Abstract
Let \(A_q(n,d)\) denote the maximum size of a q-ary code with size n and minimum distance d. For most values of n and d, only lower and upper bounds on \(A_q(n,d)\) are known. In this paper we present 19 new lower bounds where \(q \in \{3,4,5\}\). The bounds are based on codes whose automorphisms are prescribed by transitive permutation groups. An exhaustive computer search was carried out to find the new codes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Agrell, E., Vardy, A., Zeger, K.: A table of upper bounds for binary codes. IEEE Trans. Inform. Theory 47, 3004–3006 (2001)
Assmus, E.F., Mattson, H.F.: New 5-designs. J. Combin. Theory 6, 122–151 (1969)
Assmus, E.F., Mattson, H.F.: On weights in quadratic-residue codes. Discrete Math. 3, 1–20 (1972)
Best, M., Brouwer, A.E., MacWilliams, F.J., Odlyzko, A.M., Sloane, N.J.A.: Bounds for binary codes of length less than 25. IEEE Trans. Inform. Theory 24, 81–93 (1978)
Bogdanova, G.T., Brouwer, A.E., Kapralov, S.N., Östergård, P.R.J.: Error-correcting codes over an alphabet of four elements. Des. Codes Cryptogr. 23, 333–342 (2001)
Boukliev, I., Kapralov, S., Maruta, T., Fukui, M.: Optimal linear codes of dimension 4 over \(F_5\). IEEE Trans. Inform. Theory 43, 308–313 (1997)
Bogdanova, G.T., Östergård, P.R.J.: Bounds on codes over an alphabet of five elements. Discrete Math. 240, 13–19 (2001)
Brouwer, A.E., Hämäläinen, H.O., Östergård, P.R.J., Sloane, N.J.A.: Bounds on mixed binary/ternary codes. IEEE Trans. Inform. Theory 44, 140–161 (1998)
Cannon, J.J., Holt, D.F.: The transitive permutation groups of degree 32. Exp. Math. 17, 307–314 (2008)
Código [pseud.], A discussion on Foros de Free1X2.com, cited by Andries Brouwer at https://www.win.tue.nl/~aeb/codes/ternary-1.html
Conway, J.H., Pless, V., Sloane, N.J.A.: Self-dual codes over GF(3) and GF(4) of length not exceeding 16. IEEE Trans. Inform. Theory 25, 312–322 (1979)
Elssel, K., Zimmermann, K.-H.: Two new nonlinear binary codes. IEEE Trans. Inform. Theory 51, 1189–1190 (2005)
Hulpke, A.: Constructing transitive permutation groups. J. Symbolic Comput. 39, 1–30 (2005)
Kaikkonen, M.K.: Codes from affine permutation groups. Des. Codes Cryptogr. 15, 183–186 (1998)
Kschischang, F.R., Subbarayan, P.: Some ternary and quaternary codes and associated sphere packings. IEEE Trans. Inform. Theory 38, 227–246 (1992)
Laaksonen, A., Östergård, P.R.J.: Constructing error-correcting binary codes using transitive permutation groups (submitted). Preprint at. arXiv:1604.06022
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes, North-Holland, Amsterdam (1977)
Niskanen, S., Östergård, P.R.J.: Cliquer User’s Guide, Version 1.0, Communications Laboratory, Helsinki University of Technology, Espoo, Finland, Technical report T48 (2003)
Östergård, P.R.J.: Two new four-error-correcting binary codes. Des. Codes Cryptogr. 36, 327–329 (2005)
Plotkin, M.: Binary codes with specified minimum distance. IRE Trans. Inform. Theory 6, 445–450 (1960)
Acknowledgments
This work was supported in part by the Academy of Finland, Project #289002.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix: Codes for the New Lower Bounds
Appendix: Codes for the New Lower Bounds
Bound: \(A_3(13,4) \ge 13122\)
Generators of G :
(1 10 35 32 31 28)(2 14 36 22 19 5)(3 25 20 4 24 21)
