The smallest length of eight-dimensional binary linear codes with prescribed minimum distance

IEEE Transactions on Information Theory

Volumn 46, Issue 4, 2000, Pages 1539-1544

  Bouyukliev, Iliya ^a,b   Jaffe, David B ^a   Vavrek, Vesselin ^a  

^a INSTITUTE OF MATHEMATICS AND INFORMATICS   (Bulgaria)

^b UNIVERSITY OF NEBRASKA LINCOLN   (United States)

Author keywords

Binary linear code; Bounds; Minimum distance

Indexed keywords

ALGORITHMS; COMPUTATIONAL METHODS; COMPUTER AIDED SOFTWARE ENGINEERING; LINEAR PROGRAMMING; MATRIX ALGEBRA; THEOREM PROVING;

EIGHT DIMENSIONAL BINARY LINEAR CODE; RESIDUAL CODE;

BINARY CODES;

EID: 0034229311     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/18.850690     Document Type: Article

References (40)

1. L. D. Baumert and R. J. McEliece, A note on the Griesmer bound, IEEE Trans. Inform. Theory, vol. IT-19, pp. 134-135, Jan. 1973.
2. B. L. Belov, A conjecture on the Griesmer boundary" (in Russian), in Optimization Methods and Their Applications, All-Union Summer Sem., Khakusy, Lake Baikal, 1972, Sibersk. Energet. Inst. Sibirsk. Otdel. Akad. Nauk SSSR, Irkutsk, 1974, pp. 100-106, 182.
3. B. L. Belov, V. N. Logačev, and V. P. Sandimirov, Construction of a class of linear binary codes achieving the Varshamov-Griesmer bound, Probl. Pered. Inform., vol. 10, pp. 36-44, 1974.
4. J. Bierbrauer and Y. Edel, New code parameters from Reed-Solomon subfield codes, IEEE Trans. Inform. Theory, vol. 43, pp. 953-968, May 1997.
5. E. L. Blokh and V. V. Zyablov, Coding of generalized concatenated codes, Probl. Inform. Transm., vol. 10, pp. 218-222, 1974.
6. I. Boukliev, S. M. Dodunekov, T. Helleseth, and Ø. Ytrehus, On the [162, 8, 80] codes, IEEE Trans. Inform. Theory, vol. 43, pp. 2055-2057, Nov. 1997.
7. 28*, in Proc. 2nd Int. Workshop on Optimal Codes and Related Topics, Sozopol, Bulgaria, June 9-15, 1998. Sofia, Bulgaria: Inst. Math. Inform., 1998, pp. 28-32.
8. I. Bouyukliev and D. B. Jaffe, Optimal binary linear codes of dimension at most seven, Discr. Math., to be published.
9. A. E. Brouwer, The linear programming bound for binary linear codes, IEEE Trans. Inform. Theory, vol. 39, pp. 677-680, Mar. 1993.
10. A. E. Brouwer and T. Verhoeff, An updated table of minimum-distance bounds for binary linear codes, IEEE Trans. Inform. Theory, vol. 39, pp. 662-677, Mar. 1993.
11. C. L. Chen, Computer results on the minimum distance of some binary cyclic codes, IEEE Trans. Inform. Theory, vol. IT-16, pp. 359-360, May 1970.
12. P. Delsarte, Bounds for unrestricted codes, by linear programming, Philips Res. Rep., vol. 27, pp. 272-289, 1972.
13. S.M. Dodunekov, T. Helleseth, N.Manev, and Ø. Ytrehus, New bounds on binary linear codes of dimension eight, IEEE Trans. Inform. Theory, vol. IT-33, pp. 917-919, Nov. 1987.
14. S. M. Dodunekov and N. L. Manev, An improvement of the Griesmer bound for some small minimum distances, Discr. Appl. Math., vol. 12, pp. 103-114, 1985.
15. _, An improvement of the Griesmer bound for some classes of distances, Probl. Inform. Transm., vol. 23, pp. 38-46, 1987.
16. J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., vol. 4, pp. 532-542, 1960.
17. P. W. Heijnen, Er bestaat geen binaire [33, 9, 13] code, Afstudeerverslag, Tech. Univ. Delft, Delft, The Netherlands, 1993.
