American Institute of Mathematical Sciences

Advanced
November  2017, 11(4): 767-775. doi: 10.3934/amc.2017056

On the performance of optimal double circulant even codes

T. Aaron Gulliver 1, and  Masaaki Harada 2,,

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

* Corresponding author: Masaaki Harada

Received  April 2016 Published  November 2017

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.

Keywords: Double circulant code, self-dual code, weight distribution, decoding error probability.
Mathematics Subject Classification: Primary: 94B05; Secondary: 94B70.
Citation: T. Aaron Gulliver, Masaaki Harada. On the performance of optimal double circulant even codes. Advances in Mathematics of Communications, 2017, 11 (4) : 767-775. doi: 10.3934/amc.2017056
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.

Table 1.  Possible weight enumerators $W_{2n, d}$
$(2n,d)$ $A_0$ $A_d$ $A_{d+2}$ $A_{d+4}$ $A_{d+6}$
$(32, 8)$ 1 $a$ $4960-8a$ $-3472+28a$ $34720-56a$
$(34, 8)$ 1 $a$ $4114-7a$ $2516+20a$ $29172-28a$
$(36, 8)$ 1 $a$ $3366-6a$ $6630+13a$ $30600-8a$
$(38, 8)$ 1 $a$ $2717-5a$ $9177+7a$ $35910+5a$
$(40, 8)$ 1 $a$ $-4a+b$ $32110+2a-10b$ $-54720+12a+45b$
$(42,10)$ 1 $a$ $26117-9a$ $-10455+35a$ $286713-75a$
$(44,10)$ 1 $a$ $21021-8a$ $19712+26a$ $250778-40a$
$(46,10)$ 1 $a$ $16744-7a$ $38709+18a$ $249458-14a$
$(48,10)$ 1 $a$ $-6a+b$ $207552+11a-12b$ $-606441+4a+66b$
$(50,10)$ 1 $a$ $-5a+b$ $166600+5a-11b$ $-271950+15a+54b$
$(52,10)$ 1 $a$ $-4a+b$ $132600-10b$ $-41990+20a+43b$
$(54,10)$ 1 $a$ $-3a+b$ $104652-4a-9b$ $107406+20a+33b$
$(56,12)$ 1 $a$ $-8a+b$ $1343034+24a-14b$ $-5765760-24a+91b$
$(58,12)$ 1 $a$ $-7a+b$ $1067838+16a-13b$ $-3224452+77b$
$(60,12)$ 1 $a$ $-6a+b$ $843030+9a-12b$ $-1454640+16a+64b$
$(62,12)$ 1 $a$ $-5a+b$ $660858+3a-11b$ $-270940+25a+52b$
$(64,12)$ 1 $a$ $-4a+b$ $-2a-10b+c$ $8707776+28a+41b-16c$
$(66,12)$ 1 $a$ $-3a+b$ $-6a-9b+c$ $6874010+26a+31b-15c$
$(68,12)$ 1 $a$ $-2a+b$ $-9a-8b+c$ $5393454+20a+22b-14c$
$(70,12)$ 1 $a$ $-a+b$ $-11a-7b+c$ $4206125+11a+14b-13c$
$(72,14)$ 1 $a$ $-6a+b$ $7a-12b+c$ $56583450+28a+62b-18c$
Table Options

