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On primitive constant dimension codes and a geometrical sunflower bound
On the performance of optimal double circulant even codes
1. | Department of Electrical and Computer Engineering, University of Victoria, P.O. Box 1700, STN CSC, Victoria, BC, Canada V8W 2Y2 |
2. | Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan |
In this note, we investigate the performance of optimal double circulant even codes which are not self-dual, as measured by the decoding error probability in bounded distance decoding. To achieve this, we classify the optimal double circulant even codes that are not self-dual which have the smallest weight distribution for lengths up to 72. We also give some restrictions on the weight distributions of (extremal) self-dual [54, 27, 10] codes with shadows of minimum weight 3. Finally, we consider the performance of extremal self-dual codes of lengths 88 and 112.
References:
[1] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[2] |
S. Bouyuklieva, A. Malevich and W. Willems,
On the performance of binary extremal self-dual codes, Adv. Math. Commun., 5 (2011), 267-274.
doi: 10.3934/amc.2011.5.267. |
[3] |
S. Bouyuklieva and P. R. J. Östergård,
New constructions of optimal self-dual binary codes of length 54, Des. Codes Cryptogr., 41 (2006), 101-109.
doi: 10.1007/s10623-006-0018-2. |
[4] |
N. Chigira, M. Harada and M. Kitazume,
Extremal self-dual codes of length 64 through neighbors and covering radii, Des. Codes Cryptogr., 42 (2007), 93-101.
|
[5] |
J. H. Conway and N. J. A. Sloane,
A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.
doi: 10.1109/18.59931. |
[6] |
R. Dontcheva and M. Harada,
New extremal self-dual codes of length 62 and related extremal self-dual codes, IEEE Trans. Inform. Theory, 48 (2002), 2060-2064.
doi: 10.1109/TIT.2002.1013144. |
[7] |
S. T. Dougherty, T. A. Gulliver and M. Harada,
Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.
doi: 10.1109/18.641574. |
[8] |
S. T. Dougherty and M. Harada,
New extremal self-dual codes of length 68, IEEE Trans. Inform. Theory, 45 (1999), 2133-2136.
doi: 10.1109/18.782158. |
[9] |
A. Faldum, J. Lafuente, G. Ochoa and W. Willems,
Error probabilities for bounded distance decoding, Des. Codes Cryptogr., 40 (2006), 237-252.
doi: 10.1007/s10623-006-0010-x. |
[10] |
T. A. Gulliver and M. Harada,
Classification of extremal double circulant formally self-dual even codes, Des. Codes Cryptogr., 11 (1997), 25-35.
doi: 10.1023/A:1008206223659. |
[11] |
T. A. Gulliver and M. Harada,
The existence of a formally self-dual even [70, 35, 14] code, Appl. Math. Lett., 11 (1998), 95-98.
doi: 10.1016/S0893-9659(97)00140-7. |
[12] |
T. A. Gulliver and M. Harada,
Classification of extremal double circulant self-dual codes of lengths 64 to 72, Des. Codes Cryptogr., 13 (1998), 257-269.
doi: 10.1023/A:1008249924142. |
[13] |
T. A. Gulliver and M. Harada,
Classification of extremal double circulant self-dual codes of lengths 74-88, Discrete Math., 306 (2006), 2064-2072.
doi: 10.1016/j.disc.2006.05.004. |
[14] |
T. A. Gulliver and M. Harada, On extremal double circulant self-dual codes of lengths 90-96, (submitted), arXiv: 1601.07343. |
[15] |
M. Harada, An extremal doubly even self-dual code of length 112, Electron. J. Combin. 15(2008), Note 33, 5 pp. |
[16] |
M. Harada, T. A. Gulliver and H. Kaneta,
Classification of extremal double-circulant self-dual codes of length up to 62, Discrete Math., 188 (1998), 127-136.
doi: 10.1016/S0012-365X(97)00250-1. |
[17] |
M. Harada and T. Nishimura,
An extremal singly even self-dual code of length 88, Adv. Math. Commun., 1 (2007), 261-267.
doi: 10.3934/amc.2007.1.261. |
[18] |
S. K. Houghten, C. W. H. Lam, L. H. Thiel and J. A. Parker,
The extended quadratic residue code is the only (48, 24, 12) self-dual doubly-even code, IEEE Trans. Inform. Theory, 49 (2003), 53-59.
doi: 10.1109/TIT.2002.806146. |
[19] |
W. C. Huffman,
On the classification and enumeration of self-dual codes, Finite Fields Appl., 11 (2005), 451-490.
