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A New Lower Bound for the Football Pool Problem for Six Matches
Journal of Combinatorial Theory, Series A 99, 175–179 (2002) doi:10.1006/jcta.2002.3260
NOTE
A New Lower Bound for the Football Pool Problem for Six Matches . sterg(ard1 Patric R. J. O Department of Computer Science and Engineering, Helsinki University of Technology, P.O. Box 5400, 02015 HUT, Finland E-mail: patric.ostergard@hut.fi
and Alfred Wassermann Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany E-mail: [email protected]
Received March 16, 2001
In the football pool problem one wants to minimize the cardinality of a ternary code, C F3n ; with covering radius one, and the size of a minimum code is denoted by sn : The smallest unsettled case is 634s6 473: The lower bound is here improved to 65 in a coordinate-by-coordinate backtrack search using the LLL algorithm and complete equivalence checking of subcodes. # 2002 Elsevier Science (USA) Key Words: covering radius; football pool problem; LLL algorithm; ternary code.
1. INTRODUCTION The football pool problem is one of the oldest and most famous problems in combinatorial coding theory. For a given length n; one wants to find a set of ternary words, C F3n ; such that for any x 2 F3n there is a word in C that differs from x in at most one coordinate. The minimum size of such a set is denoted by sn : This is a special case of the more general problem of finding Kq ðn; RÞ; the minimum size of q-ary codes of length n and covering radius R: Obviously, sn ¼ K3 ðn; 1Þ: For 14n43 it is not difficult to prove that the corresponding values of sn are 1, 3, and 5, respectively. The ternary Hamming code of length 4 gives s4 ¼ 9; and by lengthening this code ðsnþ1 43sn Þ we get s5 427: Actually 1
Supported by the Academy of Finland.
175 0097-3165/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.
176
NOTE
s5 ¼ 27; which is proved in [2]. This is the largest n for which sn is known, apart from the lengths for which ternary Hamming codes exist. This first open case s6 has received a lot of attention along the years. Lengthening gives s6 43s5 ¼ 81; which was improved to 79 in [12] and later to 74 and 73 in [13] and [3], respectively. A combinatorial proof of s6 473 is given in [7]. It should be mentioned that this record was obtained even earlier by constructors of football pool systems who, unfortunately, did not publish it in the mathematical literature. As for lower bounds, the Hamming bound gives s6 5 729=13 ¼ 57: The best lower bound in the literature [12] is s6 563: However, as the ‘‘proof’’ of this bound restricts to the comment ‘‘By methods similar to [those used in [2]], we could show that s6 563,’’ this result is often ignored. A welldocumented proof of s6 560 is published in [1]. In this work, we improve the lower bound on s6 by proving nonexistence of codes. This is done in a coordinate-by-coordinate backtrack search, which is described in Section 2. In this search, the LLL algorithm and equivalence checking of subcodes play central roles. Application of the LLL algorithm is discussed in Section 3. The results of the search leading to the new bound s6 565 are presented in Section 4.
2. CONSTRUCTING COVERING CODES A code in F3n with M words can be expressed using an n M matrix with entries from f0; 1; 2g: The fact that there are many more columns than rows for the instances in this work suggests a row-by-row (that is, coordinate-bycoordinate) approach for our backtrack search. The search described here is closely related to that in [8], which in turn is based on old ideas from [2, 9]. Several new ideas, including the use of the LLL algorithm, have been incorporated here to enhance the search method. The covering property of the code to be constructed can be expressed using a set of linear (in)equalities. Let Qx F3n denote the set of 3n m words whose first m ¼ j x j coordinates coincide with x: Moreover, given a code C; let Mx ¼ jQx \ Cj; and let BðxÞ ¼ fy 2 F3m j dðx; yÞ ¼ 1g: The covering property}all words in Qx must be covered by a codeword}now gives the inequalities X My 53n m ; x 2 F3m ð1Þ ð2n 2m þ 1ÞMx þ y2BðxÞ
and, obviously, Mz0 þ Mz1 þ Mz2 ¼ Mz :
ð2Þ
NOTE
177
Starting from m ¼ 1; we now find all solutions to (1), (2), and if we have a solution for m ¼ n; then sn 4M: As such, this approach leads to an abundance of intermediate codes. It is, however, only necessary to consider inequivalent subcodes in this search, and rejection of equivalent copies of codes gives a considerable speed-up. Two ternary codes are equivalent if one can be obtained from the other by a permutation of the coordinates and by coordinate-wise permutations of the values in F3 : Equivalence tests are here carried out on all subcodes by using an orderly algorithm [6] and a transformation of the codes into graphs that are fed into the graph isomorphism program nauty [4]; see [8]. Finally, we also require that Mi 43n m as no codeword occurs more than once in an optimal code.