(6 18 15 27 23 9)(7 17 37 34 29 12)(8 16 38 33 30 11),
(1 25 22 21 5 17)(2 24 32 3 23 33)(4 27 38 35 8 31)
(6 16 36 7 28 11)(9 34 18 30 14 12)(10 20 15 37 19 29)
Orbit representatives:
1112000000000, 2001001000000, 0110201000000, 0220021000000,
0202002000000, 1201000000001, 2111010000001, 2022110000001,
2200120000001, 0002000100001, 1011100000002, 2220010000002,
0212110000002, 1202011000002, 2200021100002
Bound: \(A_3(15,4) \ge 83106\)
Generators of G :
(1 41 6)(2 45 7 5 42 25)(3 4 8 9 28 44)(10 32 30 37 35 27)(11 21 31)
(12 40 17 15 22 20)(13 14 18 19 38 39)(16 26 36)(23 24 43 29 33 34),
(1 2 10 44)(3 33)(4 11 7 35)(5 19 26 37)(6 27 45 39)(8 43)(9 36 42 30)
(12 15 24 21)(13 38)(14 31 32 25)(16 17 40 29)(18)(20 34 41 22)(23 28)
Orbit representatives:
200101000000000, 111201000000000, 022202000000000, 120112000000000,
222210100000000, 012120100000000, 102200200000000, 021111010000000,
020022110000000, 220100210000000, 210021210000000, 201210020000000,
001022020000000, 222001001000000, 210201201000000, 011012011000000,
012021111000000, 211001121000000, 201222221000000, 222211012000000,
210101112000000, 211110222000000, 001102222000000, 000000110100000,
221021021100000, 010112112200000
Bound: \(A_3(15,5) \ge 7812\)
Generators of G :
(1 13 34 22)(2 6 23 39 17 21 38 9 32 36 8 24)
(3 14 27 26 33 44 42 41 18 29 12 11)(4 7 16 43)
(5 45 35 30 20 15)(19 37 31 28),
(1 45 41 25 21 5)(2 4 12 29 7 9)(3 23 28)
(6 35 16 30 26 10)(8 13 18)(11 40 36 20 31 15)
(14 22 24 17 34 27)(19 42 44 37 39 32)(33 38 43)
Orbit representatives:
212221100000000, 002122200000000, 210110001000000, 112100102000000,
121212212000000, 000002022010000, 010202102122000
Generators of H :
(16 4 7 10 43)(31 19 37 40 28)(2 35 38 26 29)(17 5 8 41 14)
(32 20 23 11 44)(3 36 39 42 15)(18 21 24 27 45)(33 6 9 12 30)
(34 22 25 13 1)
Orbit representatives:
000011021101212, 000022002020201, 000111111111010, 000112122210020,
000122202220010, 000201021012210, 000201122122112, 000210120202021,
000220012002020, 000220120210100, 001010010100011, 001011112001020,
001022001012200, 001110111110022, 001121201222012, 001122121212102,
001200121121111, 001201102011212, 002010011001022, 002021000011202,
002022120122210, 002121120211101, 002200101010211, 002200220120102,
010002012000221, 010012110022220, 010102212200000, 010200100220120,
010211001022200, 010211102102102, 010212110100100, 011000101100201,
011001011002220, 011101210120002, 011102101222122, 011211112021202,
011211201101120, 012001010021222, 012002100102200, 012010111020221,
012210111101101, 012210200100122, 020010022000211, 020110112200110,
020112221101020, 021012021002210, 021111221212022, 022010020011212,
022110220211021, 100001221212110, 100020212202220, 100102000212100,
100111211001212, 100112210200221, 100120202210101, 101000220211112,
101101002211102, 101110220000111, 102000001210111, 102002222010110,
102110011200222, 110011201222100, 110100212220121, 110112010222120,
111010200221102, 111101211202120, 111111012221122, 112012202220101,
112100210201122, 120112222210111, 121112221202110, 122111220201112
Bound: \(A_3(15,6) \ge 3321\)
Generators of G :
(1 33 20 7 24 11 13 45 17 34 36 8 40 27 44)
(2 4 6 23 10 42 29 31 18 5 22 39 26 28 15)
(3 35 37 9 41 43 30 32 19 21 38 25 12 14 16),
