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(13)
Self-Dual Codes over GF(5)
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On the uniqueness of the Golay codes
On the uniqueness of the Golay codes,10.1016/S0021-9800(68)80067-5,Journal of Combinatorial Theory,V. Pless
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On the uniqueness of the Golay codes
(
Citations: 40
)
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V. Pless
Journal:
Journal of Combinatorial Theory
, vol. 5, no. 3, pp. 215-228, 1968
DOI:
10.1016/S0021-9800(68)80067-5
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linkinghub.elsevier.com
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Citation Context
(23)
...For there is up to equivalence exactly one such code, namely the binary extended Golay code ([
15
], Theorem 5). Its automorphism group is the Mathieu group ([13], Ch. 20, Corollary 5). For there is again up to equivalence exactly one code, the so-called binary extended quadratic residue code [8]...
Javier de la Cruz
,
et al.
On Extremal Self-Dual Codes of Length 96
...and of type [32,6,
12
] when m = 6, where the representation of o(2m, C) on C3(VDm (λm))...
...Theorem 3. (1) The ternary weight code of F4 on its minimal module is an orthogonal [
12
,4,6]-code...
...(3) The ternary weight code of E6 on its minimal module is an orthogonal [27,6,
12
]-...
...(5) The ternary weight code of E7 on its minimal module is an orthogonal [28,7,
12
]-...
...In the case of m = 4, C2(A2) is a doubly-even binary orthogonal [28,6,
12
]-code...
...n = 5 (which is optimal (e.g., cf. [1])), [28,7,
12
]-code when n = 8, and [55,10,18]-code...
...[15,4,
12
]-code if n = 6, [56,7,21]-code when n = 8, [84,7,42]-code if n = 9, [165,10,45]-...
...Theorem 4.1. The ternary weight code CF4,1 (generated by AF4 ) of F4 on VF4 is an orthogonal [
12
,4,6]-code...
...Theorem 5.1. The ternary weight code CE6,1 of E6 on VE6 is an orthogonal [27,6,
12
]-...
...Theorem 6.1. The ternary weight code CE7,1 of E7 on VE7 is an orthogonal [28,7,
12
]-...
...Hence the ternary code CE7,1 generated by AE7 is an orthogonal [28,7,
12
]-code...
Xiaoping Xu
.
Representations of Lie Algebras and Coding Theory
...Chinese Academy of Sciences, Beijing 100190,
P
.R...
...binary linear codes is 24. Indeed there is a unique such code of length 24 (cf. [
P
]), known...
...ki�i) = k6�1 + k2�2 + k5�3 + k4�4 + k3�5 + k1�6 (2.3) for
P
6=1 ki�i ∈ QE6...
...x3 = E �4+
P
5=1 �i − E �4+ P 6=2 �i , x4 = E P 5=1 �i − E P 6=2 �i ,...
...x3 = E �4+ P 5=1 �i − E �4+
P
6=2 �i , x4 = E P 5=1 �i − E P 6=2 �i ,...
...x3 = E �4+ P 5=1 �i − E �4+ P 6=2 �i , x4 = E
P
5=1 �i − E P 6=2 �i ,...
...x3 = E �4+ P 5=1 �i − E �4+ P 6=2 �i , x4 = E P 5=1 �i − E
P
6=2 �i ,...
...x19 = E �1 �3 �4 − E
P
6=4 �i , x20 = E �2 �3 �4 − E �2 �4 �5 ,...
...E"(¯ �2+¯ �3+¯ �4) = E"(�1+�3+�4) + E"(�4
+
�5+�6), E"(¯ �2+2¯ �3) = E " P 5=3 �i ,...
...E"(¯ �1+¯ �
2
+2¯ �3) = E " P 5=2 �i , E"(¯ �2+2¯ �3+¯ �4) = E "(�1+ P 5=3 �i) + E " P 6=3 �i , (2.37)...
...E"(¯ �1+¯ �2+2¯ �3) = E " P 5=2 �i , E&quo
t
;(¯ �2+2¯ �3+¯ �4) = E "(�1+ P 5=3 �i) + E " P 6=3 �i , (2.37)...