doi: 10.1016/j.ffa.2005.05.012. |
[20] |
W. C. Huffman and V. D. Tonchev,
The existence of extremal self-dual [50, 25, 10] codes and quasi-symmetric 2-(49, 9, 6) designs, Des. Codes Cryptogr., 6 (1996), 97-106.
doi: 10.1007/BF01398008. |
[21] |
C. L. Mallows and N. J. A. Sloane,
An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200.
doi: 10.1016/S0019-9958(73)90273-8. |
[22] |
E. Rains and N. J. A. Sloane, Self-dual codes, Handbook of Coding Theory, V. S. Pless and W. C. Huffman (Editors), Elsevier, Amsterdam, 1998,177-294. |
[23] |
R. Russeva and N. Yankov,
On binary self-dual codes of lengths 60, 62, 64 and 66 having an automorphism of order 9, Des. Codes Cryptogr., 45 (2007), 335-346.
doi: 10.1007/s10623-007-9127-9. |
[24] |
N. J. A. Sloane, Is there a (72, 36) d=16 self-dual code? IEEE Trans. Inform. Theory 19(1973), p251. |
[25] |
N. Yankov and M. H. Lee,
New binary self-dual codes of lengths 50-60, Des. Codes Cryptogr., 73 (2014), 983-996.
doi: 10.1007/s10623-013-9839-y. |
[26] |
N. Yankov, M. H. Lee, M. Gürel and M. Ivanova,
Self-dual codes with an automorphism of order 11, IEEE Trans. Inform. Theory, 61 (2015), 1188-1193.
doi: 10.1109/TIT.2015.2396915. |
[27] |
N. Yankov and R. Russeva,
Binary self-dual codes of lengths 52 to 60 with an automorphism of order 7 or 13, IEEE Trans. Inform. Theory, 57 (2011), 7498-7506.
doi: 10.1109/TIT.2011.2155619. |
show all references
References:
[1] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system Ⅰ: The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[2] |
S. Bouyuklieva, A. Malevich and W. Willems,
On the performance of binary extremal self-dual codes, Adv. Math. Commun., 5 (2011), 267-274.
doi: 10.3934/amc.2011.5.267. |
[3] |
S. Bouyuklieva and P. R. J. Östergård,
New constructions of optimal self-dual binary codes of length 54, Des. Codes Cryptogr., 41 (2006), 101-109.
doi: 10.1007/s10623-006-0018-2. |
[4] |
N. Chigira, M. Harada and M. Kitazume,
Extremal self-dual codes of length 64 through neighbors and covering radii, Des. Codes Cryptogr., 42 (2007), 93-101.
|
[5] |
J. H. Conway and N. J. A. Sloane,
A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333.
doi: 10.1109/18.59931. |
[6] |
R. Dontcheva and M. Harada,
New extremal self-dual codes of length 62 and related extremal self-dual codes, IEEE Trans. Inform. Theory, 48 (2002), 2060-2064.
doi: 10.1109/TIT.2002.1013144. |
[7] |
S. T. Dougherty, T. A. Gulliver and M. Harada,
Extremal binary self-dual codes, IEEE Trans. Inform. Theory, 43 (1997), 2036-2047.
doi: 10.1109/18.641574. |
[8] |
S. T. Dougherty and M. Harada,
New extremal self-dual codes of length 68, IEEE Trans. Inform. Theory, 45 (1999), 2133-2136.
doi: 10.1109/18.782158. |
[9] |
A. Faldum, J. Lafuente, G. Ochoa and W. Willems,
Error probabilities for bounded distance decoding, Des. Codes Cryptogr., 40 (2006), 237-252.
doi: 10.1007/s10623-006-0010-x. |
[10] |
T. A. Gulliver and M. Harada,
Classification of extremal double circulant formally self-dual even codes, Des. Codes Cryptogr., 11 (1997), 25-35.
doi: 10.1023/A:1008206223659. |
[11] |
T. A. Gulliver and M. Harada,
The existence of a formally self-dual even [70, 35, 14] code, Appl. Math. Lett., 11 (1998), 95-98.
doi: 10.1016/S0893-9659(97)00140-7. |
[12] |
T. A. Gulliver and M. Harada,
Classification of extremal double circulant self-dual codes of lengths 64 to 72, Des. Codes Cryptogr., 13 (1998), 257-269.
doi: 10.1023/A:1008249924142. |
[13] |
T. A. Gulliver and M. Harada,
Classification of extremal double circulant self-dual codes of lengths 74-88, Discrete Math., 306 (2006), 2064-2072.
