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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

T. A. Gulliver and M. Harada, On extremal double circulant self-dual codes of lengths 90-96, (submitted), arXiv: 1601.07343. Google Scholar

[15]

M. Harada, An extremal doubly even self-dual code of length 112, Electron. J. Combin. 15(2008), Note 33, 5 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

N. J. A. Sloane, Is there a (72, 36) d=16 self-dual code? IEEE Trans. Inform. Theory 19(1973), p251.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

T. A. Gulliver and M. Harada, On extremal double circulant self-dual codes of lengths 90-96, (submitted), arXiv: 1601.07343. Google Scholar

[15]

M. Harada, An extremal doubly even self-dual code of length 112, Electron. J. Combin. 15(2008), Note 33, 5 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

N. J. A. Sloane, Is there a (72, 36) d=16 self-dual code? IEEE Trans. Inform. Theory 19(1973), p251.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 $a$ $-6a+b$ $7a-12b+c$ $56583450+28a+62b-18c$
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$(2n,d)$ $A_0$ $A_d$ $A_{d+2}$ $A_{d+4}$ $A_{d+6}$
$(32, 8)$ 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 -
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$2n$ $d_{P}$ $A_{d_{P}}$ $N_{P}$ $d_{B}$ $A_{d_{B}}$ $N_{B}$ $d_{SD}$ $A_{d_{SD}}$
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 $(8064,127809,1202464)$
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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 $(8190,126952,1204560)$
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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 $(8190,126952,1204560)$
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