3. LLL-ENUMERATION In order to compute the subcodes for the cases m44 we used an exhaustive enumeration algorithm based on lattice basis reduction as described in [10, 11]. This algorithm solves exhaustively Diophantine linear systems of equations A x ¼ b where the integer vector x is bounded by l4x4u: The algorithm was already successfully applied to the construction of combinatorial designs, see [10]. To solve the linear system of (in)equalities (1) and (2) we introduce slack variables and get a system of 3m þ 3m 1 equations: X ð2n 2m þ 1ÞMx þ My sx ¼ 3n m ; x 2 F3m ; ð3Þ y2BðxÞ
Mz0 þ Mz1 þ Mz2 ¼ Mz :
ð4Þ
Both the variables Mx and the slack variables sx are required to be nonnegative. Obvious upper bounds for the variables Mx are the right-hand sides of their corresponding Eq. (2). Since the enumeration algorithm works best if there are tight bounds on the variables, we need also good upper bounds for the slack variables. One approach to get an upper bound for sx ; x 2 F3m ; is to replace in (3) the variables Mx ; My by their corresponding upper bounds. However, improved upper bounds for the slack variables and substantial speed-up can be achieved in the following way. Adding Eqs. (3) for z0; z1 and z2 and using Mz0 þ Mz1 þ Mz2 ¼ Mz we get X ð2n 2m þ 3ÞMz þ My sz0 sz1 sz2 ¼ 3n mþ1 y2BðzÞ
178
NOTE
and by comparing this with (3) for z and m0 ¼ m 1; we have sz0 þ sz1 þ sz2 ¼ sz : Consequently, szi 4sz : These upper bounds are good enough for a fast solution of the systems of equations. The algorithm computes via lattice basis reduction an integer basis of the kernel of system (3), (4), where the right-hand side is regarded as variable in order to have an homogeneous system. Then all integer linear combinations of the basis vectors are enumerated and tested if they are solutions of (3), (4). Working in the kernel of the system has the advantage that the enumeration time does not depend on the size of the entries in the righthand side of (3), (4). This makes the algorithm ideally suited for large values on the right-hand side, in our case for the values m ¼ 1; . . . ; 4: For m ¼ 5; 6; the fastest algorithm was obtained by skipping the LLL part and directly applying an exhaustive enumeration algorithm.
4. THE RESULTS The number of inequivalent subcodes obtained using the method discussed in the previous section are given in Table 1 for 574M464: The CPU time needed to solve the case M ¼ 64 was about 2 weeks using a network of twenty-six 400- and 500-MHz PC computers. The search was distributed over this network with the batch system autoson [5]. Theorem 1. s6 565: Proof.
Follows from the zero in the entry m ¼ 6; M ¼ 64 of Table 1.