doi: 10.1016/j.disc.2006.05.004. |
[14] |
T. A. Gulliver and M. Harada, On extremal double circulant self-dual codes of lengths 90-96, (submitted), arXiv: 1601.07343. |
[15] |
M. Harada, An extremal doubly even self-dual code of length 112, Electron. J. Combin. 15(2008), Note 33, 5 pp. |
[16] |
M. Harada, T. A. Gulliver and H. Kaneta,
Classification of extremal double-circulant self-dual codes of length up to 62, Discrete Math., 188 (1998), 127-136.
doi: 10.1016/S0012-365X(97)00250-1. |
[17] |
M. Harada and T. Nishimura,
An extremal singly even self-dual code of length 88, Adv. Math. Commun., 1 (2007), 261-267.
doi: 10.3934/amc.2007.1.261. |
[18] |
S. K. Houghten, C. W. H. Lam, L. H. Thiel and J. A. Parker,
The extended quadratic residue code is the only (48, 24, 12) self-dual doubly-even code, IEEE Trans. Inform. Theory, 49 (2003), 53-59.
doi: 10.1109/TIT.2002.806146. |
[19] |
W. C. Huffman,
On the classification and enumeration of self-dual codes, Finite Fields Appl., 11 (2005), 451-490.
doi: 10.1016/j.ffa.2005.05.012. |
[20] |
W. C. Huffman and V. D. Tonchev,
The existence of extremal self-dual [50, 25, 10] codes and quasi-symmetric 2-(49, 9, 6) designs, Des. Codes Cryptogr., 6 (1996), 97-106.
doi: 10.1007/BF01398008. |
[21] |
C. L. Mallows and N. J. A. Sloane,
An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200.
doi: 10.1016/S0019-9958(73)90273-8. |
[22] |
E. Rains and N. J. A. Sloane, Self-dual codes, Handbook of Coding Theory, V. S. Pless and W. C. Huffman (Editors), Elsevier, Amsterdam, 1998,177-294. |
[23] |
R. Russeva and N. Yankov,
On binary self-dual codes of lengths 60, 62, 64 and 66 having an automorphism of order 9, Des. Codes Cryptogr., 45 (2007), 335-346.
doi: 10.1007/s10623-007-9127-9. |
[24] |
N. J. A. Sloane, Is there a (72, 36) d=16 self-dual code? IEEE Trans. Inform. Theory 19(1973), p251. |
[25] |
N. Yankov and M. H. Lee,
New binary self-dual codes of lengths 50-60, Des. Codes Cryptogr., 73 (2014), 983-996.
doi: 10.1007/s10623-013-9839-y. |
[26] |
N. Yankov, M. H. Lee, M. Gürel and M. Ivanova,
Self-dual codes with an automorphism of order 11, IEEE Trans. Inform. Theory, 61 (2015), 1188-1193.
doi: 10.1109/TIT.2015.2396915. |
[27] |
N. Yankov and R. Russeva,
Binary self-dual codes of lengths 52 to 60 with an automorphism of order 7 or 13, IEEE Trans. Inform. Theory, 57 (2011), 7498-7506.
doi: 10.1109/TIT.2011.2155619. |
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32 | 8 | 2 | 8 | 1 | 8 | 364 | [5] | |||
34 | 8 | 15 | 8 | 10 | 6 | - | ||||
36 | 8 | 4 | 8 | 3 | 8 | 225 | [5] | |||
38 | 8 | 1 | 8 | 1 | 8 | 171 | [5] | |||
40 | 8 | 1 | 8 | 2 | 8 | 125 | [5] | |||
42 | 10 | 2 | 10 | 1 | 8 | - | ||||
44 | 10 | 1 | 10 | 3 | 8 | - | ||||
46 | 10 | 1 | 10 | 2 | 10 | 1012 | [5] | |||
48 | 10 | 1 | 10 | 1 | 12 | 17296 | [5] | |||
50 | 10 | 1 | 10 | 1 | 10 | 196 | [20] | |||
52 | 10 | 1 | 10 | 1 | 10 | 250 | [5] | |||
54 | 10 | 1 | 10 | 1 | 10 | 7-135 | [3], [5] | |||
56 | 12 | 1 | 10 | 1 | 12 | 4606-8190 | [5] | |||
58 | 12 | 1 | 12 | 1 | 10 | - | ||||
60 | 12 | 1 | 12 | 1 | 12 | 2555 | [23] | |||
62 | 12 | 1 | 12 | 1 | 12 | 1860 | [6] | |||
64 | 12 | 1 | 12 | 1 | 12 | 1312 | [4] | |||
66 | 12 | 1 | 12 | 1 | 12 | 858 | [5] (see [7]) | |||