(1 36 11)(2 10 42 5 22 15)(3 29 43 39 38 4)
(6 26 31)(7 45 17 40 27 20)(8 19 33 14 13 9)
(12 35 37 30 32 25)(16 21 41)(18 44 28 24 23 34)
Orbit representatives:
022102200000000, 111000121000000, 222222012000000, 110012000100000,
101120020200000
Bound: \(A_3(16,7) \ge 1026\)
Generators of G :
(1 21 18 38)(2 22 17 37)(3 23 36 8)(4 24 35 7)(5 34 6 33)
(9 29 26 46)(10 30 25 45)(11 31 44 16)(12 32 43 15)
(13 42 14 41)(19 39 20 40)(27 47 28 48),
(1 35 5)(2 36 6)(3 37 33)(4 38 34)(9 43 13)(10 44 14)
(11 45 41)(12 46 42)(17 19 21)(18 20 22)(25 27 29)(26 28 30)
Orbit representatives:
2000000020000000, 0122222201000000, 2221220000100000,
2010010012212000, 0112121210101010, 2001010102101010,
0221120112201010, 1012022020102010, 2002122110201020
Bound: \(A_3(16,8) \ge 387\)
Generators of G :
(1 20 7 8 5 38)(2 3 18 35 34 19)(4 23 40 21 22 17)(6 33 36 39 24 37)
(9 28 15 16 13 46)(10 11 26 43 42 27)(12 31 48 29 30 25)(14 41 44 47 32 45),
(1 23 8 3 34 22 36)(2 38 4 33 7 24 35)(6 20 17 39 40 19 18)
(9 31 16 11 42 30 44)(10 46 12 41 15 32 43)(13)(14 28 25 47 48 27 26)
Orbit representatives:
2200210222100000, 0201000120210000, 0000221122221100,
0202211002022110, 2020120102022110, 1111002202022110
Bound: \(A_4(8,4) \ge 352\)
Generators of G :
(1 20)(2 19)(3 26 11 18 27 10)(4 25 12 17 28 9)
(5 24)(6 23)(7 30 15 22 31 14)(8 29 16 21 32 13),
(1 3)(2 4)(5 7)(6 8)(9 27)(10 28)(11 25)(12 26)(13 31)
(14 32)(15 29)(16 30)(17 19)(18 20)(21 23)(22 24),
(1 17 9)(2 18 10)(3 11 19)(4 12 20)(5 21 13)(6 22 14)(7 15 23)(8 16 24)
Orbit representatives:
30100000, 21320000, 22002200, 11112200, 10201010, 01021010, 21212010,
12122010, 33003010, 03322110, 30232110, 30321210, 03231210, 11220310
Bound: \(A_4(9,4) \ge 1152\)
Generators of G :
(1 8 28 35 10 17 19 26)(2 7 29 34 11 16 20 25)
(3 33 21 6 12 24 30 15)(4 32 22 5 13 23 31 14),
(1 32 30 34 10 23 21 25)(2 33 31 35 11 24 22 26)
(3 16 28 5 12 7 19 14)(4 17 29 6 13 8 20 15),
(1 13)(2 12)(3 11)(4 10)(5 24)(6 23)(7 35)(8 34)(14 33)
(15 32)(16 26)(17 25)(19 22)(20 21)(28 31)(29 30)
Orbit representatives:
210020000, 022220000, 203220100, 202030300, 332020001, 311030101,
123230101, 130220201, 111220002, 013020102, 320220102, 031230302,
112030003, 221230003, 121020103, 303020203
Bound: \(A_4(9,6) \ge 76\)
Generators of G :
(1 30 24 14)(2 35 25 36)(3 33 23 19)(4 13)(5 10 12 6)(7 9 11 8)
(15 32 28 21)(16 18 29 17)(20 26 34 27)(22 31),
(1 11 34)(2 7 28)(3 22 17)(4 8 30)(5 9 33)(6 32 27)(10 29 16)
(12 31 35)(13 26 21)(14 18 24)(15 23 36)(19 20 25)
Orbit representatives:
221012000, 000322200
Bound: \(A_4(10,3) \ge 24576\)
Generators of G :
(1 37 31 7)(2 26 12 36)(3 5)(4 14)(6 22 16 32)(8 40 18 30)(9 19)
(10 38 20 28)(11 27 21 17)(13 15)(23 35)(24 34)(25 33)(29 39),
(1 6 21 36)(2 35)(3 4 13 14)(5 22)(7 10)(8 39 18 29)(9 38 19 28)
(11 16 31 26)(12 25)(15 32)(17 20)(23 34 33 24)(27 30)(37 40)
Orbit representatives:
1310000000, 3120220000
Bound: \(A_4(10,4) \ge 4192\)
Generators of G :
(1 29 4 6 11 39 24 36)(2 30 3 5 12 40 23 35)(7 38 27 18)(8 37 28 17)
(9 14 16 21 19 34 26 31)(10 13 15 22 20 33 25 32),
(1 19)(2 20)(3 7 23 27 33 37)(4 8 24 28 34 38)(5 36 25 6 35 26)
(9 11 29 31 39 21)(10 12 30 32 40 22)(13 17)(14 18)(15 16)
Orbit representatives:
0000000000, 3311000000, 0110301000, 1001301000, 2332301000,
1032121000, 1210303010
Bound: \(A_4(11,3) \ge 77056\)