]
TABLE 1 Number of Inequivalent Subcodes m
0 1 2 3 4 5 6
M 57
58
59
60
61
62
63
64
1 1 0 0 0 0 0
1 1 0 0 0 0 0
1 2 2 1 0 0 0
1 3 11 10 6 0 0
1 4 34 99 176 1 0
1 5 88 831 7 652 21 0
1 12 208 5 728 332 622 815 0
1 14 423 33 397 12 139 972 46 834 0
NOTE
179
REFERENCES 1. L. Habsieger, Lower bounds for q-ary coverings by spheres of radius one, J. Combin. Theory Ser. A 67 (1994), 199–222. 2. H. J. L. Kamps and J. H. van Lint, The football pool problem for 5 matches, J. Combin. Theory 3 (1967), 315–325. 3. P. J. M. van Laarhoven, E. H. L. Aarts, J. H. van Lint, and L. T. Wille, New upper bounds for the football pool problem for 6, 7, and 8 matches, J. Combin. Theory Ser. A 52 (1989), 304–312. 4. B. D. McKay, ‘‘nauty User’s Guide,’’ version 1.5, Technical Report TR-CS-90-02, Computer Science Department, Australian National University, 1990. 5. B. D. McKay, ‘‘autoson}A Distributed Batch System for UNIX Workstation Networks,’’ version 1.3, Technical Report TR-CS-96-03, Computer Science Department, Australian National University, 1996. 6. B. D. McKay, Isomorph-free exhaustive generation, J. Algorithms 26 (1998), 306–324. . sterg(ard, A combinatorial proof for the football pool problem for six matches, 7. P. R. J. O J. Combin. Theory Ser. A 76 (1996), 160–163. . sterg(ard and U. Blass, On the size of optimal binary codes of length 9 and 8. P. R. J. O covering radius 1, IEEE Trans. Inform. Theory 47 (2001), 2556–2557. 9. R. G. Stanton and J. G. Kalbfleisch, Intersection inequalities for the covering problem, SIAM J. Appl. Math. 17 (1969), 1311–1316. 10. A. Wassermann, Finding simple t-designs with enumeration techniques, J. Combin. Des. 6 (1998), 79–90. 11. A. Wassermann, Attacking the market split problem with lattice basis reduction, J. Comb. Optim. 6 (2002), 5–16. 12. E. W. Weber, On the football pool problem for 6 matches: A new upper bound, J. Combin. Theory Ser. A 35 (1983), 106–108. 13. L. T. Wille, The football pool problem for 6 matches: A new upper bound obtained by simulated annealing, J. Combin. Theory Ser. A 45 (1987), 171–177.
NOTE
A New Lower Bound for the Football Pool Problem for Six Matches . sterg(ard1 Patric R. J. O Department of Computer Science and Engineering, Helsinki University of Technology, P.O. Box 5400, 02015 HUT, Finland E-mail: patric.ostergard@hut.fi
and Alfred Wassermann Department of Mathematics, University of Bayreuth, 95440 Bayreuth, Germany E-mail: [email protected]
Received March 16, 2001
In the football pool problem one wants to minimize the cardinality of a ternary code, C F3n ; with covering radius one, and the size of a minimum code is denoted by sn : The smallest unsettled case is 634s6 473: The lower bound is here improved to 65 in a coordinate-by-coordinate backtrack search using the LLL algorithm and complete equivalence checking of subcodes. # 2002 Elsevier Science (USA) Key Words: covering radius; football pool problem; LLL algorithm; ternary code.
1. INTRODUCTION The football pool problem is one of the oldest and most famous problems in combinatorial coding theory. For a given length n; one wants to find a set of ternary words, C F3n ; such that for any x 2 F3n there is a word in C that differs from x in at most one coordinate. The minimum size of such a set is denoted by sn : This is a special case of the more general problem of finding Kq ðn; RÞ; the minimum size of q-ary codes of length n and covering radius R: Obviously, sn ¼ K3 ðn; 1Þ: For 14n43 it is not difficult to prove that the corresponding values of sn are 1, 3, and 5, respectively. The ternary Hamming code of length 4 gives s4 ¼ 9; and by lengthening this code ðsnþ1 43sn Þ we get s5 427: Actually 1
Supported by the Academy of Finland.
175 0097-3165/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.
