# American Institute of Mathematical Sciences

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

## 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

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.

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
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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 $a$ $-6a+b$ $7a-12b+c$ $56583450+28a+62b-18c$
 $(2n,d)$ $A_0$ $A_d$ $A_{d+2}$ $A_{d+4}$ $A_{d+6}$ $(32, 8)$ 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 $a$ $-6a+b$ $7a-12b+c$ $56583450+28a+62b-18c$
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 -
 $2n$ $d_{P}$ $A_{d_{P}}$ $N_{P}$ $d_{B}$ $A_{d_{B}}$ $N_{B}$ $d_{SD}$ $A_{d_{SD}}$ 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 -
Pure double circulant even codes satisfying (C1)-(C3)
 Code First 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 $(8064,127809,1202464)$
 Code First 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 $(8064,127809,1202464)$
Bordered double circulant even codes satisfying (C1)-(C3)
 Code First 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 $(8190,126952,1204560)$
 Code First 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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