18. T. Helleseth, A characterization of codes meeting the Griesmer bound, Inform. Contr., vol. 50, pp. 128-159, 1981.
19. _, New constructions of codes meeting the Griesmer bound, IEEE Trans. Inform. Theory, vol. IT-29, pp. 434-439, May 1983.
20. T. Helleseth and H. van Tilborg, A new class of codes meeting the Griesmer bound, IEEE Trans. Inform. Theory, vol. IT-27, pp. 548-555, Sept. 1981.
21. T. Helleseth and Ø. Ytrehus, How to find a [33, 8, 14] code, Reports in Informatics (University of Bergen), vol. 41, 1989.
22. D. B. Jaffe. Binary linear codes: New results on nonexistence, preprint (ongoing work). [Online] http://www.math.unl.edu/djaffe/codes/code. ps.gz or code.dvi.gz; see ... djaffe/binary/codeform.html for an online database, which is more frequently updated.
23. _, Optimal binary linear codes of length ≤ 30, Discr. Math., to be published.
24. _, A brief tour of split linear programming, in Proc. AAECC 12 (Lecture Notes in Computer Science), T. Mora and H. Mattson, Eds. Springer-Verlag: New York, 1997, vol. 1255, pp. 164-173.
25. D. B. Jaffe and J. Simonis, New binary linear codes which are dual transforms of good codes, IEEE Trans. Inform. Theory, vol. 45, pp. 2136-2137, Sept. 1999.
26. B. K. Kostova and N. L. Manev, A [25, 8, 10] code does not exist, in C. R. Acad. Bulg. Sci., 1990, vol. 43, pp. 41-44.
27. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977.
28. B. D. McKay, Practical graph isomorphism, Congr. Numer., vol. 30, pp. 45-87, 1981.
29. _, (1990) Nauty User's Guide (Version 1.5). [Online] http://cs.anu. edu.au/people/bdm/nauty19/nauty19p.tar.Z.
30. P. Piret, Good linear codes of lengths 27 and 28, IEEE Trans. Inform. Theory, vol. IT-26, p. 227, Mar. 1980.
31. A. Said and R. Palazzo, Using combinatorial optimization to design good unit-memory convolutional codes, IEEE Trans. Inform. Theory, vol. 39, pp. 1100-1108, May 1993.
32. J. Simonis, MacWilliams identities and coordinate partitions, Linear Alg. Appl., vol. 216, pp. 81-91, 1995.
33. G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. Contr., vol. 8, pp. 170-179, 1965.
34. Y. Sugiyama, M. Kasahara, S. Hirasawa, and T. Namekawa, Further results on Goppa codes and their applications to constructing efficient binary codes, IEEE Trans. Inform. Theory, vol. IT-22, pp. 518-526, Sept. 1976.
35. H. van Tilborg, On quasicyclic codes with rate 1/m, IEEE Trans. Inform. Theory, vol. IT-24, pp. 628-630, Sept. 1978.
36. _, A proof of the nonexistence of a binary (55, 7, 26) code, Techn. Hogeschool Eindhoven, Eindhoven, The Netherlands, Tech. Rep. 79-WSK-09, 1979.
37. _, The smallest length of binary 7-dimensional linear codes with prescribed minimum distance, Discr. Math., vol. 33, pp. 197-207, 1981.
38. S. Topalova, Construction and investigation of combinatorial designs with given automorphisms, PhD dissertation, Inst. Math. and Informatics, Bulgarian Acad. Sci., Bulgaria, 1998.
39. Ø. Ytrehus and T. Helleseth, There is no binary [25, 8, 10] code, IEEE Trans. Inform. Theory, vol. 36, pp. 695-696, May 1990.
40. Reference [10] supplemented by on-line updates: information regarding [n, k, d] codes with fixed n, k. [Online] Available: http:// www.win.tue.nl/aeb/voorlincod.html