |
35 | $\alpha^{87}$ | 4 | 8 | [0, 1, 4, 8, 5, 19, 6, 24, 7, 49, 8, 57, 9, 42, 10, 32, 11, 14, 12, 6, 13, 3, 15, 1] |
22 | $\alpha^{74}$ | 4 | 8 | [0, 1, 4, 8, 5, 19, 6, 27, 7, 46, 8, 55, 9, 44, 10, 28, 11, 18, 12, 8, 13, 1, 14, 1] |
71 | $\alpha^{181}$ | 4 | 8 | [0, 1, 4, 8, 5, 19, 6, 27, 7, 46, 8, 55, 9, 44, 10, 28, 11, 18, 12, 8, 13, 1, 14, 1] |
34 | $\alpha^{86}$ | 4 | 9 | [0, 1, 4, 9, 5, 20, 6, 20, 7, 48, 8, 63, 9, 40, 10, 28, 11, 16, 12, 7, 13, 4] |
59 | $\alpha^{169}$ | 4 | 9 | [0, 1, 4, 9, 5, 20, 6, 20, 7, 48, 8, 63, 9, 40, 10, 28, 11, 16, 12, 7, 13, 4] |
32 | $\alpha^{84}$ | 4 | 9 | [0, 1, 4, 9, 5, 20, 6, 20, 7, 48, 8, 63, 9, 40, 10, 28, 11, 16, 12, 7, 13, 4] |
61 | $\alpha^{171}$ | 4 | 9 | [0, 1, 4, 9, 5, 20, 6, 20, 7, 48, 8, 63, 9, 40, 10, 28, 11, 16, 12, 7, 13, 4] |
60 | $\alpha^{170}$ | 4 | 10 | [0, 1, 4, 10, 5, 18, 6, 18, 7, 51, 8, 63, 9, 42, 10, 30, 11, 12, 12, 6, 13, 4, 15, 1] |
33 | $\alpha^{85}$ | 4 | 10 | [0, 1, 4, 10, 5, 18, 6, 18, 7, 51, 8, 63, 9, 42, 10, 30, 11, 12, 12, 6, 13, 4, 15, 1] |
13 | $\alpha^{56}$ | 4 | 10 | [0, 1, 4, 10, 5, 12, 6, 25, 7, 60, 8, 57, 9, 36, 10, 22, 11, 20, 12, 12, 14, 1] |
80 | $\alpha^{199}$ | 4 | 10 | [0, 1, 4, 10, 5, 12, 6, 25, 7, 60, 8, 57, 9, 36, 10, 22, 11, 20, 12, 12, 14, 1] |
42 | $\alpha^{113}$ | 4 | 11 | [0, 1, 4, 11, 5, 14, 6, 24, 7, 54, 8, 53, 9, 42, 10, 30, 11, 18, 12, 7, 14, 2] |
43 | $\alpha^{114}$ | 4 | 11 | [0, 1, 4, 11, 5, 14, 6, 25, 7, 52, 8, 51, 9, 48, 10, 30, 11, 12, 12, 9, 13, 2, 14, 1] |
50 | $\alpha^{141}$ | 4 | 11 | [0, 1, 4, 11, 5, 14, 6, 25, 7, 52, 8, 51, 9, 48, 10, 30, 11, 12, 12, 9, 13, 2, 14, 1] |
51 | $\alpha^{142}$ | 4 | 11 | [0, 1, 4, 11, 5, 14, 6, 24, 7, 54, 8, 53, 9, 42, 10, 30, 11, 18, 12, 7, 14, 2] |
78 | $\alpha^{197}$ | 4 | 13 | [0, 1, 4, 13, 5, 10, 6, 24, 7, 58, 8, 49, 9, 46, 10, 30, 11, 14, 12, 9, 14, 2] |
19 | $\alpha^{62}$ | 4 | 13 | [0, 1, 4, 13, 5, 11, 6, 36, 7, 36, 8, 47, 9, 58, 10, 28, 11, 20, 12, 3, 13, 3] |
15 | $\alpha^{58}$ | 4 | 13 | [0, 1, 4, 13, 5, 10, 6, 24, 7, 58, 8, 49, 9, 46, 10, 30, 11, 14, 12, 9, 14, 2] |
14 | $\alpha^{57}$ | 4 | 13 | [0, 1, 4, 13, 5, 7, 6, 28, 7, 53, 8, 63, 9, 38, 10, 20, 11, 26, 12, 3, 13, 3, 15, 1] |
74 | $\alpha^{193}$ | 4 | 13 | [0, 1, 4, 13, 5, 11, 6, 36, 7, 36, 8, 47, 9, 58, 10, 28, 11, 20, 12, 3, 13, 3] |
79 | $\alpha^{198}$ | 4 | 13 | [0, 1, 4, 13, 5, 7, 6, 28, 7, 53, 8, 63, 9, 38, 10, 20, 11, 26, 12, 3, 13, 3, 15, 1] |
18 | $\alpha^{61}$ | 4 | 14 | [0, 1, 4, 14, 5, 9, 6, 38, 7, 34, 8, 43, 9, 68, 10, 26, 11, 14, 12, 6, 13, 3] |
75 | $\alpha^{194}$ | 4 | 14 | [0, 1, 4, 14, 5, 9, 6, 38, 7, 34, 8, 43, 9, 68, 10, 26, 11, 14, 12, 6, 13, 3] |
17 | $\alpha^{60}$ | 4 | 15 | [0, 1, 4, 15, 5, 8, 6, 35, 7, 38, 8, 45, 9, 62, 10, 28, 11, 18, 12, 3, 13, 2, 14, 1] |