68 | 12 | 1 | 12 | 1 | 12 | 442-486 | [7], [26] | |||
70 | 12 | 1 | 14 | 12172 | 1 | 12-14 | - | |||
72 | 14 | 1 | 14 | 1 | 12-16 | - |
|
||||||||||
32 | 8 | 2 | 8 | 1 | 8 | 364 | [5] | |||
34 | 8 | 15 | 8 | 10 | 6 | - | ||||
36 | 8 | 4 | 8 | 3 | 8 | 225 | [5] | |||
38 | 8 | 1 | 8 | 1 | 8 | 171 | [5] | |||
40 | 8 | 1 | 8 | 2 | 8 | 125 | [5] | |||
42 | 10 | 2 | 10 | 1 | 8 | - | ||||
44 | 10 | 1 | 10 | 3 | 8 | - | ||||
46 | 10 | 1 | 10 | 2 | 10 | 1012 | [5] | |||
48 | 10 | 1 | 10 | 1 | 12 | 17296 | [5] | |||
50 | 10 | 1 | 10 | 1 | 10 | 196 | [20] | |||
52 | 10 | 1 | 10 | 1 | 10 | 250 | [5] | |||
54 | 10 | 1 | 10 | 1 | 10 | 7-135 | [3], [5] | |||
56 | 12 | 1 | 10 | 1 | 12 | 4606-8190 | [5] | |||
58 | 12 | 1 | 12 | 1 | 10 | - | ||||
60 | 12 | 1 | 12 | 1 | 12 | 2555 | [23] | |||
62 | 12 | 1 | 12 | 1 | 12 | 1860 | [6] | |||
64 | 12 | 1 | 12 | 1 | 12 | 1312 | [4] | |||
66 | 12 | 1 | 12 | 1 | 12 | 858 | [5] (see [7]) | |||
68 | 12 | 1 | 12 | 1 | 12 | 442-486 | [7], [26] | |||
70 | 12 | 1 | 14 | 12172 | 1 | 12-14 | - | |||
72 | 14 | 1 | 14 | 1 | 12-16 | - |
Code | First row | ||
|
(1100101100110101) | 8 | |
(1110110100010011) | 8 | ||
(11111110001000100) | 8 | ||
(11100000111010110) | 8 | ||
(11110101101101100) | 8 | ||
(11110011101101010) | 8 | ||
(10001011101100000) | 8 | ||
(10001100110010100) | 8 | ||
(11101110110100000) | 8 | ||
(10100101100011110) | 8 | ||
(10100100110010001) | 8 | ||
(10101010011111000) | 8 | ||
(10001110000100110) | 8 | ||
(11010010010001111) | 8 | ||
(10001100001110100) | 8 | ||
(11011010100001101) | 8 | ||
(11100001101010011) | 8 | ||
(101011110110000001) | 8 | ||
(111100001000010111) | 8 | ||
(100110111010010001) | 8 | ||
(100001010110111100) | 8 | ||
(1111000001001010110) | 8 | ||
(10101101111101111000) | 8 | ||
(100001101101110010110) | 10 | ||
(101010010101110110111) | 10 | ||
(1001111111001011011011) | 10 | ||
(11001011010111100000001) | 10 | ||
(110111000101111101110100) | 10 | ||
(1000100001011001001011101) | 10 | ||
(10001010100011011011000001) | 10 | ||
(111000000011101101100010011) | 10 | ||
(1001100011110101110111110100) | 12 | ||
(11011000010100000000110011010) | 12 | ||
(100000101101110000100111010001) | 12 | ||
(0010100111101100111111010000000) | 12 | ||
(10101000110010111100110100000000) | 12 | ||
(100100010010000101111011100100000) | 12 | ||
(1001001011010110101010101011000000) | 12 | ||
(01011011100110100101110000110000000) | 12 | ||
(101101101101001101001101111100010000) | 14 |
Code | First row | ||
|
(1100101100110101) | 8 | |
(1110110100010011) | 8 | ||
(11111110001000100) | 8 | ||
(11100000111010110) | 8 | ||
(11110101101101100) | 8 | ||
(11110011101101010) | 8 | ||
(10001011101100000) | 8 | ||
(10001100110010100) | 8 | ||
(11101110110100000) | 8 | ||
(10100101100011110) | 8 | ||
(10100100110010001) | 8 | ||
(10101010011111000) | 8 | ||
(10001110000100110) | 8 | ||
(11010010010001111) | 8 | ||
(10001100001110100) | 8 | ||
(11011010100001101) | 8 | ||
(11100001101010011) | 8 | ||
(101011110110000001) | 8 | ||
(111100001000010111) | 8 | ||
(100110111010010001) | 8 | ||
(100001010110111100) | 8 | ||
(1111000001001010110) | 8 | ||
(10101101111101111000) | 8 | ||
(100001101101110010110) | 10 | ||