176
NOTE
s5 ¼ 27; which is proved in [2]. This is the largest n for which sn is known, apart from the lengths for which ternary Hamming codes exist. This first open case s6 has received a lot of attention along the years. Lengthening gives s6 43s5 ¼ 81; which was improved to 79 in [12] and later to 74 and 73 in [13] and [3], respectively. A combinatorial proof of s6 473 is given in [7]. It should be mentioned that this record was obtained even earlier by constructors of football pool systems who, unfortunately, did not publish it in the mathematical literature. As for lower bounds, the Hamming bound gives s6 5 729=13 ¼ 57: The best lower bound in the literature [12] is s6 563: However, as the ‘‘proof’’ of this bound restricts to the comment ‘‘By methods similar to [those used in [2]], we could show that s6 563,’’ this result is often ignored. A welldocumented proof of s6 560 is published in [1]. In this work, we improve the lower bound on s6 by proving nonexistence of codes. This is done in a coordinate-by-coordinate backtrack search, which is described in Section 2. In this search, the LLL algorithm and equivalence checking of subcodes play central roles. Application of the LLL algorithm is discussed in Section 3. The results of the search leading to the new bound s6 565 are presented in Section 4.
2. CONSTRUCTING COVERING CODES A code in F3n with M words can be expressed using an n M matrix with entries from f0; 1; 2g: The fact that there are many more columns than rows for the instances in this work suggests a row-by-row (that is, coordinate-bycoordinate) approach for our backtrack search. The search described here is closely related to that in [8], which in turn is based on old ideas from [2, 9]. Several new ideas, including the use of the LLL algorithm, have been incorporated here to enhance the search method. The covering property of the code to be constructed can be expressed using a set of linear (in)equalities. Let Qx F3n denote the set of 3n m words whose first m ¼ j x j coordinates coincide with x: Moreover, given a code C; let Mx ¼ jQx \ Cj; and let BðxÞ ¼ fy 2 F3m j dðx; yÞ ¼ 1g: The covering property}all words in Qx must be covered by a codeword}now gives the inequalities X My 53n m ; x 2 F3m ð1Þ ð2n 2m þ 1ÞMx þ y2BðxÞ
and, obviously, Mz0 þ Mz1 þ Mz2 ¼ Mz :
ð2Þ
NOTE
177
Starting from m ¼ 1; we now find all solutions to (1), (2), and if we have a solution for m ¼ n; then sn 4M: As such, this approach leads to an abundance of intermediate codes. It is, however, only necessary to consider inequivalent subcodes in this search, and rejection of equivalent copies of codes gives a considerable speed-up. Two ternary codes are equivalent if one can be obtained from the other by a permutation of the coordinates and by coordinate-wise permutations of the values in F3 : Equivalence tests are here carried out on all subcodes by using an orderly algorithm [6] and a transformation of the codes into graphs that are fed into the graph isomorphism program nauty [4]; see [8]. Finally, we also require that Mi 43n m as no codeword occurs more than once in an optimal code.
3. LLL-ENUMERATION In order to compute the subcodes for the cases m44 we used an exhaustive enumeration algorithm based on lattice basis reduction as described in [10, 11]. This algorithm solves exhaustively Diophantine linear systems of equations A x ¼ b where the integer vector x is bounded by l4x4u: The algorithm was already successfully applied to the construction of combinatorial designs, see [10]. To solve the linear system of (in)equalities (1) and (2) we introduce slack variables and get a system of 3m þ 3m 1 equations: X ð2n 2m þ 1ÞMx þ My sx ¼ 3n m ; x 2 F3m ; ð3Þ y2BðxÞ
Mz0 þ Mz1 þ Mz2 ¼ Mz :
ð4Þ
Both the variables Mx and the slack variables sx are required to be nonnegative. Obvious upper bounds for the variables Mx are the right-hand sides of their corresponding Eq. (2). Since the enumeration algorithm works best if there are tight bounds on the variables, we need also good upper bounds for the slack variables. One approach to get an upper bound for sx ; x 2 F3m ; is to replace in (3) the variables Mx ; My by their corresponding upper bounds. However, improved upper bounds for the slack variables and substantial speed-up can be achieved in the following way. Adding Eqs. (3) for z0; z1 and z2 and using Mz0 þ Mz1 þ Mz2 ¼ Mz we get X ð2n 2m þ 3ÞMz þ My sz0 sz1 sz2 ¼ 3n mþ1 y2BðzÞ
178
NOTE
and by comparing this with (3) for z and m0 ¼ m 1; we have sz0 þ sz1 þ sz2 ¼ sz : Consequently, szi 4sz : These upper bounds are good enough for a fast solution of the systems of equations. The algorithm computes via lattice basis reduction an integer basis of the kernel of system (3), (4), where the right-hand side is regarded as variable in order to have an homogeneous system. Then all integer linear combinations of the basis vectors are enumerated and tested if they are solutions of (3), (4). Working in the kernel of the system has the advantage that the enumeration time does not depend on the size of the entries in the righthand side of (3), (4). This makes the algorithm ideally suited for large values on the right-hand side, in our case for the values m ¼ 1; . . . ; 4: For m ¼ 5; 6; the fastest algorithm was obtained by skipping the LLL part and directly applying an exhaustive enumeration algorithm.