Download as excel

$(2n,d)$ $A_0$ $A_d$ $A_{d+2}$ $A_{d+4}$ $A_{d+6}$
$(32, 8)$ 1 $a$ $4960-8a$ $-3472+28a$ $34720-56a$
$(34, 8)$ 1 $a$ $4114-7a$ $2516+20a$ $29172-28a$
$(36, 8)$ 1 $a$ $3366-6a$ $6630+13a$ $30600-8a$
$(38, 8)$ 1 $a$ $2717-5a$ $9177+7a$ $35910+5a$
$(40, 8)$ 1 $a$ $-4a+b$ $32110+2a-10b$ $-54720+12a+45b$
$(42,10)$ 1 $a$ $26117-9a$ $-10455+35a$ $286713-75a$
$(44,10)$ 1 $a$ $21021-8a$ $19712+26a$ $250778-40a$
$(46,10)$ 1 $a$ $16744-7a$ $38709+18a$ $249458-14a$
$(48,10)$ 1 $a$ $-6a+b$ $207552+11a-12b$ $-606441+4a+66b$
$(50,10)$ 1 $a$ $-5a+b$ $166600+5a-11b$ $-271950+15a+54b$
$(52,10)$ 1 $a$ $-4a+b$ $132600-10b$ $-41990+20a+43b$
$(54,10)$ 1 $a$ $-3a+b$ $104652-4a-9b$ $107406+20a+33b$
$(56,12)$ 1 $a$ $-8a+b$ $1343034+24a-14b$ $-5765760-24a+91b$
$(58,12)$ 1 $a$ $-7a+b$ $1067838+16a-13b$ $-3224452+77b$
$(60,12)$ 1 $a$ $-6a+b$ $843030+9a-12b$ $-1454640+16a+64b$
$(62,12)$ 1 $a$ $-5a+b$ $660858+3a-11b$ $-270940+25a+52b$
$(64,12)$ 1 $a$ $-4a+b$ $-2a-10b+c$ $8707776+28a+41b-16c$
$(66,12)$ 1 $a$ $-3a+b$ $-6a-9b+c$ $6874010+26a+31b-15c$
$(68,12)$ 1 $a$ $-2a+b$ $-9a-8b+c$ $5393454+20a+22b-14c$
$(70,12)$ 1 $a$ $-a+b$ $-11a-7b+c$ $4206125+11a+14b-13c$
$(72,14)$ 1 $a$ $-6a+b$ $7a-12b+c$ $56583450+28a+62b-18c$
Table 2.  Double circulant even codes satisfying (C1)-(C3)
$2n$ $d_{P}$ $A_{d_{P}}$ $N_{P}$ $d_{B}$ $A_{d_{B}}$ $N_{B}$ $d_{SD}$ $A_{d_{SD}}$
32 8 $348$ 2 8 $ 300$ 1 8 364 [5]
34 8 $272$ 15 8 $ 272$ 10 6 -
36 8 $153$ 4 8 $ 153$ 3 8 225 [5]
38 8 $ 76$ 1 8 $ 72$ 1 8 171 [5]
40 8 $ 25$ 1 8 $ 38$ 2 8 125 [5]
42 10 $1680$ 2 10 $1682$ 1 8 -
44 10 $1144$ 1 10 $1267$ 3 8 -
46 10 $851$ 1 10 $ 858$ 2 10 1012 [5]
48 10 $480$ 1 10 $ 575$ 1 12 17296 [5]
50 10 $325$ 1 10 $ 356$ 1 10 196 [20]
52 10 $156$ 1 10 $ 150$ 1 10 250 [5]
54 10 $ 27$ 1 10 $ 52$ 1 10 7-135 [3], [5]
56 12 $4060$ 1 10 $ 3$ 1 12 4606-8190 [5]
58 12 $3161$ 1 12 $3227$ 1 10 -
60 12 $2095$ 1 12 $2146$ 1 12 2555 [23]
62 12 $1333$ 1 12 $1290$ 1 12 1860 [6]
64 12 $544$ 1 12 $806$ 1 12 1312 [4]
66 12 $374$ 1 12 $480$ 1 12 858 [5] (see [7])
68 12 $136$ 1 12 $165$ 1 12 442-486 [7], [26]
70 12 $35$ 1 14 12172 1 12-14 -
72 14 $8064$ 1 14 $8190$ 1 12-16 -
Table Options