(101010010101110110111) | 10 | ||
(1001111111001011011011) | 10 | ||
(11001011010111100000001) | 10 | ||
(110111000101111101110100) | 10 | ||
(1000100001011001001011101) | 10 | ||
(10001010100011011011000001) | 10 | ||
(111000000011101101100010011) | 10 | ||
(1001100011110101110111110100) | 12 | ||
(11011000010100000000110011010) | 12 | ||
(100000101101110000100111010001) | 12 | ||
(0010100111101100111111010000000) | 12 | ||
(10101000110010111100110100000000) | 12 | ||
(100100010010000101111011100100000) | 12 | ||
(1001001011010110101010101011000000) | 12 | ||
(01011011100110100101110000110000000) | 12 | ||
(101101101101001101001101111100010000) | 14 |
Code | First row | ||
|
(100101010001111) | 8 | |
(1001101010001101) | 8 | ||
(1110111100010110) | 8 | ||
(1010100111011101) | 8 | ||
(1000110111011110) | 8 | ||
(1110010011010001) | 8 | ||
(1101101100101000) | 8 | ||
(1001001100111010) | 8 | ||
(1110000111110110) | 8 | ||
(1110000111011110) | 8 | ||
(1001010011010011) | 8 | ||
(11001011010011101) | 8 | ||
(11011100001010111) | 8 | ||
(10001000101011011) | 8 | ||
(110000101101101000) | 8 | ||
(1100000111101000100) | 8 | ||
(1010011001110001110) | 8 | ||
(10011111001111010010) | 10 | ||
(101010000011101100110) | 10 | ||
(111100011011101010111) | 10 | ||
(110000111111101101101) | 10 | ||
(1110100010011100011000) | 10 | ||
(1111100111111001000101) | 10 | ||
(11010101000010011100010) | 10 | ||
(111110011001100111100010) | 10 | ||
(1010001000101001100100101) | 10 | ||
(11101011011000000010001110) | 10 | ||
(100111100001001000000100011) | 10 | ||
(1101101000010100111100110111) | 12 | ||
(11001101111100101010111101100) | 12 | ||
(110010100011110110110000000000) | 12 | ||
(1000010101011010011011010000000) | 12 | ||
(10101110111101100111111011010000) | 12 | ||
(100011110101110110010101010100000) | 12 | ||
(1101000101110100101011110000000000) | 14 | ||
(10011110101111100101111001110111000) | 14 |
Code | First row | ||
|
(100101010001111) | 8 | |
(1001101010001101) | 8 | ||
(1110111100010110) | 8 | ||
(1010100111011101) | 8 | ||
(1000110111011110) | 8 | ||
(1110010011010001) | 8 | ||
(1101101100101000) | 8 | ||
(1001001100111010) | 8 | ||
(1110000111110110) | 8 | ||
(1110000111011110) | 8 | ||
(1001010011010011) | 8 | ||
(11001011010011101) | 8 | ||
(11011100001010111) | 8 | ||
(10001000101011011) | 8 | ||
(110000101101101000) | 8 | ||
(1100000111101000100) | 8 | ||
(1010011001110001110) | 8 | ||
(10011111001111010010) | 10 | ||
(101010000011101100110) | 10 | ||
(111100011011101010111) | 10 | ||
(110000111111101101101) | 10 | ||
(1110100010011100011000) | 10 | ||
(1111100111111001000101) | 10 | ||
(11010101000010011100010) | 10 | ||
(111110011001100111100010) | 10 | ||
(1010001000101001100100101) | 10 | ||
(11101011011000000010001110) | 10 | ||
(100111100001001000000100011) | 10 | ||
(1101101000010100111100110111) | 12 | ||
(11001101111100101010111101100) | 12 | ||
(110010100011110110110000000000) | 12 | ||
(1000010101011010011011010000000) | 12 | ||
(10101110111101100111111011010000) | 12 | ||
(100011110101110110010101010100000) | 12 | ||
(1101000101110100101011110000000000) | 14 | ||
(10011110101111100101111001110111000) | 14 |
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