77 | $\alpha^{196}$ | 4 | 15 | [0, 1, 4, 15, 5, 8, 6, 30, 7, 43, 8, 53, 9, 54, 10, 26, 11, 20, 12, 3, 13, 2, 15, 1] |
16 | $\alpha^{59}$ | 4 | 15 | [0, 1, 4, 15, 5, 8, 6, 30, 7, 43, 8, 53, 9, 54, 10, 26, 11, 20, 12, 3, 13, 2, 15, 1] |
76 | $\alpha^{195}$ | 4 | 15 | [0, 1, 4, 15, 5, 8, 6, 35, 7, 38, 8, 45, 9, 62, 10, 28, 11, 18, 12, 3, 13, 2, 14, 1] |
As shown in our paper, the codes satifying two conditions are optimal:
Note that we use two leakage detection metrics SNR (signal-to-noise ratio) and MI (mutual information), and one leakage exploitation metric SR (success rate) to assess the side-channel resistance of IPM with different codes.
As a result of Tab. II, we conclude that the optimal codes for IPM are genetated by $\mathbf{H}=[L_2, 1]$ where $L_2\in\{\alpha^8, \alpha^{126}, \alpha^{127}, \alpha^{128}, \alpha^{129}, \alpha^{247}\}$.
The six optimal codes are categoried into three classes by equivalence, namely there are three nonequivalent optimal codes.
Specifically:
BKLC is the short of Best Known Linear Code.
Note that the code $[16, 8, 5]$ is unique.
The BKLC is defined as An [n, k] linear code $\mathcal{C}$ is said to be a best known linear [n, k] code (BKLC) if $\mathcal{C}$ has the highest minimum weight among all known [n, k] linear codes.See definition from Magma.
bklc_index = [255]
cm_3 = sns.light_palette("red", as_cmap=True, reverse=True)
df_bklc = pd.DataFrame({'$L_2$': [' --'], '$d_{\mathcal{D}}^\perp$': d_all[-1], '$B_{d_{\mathcal{D}}^\perp}$': B_all[-1],
'Weight Enumerators': wd[bklc_index]})
(df_bklc.style
.background_gradient(cmap=cm_3, subset=['$d_{\mathcal{D}}^\perp$', '$B_{d_{\mathcal{D}}^\perp}$'])
.set_caption('Tab. III One BKLC code for IPM with $d_{\mathcal{D}}^\perp=5$.')
.set_table_styles(styles))
$L_2$ | $d_{\mathcal{D}}^\perp$ | $B_{d_{\mathcal{D}}^\perp}$ | Weight Enumerators | |
---|---|---|---|---|
0 | -- | 5 | 24 | [0, 1, 5, 24, 6, 44, 7, 40, 8, 45, 9, 40, 10, 28, 11, 24, 12, 10] |
We can see that this BKLC code is better than all linear codes in IPM. It is interesting to notice that the BKLC code $[16, 8, 5]$ cannot be used in IPM, since it cannot be generated by $\mathbf{H}^\perp=[1, L_2]$ with any $L_2\in\mathbb{F}_{2^8}$.
The generator matrix of the dual code of this BKLC code is: $$ \mathbf{H}_{BKLC}^\perp= \left( \begin{matrix} 1& 1& 0& 1& 0& 0& 1& 0& 1& 0& 0& 0& 0& 0& 0& 0 \\ 1& 0& 1& 1& 1& 0& 1& 1& 0& 1& 0& 0& 0& 0& 0& 0 \\ 1& 0& 0& 0& 1& 1& 1& 1& 0& 0& 1& 0& 0& 0& 0& 0 \\ 0& 1& 0& 0& 0& 1& 1& 1& 0& 0& 0& 1& 0& 0& 0& 0 \\ 1& 1& 1& 1& 0& 0& 0& 1& 0& 0& 0& 0& 1& 0& 0& 0 \\ 0& 1& 1& 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0& 0 \\ 1& 1& 1& 0& 1& 1& 1& 0& 0& 0& 0& 0& 0& 0& 1& 0 \\ 1& 0& 1& 0& 0& 1& 0& 1& 0& 0& 0& 0& 0& 0& 0& 1 \end{matrix} \right) \normalsize\in \mathbb{F}_2^{8\times 12} $$