Download as excel

$2n$ $d_{P}$ $A_{d_{P}}$ $N_{P}$ $d_{B}$ $A_{d_{B}}$ $N_{B}$ $d_{SD}$ $A_{d_{SD}}$
32 8 $348$ 2 8 $ 300$ 1 8 364 [5]
34 8 $272$ 15 8 $ 272$ 10 6 -
36 8 $153$ 4 8 $ 153$ 3 8 225 [5]
38 8 $ 76$ 1 8 $ 72$ 1 8 171 [5]
40 8 $ 25$ 1 8 $ 38$ 2 8 125 [5]
42 10 $1680$ 2 10 $1682$ 1 8 -
44 10 $1144$ 1 10 $1267$ 3 8 -
46 10 $851$ 1 10 $ 858$ 2 10 1012 [5]
48 10 $480$ 1 10 $ 575$ 1 12 17296 [5]
50 10 $325$ 1 10 $ 356$ 1 10 196 [20]
52 10 $156$ 1 10 $ 150$ 1 10 250 [5]
54 10 $ 27$ 1 10 $ 52$ 1 10 7-135 [3], [5]
56 12 $4060$ 1 10 $ 3$ 1 12 4606-8190 [5]
58 12 $3161$ 1 12 $3227$ 1 10 -
60 12 $2095$ 1 12 $2146$ 1 12 2555 [23]
62 12 $1333$ 1 12 $1290$ 1 12 1860 [6]
64 12 $544$ 1 12 $806$ 1 12 1312 [4]
66 12 $374$ 1 12 $480$ 1 12 858 [5] (see [7])
68 12 $136$ 1 12 $165$ 1 12 442-486 [7], [26]
70 12 $35$ 1 14 12172 1 12-14 -
72 14 $8064$ 1 14 $8190$ 1 12-16 -
Table 3.  Pure double circulant even codes satisfying (C1)-(C3)
CodeFirst row $d$ $(A_d,A_{d+2},A_{d+4})$
$P_{32,1}$ (1100101100110101) 8 $(348,2176,6272)$
$P_{32,2}$ (1110110100010011) 8 $(348,2176,6272)$
$P_{34,1}$ (11111110001000100) 8 $(272,2210,7956)$
$P_{34,2}$ (11100000111010110) 8 $(272,2210,7956)$
$P_{34,3}$ (11110101101101100) 8 $(272,2210,7956)$
$P_{34,4}$ (11110011101101010) 8 $(272,2210,7956)$
$P_{34,5}$ (10001011101100000) 8 $(272,2210,7956)$
$P_{34,6}$ (10001100110010100) 8 $(272,2210,7956)$
$P_{34,7}$ (11101110110100000) 8 $(272,2210,7956)$
$P_{34,8}$ (10100101100011110) 8 $(272,2210,7956)$
$P_{34,9}$ (10100100110010001) 8 $(272,2210,7956)$
$P_{34,10}$ (10101010011111000) 8 $(272,2210,7956)$
$P_{34,11}$ (10001110000100110) 8 $(272,2210,7956)$
$P_{34,12}$ (11010010010001111) 8 $(272,2210,7956)$
$P_{34,13}$ (10001100001110100) 8 $(272,2210,7956)$
$P_{34,14}$ (11011010100001101) 8 $(272,2210,7956)$
$P_{34,15}$ (11100001101010011) 8 $(272,2210,7956)$
$P_{36,1}$ (101011110110000001) 8 $(153,2448,8619)$
$P_{36,2}$ (111100001000010111) 8 $(153,2448,8619)$
$P_{36,3}$ (100110111010010001) 8 $(153,2448,8619)$
$P_{36,4}$ (100001010110111100) 8 $(153,2448,8619)$
$P_{38}$ (1111000001001010110) 8 $(76,2337,9709)$
$P_{40}$ (10101101111101111000) 8 $(25,2080,10360)$
$P_{42,1}$ (100001101101110010110) 10 $(1680,10997,48345)$
$P_{42,2}$ (101010010101110110111) 10 $(1680,10997,48345)$
$P_{44}$ (1001111111001011011011) 10 $(1144,11869,49456)$
$P_{46}$ (11001011010111100000001) 10 $(851,10787,54027)$
$P_{48}$ (110111000101111101110100) 10 $(480,10384,53664)$
$P_{50}$ (1000100001011001001011101) 10 $(325,8650,55200)$
$P_{52}$ (10001010100011011011000001) 10 $(156,7267,53690)$
$P_{54}$ (111000000011101101100010011) 10 $(27,6030,49545)$
$P_{56}$ (1001100011110101110111110100) 12 $(4060,49420,293874)$
$P_{58}$ (11011000010100000000110011010) 12 $(3161,41412,292407)$
$P_{60}$ (100000101101110000100111010001) 12 $(2095,37320,263205)$
$P_{62}$ (0010100111101100111111010000000) 12 $(1333,30597,254975)$
$P_{64}$ (10101000110010111100110100000000) 12 $(544,34304,115756)$
$P_{66}$ (100100010010000101111011100100000) 12 $(374,20163,203808)$
$P_{68}$ (1001001011010110101010101011000000) 12 $(136,15606,176936)$
$P_{70}$ (01011011100110100101110000110000000) 12 $(35,11550,151130)$
$P_{72}$ (101101101101001101001101111100010000) 14 $(8064,127809,1202464)$
Table Options