Generators of G :
(1 32 23 10)(2 31)(3 41 14 19)(4 7 37 40)(5 6 16 28)(8 25 30 36)
(9 35)(12 21 34 43)(13 42)(15 18 26 29)(17 27 39 38)(20 24),
(1 35)(2 34 24 23 13 12)(3 10 36 43 14 32)(4 9 26 31 37 42)
(5 30 27 19 38 8)(6 29 39 40 17 7)(15 20)(16 41)(18 28)(21 25)
Orbit representatives:
10120000000, 02130000000, 23330000000, 02312000000, 12000000001,
31220000001, 20130000001, 33000000002, 10200000002, 01120000002,
11100000003, 22300000003, 30002000003
Bound: \(A_5(8,4) \ge 1225\)
Generators of G :
(1 20 40 10 27 7 17 36 16 26 3 23 33 12 32 2 19 39 9 28 8 18 35 15 25 4 24 34 11 31) (5 22 37 14 29 6 21 38 13 30),
(1 28 5 26 3 30)(2 27 6 25 4 29)(7 32)(8 31)(9 20 13 18 11 22)
(10 19 14 17 12 21)(15 24)(16 23)(33 36 37 34 35 38)(39 40)
Orbit representatives:
00000000, 41131000, 02241000, 24332000, 43411010, 34212010,
01010110, 02020220
Bound: \(A_5(8,5) \ge 165\)
Generators of G :
(1 32 19 5 33 24 11 37 25 16 3 29 17 8 35 21 9 40 27 13)
(2 31 20 6 34 23 12 38 26 15 4 30 18 7 36 22 10 39 28 14),
(1 10 17 26 33 2 9 18 25 34)(3 12 19 28 35 4 11 20 27 36)
(5 14 21 30 37 6 13 22 29 38)(7 16 23 32 39 8 15 24 31 40)
Orbit representatives:
33330000, 40304100, 13013200, 12340210, 30134210
Bound: \(A_5(9,4) \ge 4375\)
Generators of G :
(1 32 10 41 19 5 28 14 37 23)(2 33 11 42 20 6 29 15 38 24)
(3 43 21 25 39 7 12 34 30 16)(4 44 22 26 40 8 13 35 31 17),
(1 3 20 13)(2 4 19 12)(5 16 6 17)(7 33 26 23)(8 32 25 24)
(10 30 11 31)(14 34 42 44)(15 35 41 43)(21 29 40 37)(22 28 39 38)
Orbit representatives:
200020000, 112040000, 104010100, 232030100, 311000001, 023010101,
342010201, 121020301, 433000002, 143040102, 231010003, 411020103,
244000004, 322020004, 402040304
Bound: \(A_5(9,5) \ge 725\)
Generators of G :
(1 44 37 26)(2 45 38 27)(3 43 39 25)(7 12 16 30)(8 10 17 28)(9 11 18 29)
(13 22 40 31)(14 23 41 32)(15 24 42 33)(19 35)(20 36)(21 34),
(1 9 13 19 27 31 37 45 4 10 18 22 28 36 40)
(2 7 15 20 25 33 38 43 6 11 16 24 29 34 42)
(3 8 14 21 26 32 39 44 5 12 17 23 30 35 41)
Orbit representatives:
444222000, 231140100, 312401100, 003121100, 123014100
Bound: \(A_5(10,4) \ge 17500\)
Generators of G :
(1 32 33 44 5)(2 3 14 35 21)(4 45 11 42 43)(6 37 38 49 10)
(7 8 19 40 26)(9 50 16 47 48)(12 13 24 25 31)
(15 41 22 23 34)(17 18 29 30 36)(20 46 27 28 39),
(1 42)(2 11)(3 5 13 45 23 35 33 25 43 15)(4 44 34 24 14)
(6 47)(7 16)(8 10 18 50 28 40 38 30 48 20)(9 49 39 29 19)
(12 21)(17 26)(22 31)(27 36)(32 41)(37 46),
(1 32 33 44 25)(2 3 14 5 21)(4 15 11 42 43)(6 37 38 49 30)
(7 8 19 10 26)(9 20 16 47 48)(12 13 24 45 31)
(17 18 29 50 36)(22 23 34 35 41)(27 28 39 40 46)
Orbit representatives:
0000000000, 1331000000, 4342110000, 4112220000, 0442030000,
3114040000, 2333240000, 1222340000
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Laaksonen, A., Östergård, P.R.J. (2017). New Lower Bounds on Error-Correcting Ternary, Quaternary and Quinary Codes. In: Barbero, Á., Skachek, V., Ytrehus, Ø. (eds) Coding Theory and Applications. ICMCTA 2017. Lecture Notes in Computer Science(), vol 10495. Springer, Cham. https://doi.org/10.1007/978-3-319-66278-7_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-66278-7_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66277-0
Online ISBN: 978-3-319-66278-7
eBook Packages: Computer ScienceComputer Science (R0)