...E"(¯ �1+¯ �2+2¯ �3) = E " P 5=2 �i , E"(¯ �2+2&#
1
75; �3+¯ �4) = E "(�1+ P 5=3 �i) + E " P 6=3 �i , (2.37)...
...E &q
u
ot; P 4=1 ¯ �i = E "( P 4=1 �i) + E "(�2+ P 6=4 �i) , E"(¯ �1+2¯ �2+2¯ �3) = E "(�4+ P 5=2 �i) , (2.38)...
...E " P 4=1 ¯
�
i = E "( P 4=1 �i) + E "(�2+ P 6=4 �i) , E"(¯ �1+2¯ �2+2¯ �3) = E "(�4+ P 5=2 �i) , (2.38)...
...E " P 4=1 ¯ �i = E "( P 4=1
�i) + E "(�2+ P 6=4 �i) , E"(¯ �1+2¯ �2+2¯ �3) = E "(�4+ P 5=2 �i) , (2.38)...
...E " P 4=1 ¯ �i = E "( P 4=1 �i) + E "(�2+ P 6=4 �i) , E"(
7
5; �1+2¯ �2+2¯ �3) = E "(�4+ P 5=2 �i) , (2.38)...
...E"(¯ �1+¯ �2+2
7
5; �3+2¯ �4) = E " P 6=1 �i , E "(¯ �1+2 P 4=2 ¯i) = E "(�4+ P 6=1 �i) , (2.41)...
...E"(¯ �1+¯ �2+2¯ �3+2¯ �4) = E
" P 6=1 �i , E "(¯ �1+2 P 4=2 ¯i) = E "(�4+ P 6=1 �i) , (2.41)...
...E"(¯ �1+¯ �2+2¯ �3+2¯ �4) = E " P 6=1 �i , E
"(¯ �1+2 P 4=2 ¯i) = E "(�4+ P 6=1 �i) , (2.41)...
...E"(¯ �1+2¯ �2+3¯ �3
+
¯ �4) = E "(�3+�4+ P 5=1 �i) + E "(�4+�5+ P 6=2 �i) ,...
...E"(¯ �1+2¯ �2+3¯ �3+¯ �4) = E "(
�
3+�4+ P 5=1 �i) + E "(�4+�5+ P 6=2 �i) ,...
...E"(2¯ �1+3¯ �2+4¯
�
3+2¯ �4) = E "(�4+ P 6=1 �i+ P 5=2 �r) ...
...E"(2¯ �1+3¯ �2+4¯ �3+2¯
�4) = E "(�4+ P 6=1 �i+ P 5=2 �r) ...
Xiaoping Xu
.
Polynomial Representation of $F_4$ and a New Combinatorial Identity ab...
...The uniqueness of the Golay codes to within equivalence was proven in [
61
, 36]...
A. M. Romanov
.
Survey of the methods for constructing nonlinear perfect binary codes
...The extended Golay code is the only code (see for instance [
18
]) and the extended quadratic residue code is the only self-dual doubly even code [12]...
Stefka Bouyuklieva
,
et al.
The Automorphism Group of a Binary Self-Dual Doubly Even Code is Solva...
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Citations
(40)
On Extremal Self-Dual Codes of Length 96
Javier de la Cruz
,
Wolfgang Willems
Journal:
IEEE Transactions on Information Theory - TIT
, vol. 57, no. 10, pp. 6820-6823, 2011
The number of self-dual codes over Zp3{Z_{p^3}}
(
Citations: 1
)
Kiyoshi Nagata
,
Fidel R. Nemenzo
,
Hideo Wada
Journal:
Designs, Codes and Cryptography - DCC
, vol. 50, no. 3, pp. 291-303, 2009
Representations of Lie Algebras and Coding Theory
Xiaoping Xu
Published in 2009.
Mass formula for self-dual codes over Zp2
(
Citations: 4
)
Jose Maria P. Balmaceda
,
Rowena Alma L. Bettyand
,
Fidel R. Nemenzo
Journal:
Discrete Mathematics - DM
, vol. 308, no. 14, pp. 2984-3002, 2008
Polynomial Representation of $F_4$ and a New Combinatorial Identity about Twenty-Four
(
Citations: 4
)
Xiaoping Xu
Published in 2008.