Download as excel

CodeFirst row $d$ $(A_d,A_{d+2},A_{d+4})$
$P_{32,1}$ (1100101100110101) 8 $(348,2176,6272)$
$P_{32,2}$ (1110110100010011) 8 $(348,2176,6272)$
$P_{34,1}$ (11111110001000100) 8 $(272,2210,7956)$
$P_{34,2}$ (11100000111010110) 8 $(272,2210,7956)$
$P_{34,3}$ (11110101101101100) 8 $(272,2210,7956)$
$P_{34,4}$ (11110011101101010) 8 $(272,2210,7956)$
$P_{34,5}$ (10001011101100000) 8 $(272,2210,7956)$
$P_{34,6}$ (10001100110010100) 8 $(272,2210,7956)$
$P_{34,7}$ (11101110110100000) 8 $(272,2210,7956)$
$P_{34,8}$ (10100101100011110) 8 $(272,2210,7956)$
$P_{34,9}$ (10100100110010001) 8 $(272,2210,7956)$
$P_{34,10}$ (10101010011111000) 8 $(272,2210,7956)$
$P_{34,11}$ (10001110000100110) 8 $(272,2210,7956)$
$P_{34,12}$ (11010010010001111) 8 $(272,2210,7956)$
$P_{34,13}$ (10001100001110100) 8 $(272,2210,7956)$
$P_{34,14}$ (11011010100001101) 8 $(272,2210,7956)$
$P_{34,15}$ (11100001101010011) 8 $(272,2210,7956)$
$P_{36,1}$ (101011110110000001) 8 $(153,2448,8619)$
$P_{36,2}$ (111100001000010111) 8 $(153,2448,8619)$
$P_{36,3}$ (100110111010010001) 8 $(153,2448,8619)$
$P_{36,4}$ (100001010110111100) 8 $(153,2448,8619)$
$P_{38}$ (1111000001001010110) 8 $(76,2337,9709)$
$P_{40}$ (10101101111101111000) 8 $(25,2080,10360)$
$P_{42,1}$ (100001101101110010110) 10 $(1680,10997,48345)$
$P_{42,2}$ (101010010101110110111) 10 $(1680,10997,48345)$
$P_{44}$ (1001111111001011011011) 10 $(1144,11869,49456)$
$P_{46}$ (11001011010111100000001) 10 $(851,10787,54027)$
$P_{48}$ (110111000101111101110100) 10 $(480,10384,53664)$
$P_{50}$ (1000100001011001001011101) 10 $(325,8650,55200)$
$P_{52}$ (10001010100011011011000001) 10 $(156,7267,53690)$
$P_{54}$ (111000000011101101100010011) 10 $(27,6030,49545)$
$P_{56}$ (1001100011110101110111110100) 12 $(4060,49420,293874)$
$P_{58}$ (11011000010100000000110011010) 12 $(3161,41412,292407)$
$P_{60}$ (100000101101110000100111010001) 12 $(2095,37320,263205)$
$P_{62}$ (0010100111101100111111010000000) 12 $(1333,30597,254975)$
$P_{64}$ (10101000110010111100110100000000) 12 $(544,34304,115756)$
$P_{66}$ (100100010010000101111011100100000) 12 $(374,20163,203808)$
$P_{68}$ (1001001011010110101010101011000000) 12 $(136,15606,176936)$
$P_{70}$ (01011011100110100101110000110000000) 12 $(35,11550,151130)$
