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An improved upper bound for the football pool problem for nine matches
Journal of Combinatorial Theory, Series A 102 (2003) 204–206
Note
An improved upper bound for the football pool problem for nine matches$ Franco Di Pasqualea and Patric R.J. O¨sterga˚rdb b
a Via Martiri della Liberta` 2, 64024 Notaresco (Teramo), Italy Department of Electrical and Communications Engineering, Helsinki University of Technology, P.O. Box 3000, 02015 HUT, Finland
Received 17 September 2002 Communicated by J.H. van Lint
Abstract The problem of determining K3 ðn; 1Þ; the minimum size of a ternary code of length n and covering radius 1, is called the football pool problem. By construction, it is here shown that K3 ð9; 1Þp1269: A code whose full automorphism group has order 648 is given, but this bound is actually achieved by a vast number of inequivalent codes. r 2003 Elsevier Science (USA). All rights reserved. Keywords: Automorphism group; Covering code; Football pool problem
A code CDZqn ; where Zq ¼ f0; 1; y; q 1g; is said to have covering radius R if R is the smallest positive integer such that dðx; CÞpR for all xAZqn : The function Kq ðn; RÞ denotes the smallest size of a code with the given parameters. See [1] for a thorough treatment of covering codes. The function K3 ðn; 1Þ has been intensively studied because of its application to gambling. Codes attaining upper bounds on this function can be used when forecasting the outcome of soccer (that is, European football) matches. Therefore, the problem of finding values of and bounds on K3 ðn; 1Þ has been coined the football pool problem [3]. The football pool problem is surveyed in [4]. For nine matches, it is currently known that 1060pK3 ð9; 1Þp1341; the lower and upper bound are from [2] and [5], respectively. (Although the code in [5] is made $
The research was supported in part by the Academy of Finland under grant 100500. E-mail addresses: [email protected] (F. Di Pasquale), patric.ostergard@hut.fi . (P.R.J. Osterg( ard). 0097-3165/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0097-3165(03)00010-4
Note / Journal of Combinatorial Theory, Series A 102 (2003) 204–206
205
obsolete by the result in the current note, we here take the opportunity to correct a misprint in that paper: the last column in [5, Eq. (2)] should be ½1 1 2 2 0 2 0T :) We shall now present a code that improves the upper bound to 1269. Two codes are said to be equivalent if one can be mapped onto the other by a permutation of the coordinates and n separate permutations of the coordinate values. Such a mapping of a code onto itself is an automorphism. All automorphisms form a group under composition; this group is called the (full) automorphism group of the code. The new code can be described in a dense form by listing generators of its automorphism group and one representative from each orbit of codewords under this group. For convenience, we present the group as a permutation group of degree 9 3 ¼ 27 acting on the set O ¼ f0; 1; y; 26g: For 0pip8 and 0pjp2; the element 3i þ jAO corresponds to value j in coordinate i: For example, the word 000110102 corresponds to the set f0; 3; 6; 10; 13; 15; 19; 21; 26g: The generators are as follows: ð3; 13; 18; 25Þð4; 12; 19; 24Þð5; 14; 20; 26Þð6; 17; 9; 21Þð7; 15; 10; 23Þð8; 16; 11; 22Þ; ð1; 10; 24; 2; 9; 26; 0; 11; 25Þð3; 21; 15; 4; 22; 17; 5; 23; 16Þð6; 18; 14; 8; 19; 13; 7; 20; 12Þ: This group has order 648 and acts transitively on the 27 points. Now the orbits in which the following representatives lie form a code with covering radius 1 (orbits of the same size are grouped together and the size is indicated): 27: 108: 162: 216:
000110102, 000010112, 000021012, 000022212, 000000011, 000000122, 000001101, 000101011.
A record-breaking code was found in a stochastic search (starting from a seed code but without imposing any symmetry on the code), and the aforementioned explanation was obtained by exploring the structure of the code found and modifying it slightly. It turns out that there is a vast number of inequivalent codes attaining the new upper bound. By removing the orbit of size 27, we get a code with 1242 codewords that cover all but 54 words, which belong to two orbits (of size 27) such that there is a one-to-one correspondence between words in these that are at distance 1 from each other. We may then take any of the two words from such a pair, or a third word that is at distance 1 from both words, to get 327 different codes, many of which are inequivalent.
References [1] G. Cohen, I. Honkala, S. Litsyn, A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997. [2] L. Habsieger, Some new lower bounds for ternary covering codes, Electron. J. Combin. 3 (2) (1996) R23 (electronic).
206
Note / Journal of Combinatorial Theory, Series A 102 (2003) 204–206
[3] H.J.L. Kamps, J.H. van Lint, The football pool problem for 5 matches, J. Combin. Theory 3 (1967) 315–325. [4] P.R.J. O¨sterga˚rd, The football pool problem, Congr. Numer. 114 (1996) 33–43. [5] P.R.J. O¨sterga˚rd, Constructing covering codes by tabu search, J. Combin. Des. 5 (1997) 71–80.