4. THE RESULTS The number of inequivalent subcodes obtained using the method discussed in the previous section are given in Table 1 for 574M464: The CPU time needed to solve the case M ¼ 64 was about 2 weeks using a network of twenty-six 400- and 500-MHz PC computers. The search was distributed over this network with the batch system autoson [5]. Theorem 1. s6 565: Proof.
Follows from the zero in the entry m ¼ 6; M ¼ 64 of Table 1.
]
TABLE 1 Number of Inequivalent Subcodes m
0 1 2 3 4 5 6
M 57
58
59
60
61
62
63
64
1 1 0 0 0 0 0
1 1 0 0 0 0 0
1 2 2 1 0 0 0
1 3 11 10 6 0 0
1 4 34 99 176 1 0
1 5 88 831 7 652 21 0
1 12 208 5 728 332 622 815 0
1 14 423 33 397 12 139 972 46 834 0
NOTE
179
REFERENCES 1. L. Habsieger, Lower bounds for q-ary coverings by spheres of radius one, J. Combin. Theory Ser. A 67 (1994), 199–222. 2. H. J. L. Kamps and J. H. van Lint, The football pool problem for 5 matches, J. Combin. Theory 3 (1967), 315–325. 3. P. J. M. van Laarhoven, E. H. L. Aarts, J. H. van Lint, and L. T. Wille, New upper bounds for the football pool problem for 6, 7, and 8 matches, J. Combin. Theory Ser. A 52 (1989), 304–312. 4. B. D. McKay, ‘‘nauty User’s Guide,’’ version 1.5, Technical Report TR-CS-90-02, Computer Science Department, Australian National University, 1990. 5. B. D. McKay, ‘‘autoson}A Distributed Batch System for UNIX Workstation Networks,’’ version 1.3, Technical Report TR-CS-96-03, Computer Science Department, Australian National University, 1996. 6. B. D. McKay, Isomorph-free exhaustive generation, J. Algorithms 26 (1998), 306–324. . sterg(ard, A combinatorial proof for the football pool problem for six matches, 7. P. R. J. O J. Combin. Theory Ser. A 76 (1996), 160–163. . sterg(ard and U. Blass, On the size of optimal binary codes of length 9 and 8. P. R. J. O covering radius 1, IEEE Trans. Inform. Theory 47 (2001), 2556–2557. 9. R. G. Stanton and J. G. Kalbfleisch, Intersection inequalities for the covering problem, SIAM J. Appl. Math. 17 (1969), 1311–1316. 10. A. Wassermann, Finding simple t-designs with enumeration techniques, J. Combin. Des. 6 (1998), 79–90. 11. A. Wassermann, Attacking the market split problem with lattice basis reduction, J. Comb. Optim. 6 (2002), 5–16. 12. E. W. Weber, On the football pool problem for 6 matches: A new upper bound, J. Combin. Theory Ser. A 35 (1983), 106–108. 13. L. T. Wille, The football pool problem for 6 matches: A new upper bound obtained by simulated annealing, J. Combin. Theory Ser. A 45 (1987), 171–177.