$P_{72}$ (101101101101001101001101111100010000) 14 $(8064,127809,1202464)$
Table 4.  Bordered double circulant even codes satisfying (C1)-(C3)
CodeFirst row $d$ $(A_d,A_{d+2},A_{d+4})$
$B_{32}$ (100101010001111) 8 $(300,2560,4928)$
$B_{34,1}$ (1001101010001101) 8 $(272,2210,7956)$
$B_{34,2}$ (1110111100010110) 8 $(272,2210,7956)$
$B_{34,3}$ (1010100111011101) 8 $(272,2210,7956)$
$B_{34,4}$ (1000110111011110) 8 $(272,2210,7956)$
$B_{34,5}$ (1110010011010001) 8 $(272,2210,7956)$
$B_{34,6}$ (1101101100101000) 8 $(272,2210,7956)$
$B_{34,7}$ (1001001100111010) 8 $(272,2210,7956)$
$B_{34,8}$ (1110000111110110) 8 $(272,2210,7956)$
$B_{34,9}$ (1110000111011110) 8 $(272,2210,7956)$
$B_{34,10}$ (1001010011010011) 8 $(272,2210,7956)$
$B_{36,1}$ (11001011010011101) 8 $(153,2448,8619)$
$B_{36,2}$ (11011100001010111) 8 $(153,2448,8619)$
$B_{36,3}$ (10001000101011011) 8 $(153,2448,8619)$
$B_{38}$ (110000101101101000) 8 $(72,2357,9681)$
$B_{40,1}$ (1100000111101000100) 8 $(38,2014,10526)$
$B_{40,2}$ (1010011001110001110) 8 $(38,2014,10526)$
$B_{42}$ (10011111001111010010) 10 $(1682,10979,48415)$
$B_{44,1}$ (101010000011101100110) 10 $(1267,10885,52654)$
$B_{44,2}$ (111100011011101010111) 10 $(1267,10885,52654)$
$B_{44,3}$ (110000111111101101101) 10 $(1267,10885,52654)$
$B_{46,1}$ (1110100010011100011000) 10 $(858,10738,54153)$
$B_{46,2}$ (1111100111111001000101) 10 $(858,10738,54153)$
$B_{48}$ (11010101000010011100010) 10 $(575,9752,55453)$
$B_{50}$ (111110011001100111100010) 10 $(356,8524,55036)$
$B_{52}$ (1010001000101001100100101) 10 $(150,7375,52850)$
$B_{54}$ (11101011011000000010001110) 10 $(52,5876,50156)$
$B_{56}$ (100111100001001000000100011) 10 $(3,4545,45477)$
$B_{58}$ (1101101000010100111100110111) 12 $(3227,40950,293463)$
$B_{60}$ (11001101111100101010111101100) 12 $(2146,36163,273876)$
$B_{62}$ (110010100011110110110000000000) 12 $(1290,30850,254428)$
$B_{64}$ (1000010101011010011011010000000) 12 $(806,25358,226982)$
$B_{66}$ (10101110111101100111111011010000) 12 $(480,19848,203112)$
$B_{68}$ (100011110101110110010101010100000) 12 $(165,15620,176099)$
$B_{70}$ (1101000101110100101011110000000000) 14 $(12172,147390,1352811)$
$B_{72}$ (10011110101111100101111001110111000) 14 $(8190,126952,1204560)$
Table Options