Note
An improved upper bound for the football pool problem for nine matches$ Franco Di Pasqualea and Patric R.J. O¨sterga˚rdb b
a Via Martiri della Liberta` 2, 64024 Notaresco (Teramo), Italy Department of Electrical and Communications Engineering, Helsinki University of Technology, P.O. Box 3000, 02015 HUT, Finland
Received 17 September 2002 Communicated by J.H. van Lint
Abstract The problem of determining K3 ðn; 1Þ; the minimum size of a ternary code of length n and covering radius 1, is called the football pool problem. By construction, it is here shown that K3 ð9; 1Þp1269: A code whose full automorphism group has order 648 is given, but this bound is actually achieved by a vast number of inequivalent codes. r 2003 Elsevier Science (USA). All rights reserved. Keywords: Automorphism group; Covering code; Football pool problem
A code CDZqn ; where Zq ¼ f0; 1; y; q 1g; is said to have covering radius R if R is the smallest positive integer such that dðx; CÞpR for all xAZqn : The function Kq ðn; RÞ denotes the smallest size of a code with the given parameters. See [1] for a thorough treatment of covering codes. The function K3 ðn; 1Þ has been intensively studied because of its application to gambling. Codes attaining upper bounds on this function can be used when forecasting the outcome of soccer (that is, European football) matches. Therefore, the problem of finding values of and bounds on K3 ðn; 1Þ has been coined the football pool problem [3]. The football pool problem is surveyed in [4]. For nine matches, it is currently known that 1060pK3 ð9; 1Þp1341; the lower and upper bound are from [2] and [5], respectively. (Although the code in [5] is made $
The research was supported in part by the Academy of Finland under grant 100500. E-mail addresses: [email protected] (F. Di Pasquale), patric.ostergard@hut.fi . (P.R.J. Osterg( ard). 0097-3165/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0097-3165(03)00010-4
Note / Journal of Combinatorial Theory, Series A 102 (2003) 204–206
205
obsolete by the result in the current note, we here take the opportunity to correct a misprint in that paper: the last column in [5, Eq. (2)] should be ½1 1 2 2 0 2 0T :) We shall now present a code that improves the upper bound to 1269. Two codes are said to be equivalent if one can be mapped onto the other by a permutation of the coordinates and n separate permutations of the coordinate values. Such a mapping of a code onto itself is an automorphism. All automorphisms form a group under composition; this group is called the (full) automorphism group of the code. The new code can be described in a dense form by listing generators of its automorphism group and one representative from each orbit of codewords under this group. For convenience, we present the group as a permutation group of degree 9 3 ¼ 27 acting on the set O ¼ f0; 1; y; 26g: For 0pip8 and 0pjp2; the element 3i þ jAO corresponds to value j in coordinate i: For example, the word 000110102 corresponds to the set f0; 3; 6; 10; 13; 15; 19; 21; 26g: The generators are as follows: ð3; 13; 18; 25Þð4; 12; 19; 24Þð5; 14; 20; 26Þð6; 17; 9; 21Þð7; 15; 10; 23Þð8; 16; 11; 22Þ; ð1; 10; 24; 2; 9; 26; 0; 11; 25Þð3; 21; 15; 4; 22; 17; 5; 23; 16Þð6; 18; 14; 8; 19; 13; 7; 20; 12Þ: This group has order 648 and acts transitively on the 27 points. Now the orbits in which the following representatives lie form a code with covering radius 1 (orbits of the same size are grouped together and the size is indicated): 27: 108: 162: 216:
000110102, 000010112, 000021012, 000022212, 000000011, 000000122, 000001101, 000101011.
A record-breaking code was found in a stochastic search (starting from a seed code but without imposing any symmetry on the code), and the aforementioned explanation was obtained by exploring the structure of the code found and modifying it slightly. It turns out that there is a vast number of inequivalent codes attaining the new upper bound. By removing the orbit of size 27, we get a code with 1242 codewords that cover all but 54 words, which belong to two orbits (of size 27) such that there is a one-to-one correspondence between words in these that are at distance 1 from each other. We may then take any of the two words from such a pair, or a third word that is at distance 1 from both words, to get 327 different codes, many of which are inequivalent.
References [1] G. Cohen, I. Honkala, S. Litsyn, A. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997. [2] L. Habsieger, Some new lower bounds for ternary covering codes, Electron. J. Combin. 3 (2) (1996) R23 (electronic).
206
Note / Journal of Combinatorial Theory, Series A 102 (2003) 204–206
[3] H.J.L. Kamps, J.H. van Lint, The football pool problem for 5 matches, J. Combin. Theory 3 (1967) 315–325. [4] P.R.J. O¨sterga˚rd, The football pool problem, Congr. Numer. 114 (1996) 33–43. [5] P.R.J. O¨sterga˚rd, Constructing covering codes by tabu search, J. Combin. Des. 5 (1997) 71–80.