Download as excel

CodeFirst row $d$ $(A_d,A_{d+2},A_{d+4})$
$B_{32}$ (100101010001111) 8 $(300,2560,4928)$
$B_{34,1}$ (1001101010001101) 8 $(272,2210,7956)$
$B_{34,2}$ (1110111100010110) 8 $(272,2210,7956)$
$B_{34,3}$ (1010100111011101) 8 $(272,2210,7956)$
$B_{34,4}$ (1000110111011110) 8 $(272,2210,7956)$
$B_{34,5}$ (1110010011010001) 8 $(272,2210,7956)$
$B_{34,6}$ (1101101100101000) 8 $(272,2210,7956)$
$B_{34,7}$ (1001001100111010) 8 $(272,2210,7956)$
$B_{34,8}$ (1110000111110110) 8 $(272,2210,7956)$
$B_{34,9}$ (1110000111011110) 8 $(272,2210,7956)$
$B_{34,10}$ (1001010011010011) 8 $(272,2210,7956)$
$B_{36,1}$ (11001011010011101) 8 $(153,2448,8619)$
$B_{36,2}$ (11011100001010111) 8 $(153,2448,8619)$
$B_{36,3}$ (10001000101011011) 8 $(153,2448,8619)$
$B_{38}$ (110000101101101000) 8 $(72,2357,9681)$
$B_{40,1}$ (1100000111101000100) 8 $(38,2014,10526)$
$B_{40,2}$ (1010011001110001110) 8 $(38,2014,10526)$
$B_{42}$ (10011111001111010010) 10 $(1682,10979,48415)$
$B_{44,1}$ (101010000011101100110) 10 $(1267,10885,52654)$
$B_{44,2}$ (111100011011101010111) 10 $(1267,10885,52654)$
$B_{44,3}$ (110000111111101101101) 10 $(1267,10885,52654)$
$B_{46,1}$ (1110100010011100011000) 10 $(858,10738,54153)$
$B_{46,2}$ (1111100111111001000101) 10 $(858,10738,54153)$
$B_{48}$ (11010101000010011100010) 10 $(575,9752,55453)$
$B_{50}$ (111110011001100111100010) 10 $(356,8524,55036)$
$B_{52}$ (1010001000101001100100101) 10 $(150,7375,52850)$
$B_{54}$ (11101011011000000010001110) 10 $(52,5876,50156)$
$B_{56}$ (100111100001001000000100011) 10 $(3,4545,45477)$
$B_{58}$ (1101101000010100111100110111) 12 $(3227,40950,293463)$
$B_{60}$ (11001101111100101010111101100) 12 $(2146,36163,273876)$
$B_{62}$ (110010100011110110110000000000) 12 $(1290,30850,254428)$
$B_{64}$ (1000010101011010011011010000000) 12 $(806,25358,226982)$
$B_{66}$ (10101110111101100111111011010000) 12 $(480,19848,203112)$
$B_{68}$ (100011110101110110010101010100000) 12 $(165,15620,176099)$
$B_{70}$ (1101000101110100101011110000000000) 14 $(12172,147390,1352811)$
$B_{72}$ (10011110101111100101111001110111000) 14 $(8190,126952,1204560)$
[1]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032
[2]

Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261
[3]

Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011
[4]

Suat Karadeniz, Bahattin Yildiz. Double-circulant and bordered-double-circulant constructions for self-dual codes over $R_2$. Advances in Mathematics of Communications, 2012, 6 (2) : 193-202. doi: 10.3934/amc.2012.6.193
[5]

Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027
[6]

T. Aaron Gulliver, Masaaki Harada, Hiroki Miyabayashi. Double circulant and quasi-twisted self-dual codes over $\mathbb F_5$ and $\mathbb F_7$. Advances in Mathematics of Communications, 2007, 1 (2) : 223-238. doi: 10.3934/amc.2007.1.223
[7]

Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503
[8]

Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393
[9]

Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433
[10]

Ken Saito. Self-dual additive $ \mathbb{F}_4 $-codes of lengths up to 40 represented by circulant graphs. Advances in Mathematics of Communications, 2019, 13 (2) : 213-220. doi: 10.3934/amc.2019014
[11]

Joe Gildea, Adrian Korban, Abidin Kaya, Bahattin Yildiz. Constructing self-dual codes from group rings and reverse circulant matrices. Advances in Mathematics of Communications, 2021, 15 (3) : 471-485. doi: 10.3934/amc.2020077
[12]

Joe Gildea, Abidin Kaya, Adam Michael Roberts, Rhian Taylor, Alexander Tylyshchak. New self-dual codes from $ 2 \times 2 $ block circulant matrices, group rings and neighbours of neighbours. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021039
[13]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889
[14]

Masaaki Harada. New doubly even self-dual codes having minimum weight 20. Advances in Mathematics of Communications, 2020, 14 (1) : 89-96. doi: 10.3934/amc.2020007
[15]

Joe Gildea, Adrian Korban, Adam M. Roberts, Alexander Tylyshchak. Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022021
[16]

Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
[17]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83
[18]

José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Information--bit error rate and false positives in an MDS code. Advances in Mathematics of Communications, 2015, 9 (2) : 149-168. doi: 10.3934/amc.2015.9.149
[19]

María Chara, Ricardo A. Podestá, Ricardo Toledano. The conorm code of an AG-code. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021018
[20]

Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (135)
  • HTML views (294)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy