A.E. Brouwer, H.O. Hämäläinen, P.R.J. Östergård & N.J.A. Sloane,(quoted as [BHÖS]) bounds on the size of mixed binary/ternary codes were given.

Bounds on Mixed Binary/Ternary Codes,

IEEE Trans. Inf. Th.44(1998) 140-161

N(m,n,d) is defined as the maximum number of vectors of length m+n with m binary and n ternary coordinates and mutual distance at least d. If n=0 or m=0, this is about binary or ternary codes, and discussed elsewhere. The case m > 0, n > 0 is the truly mixed case discussed here. The cases d ≤ 2 or d ≥ m+n−1 are easy and have been settled completely. Here we copy the tables from the reference above and add some modern improvements.

The covering problem now asks: How many football pool forms do I have to fill in order to be sure that at least one of them has not more than e wrong predictions? See Kéri for tables on covering codes.

The packing problem asks: How many football pool forms can I fill so that for no possible outcome there are two forms, both with at most e wrong predictions?

The covering problem is interesting if one wants to be sure to get a prize. The packing problem, if one does not want to waste money by covering the same possibility twice. The function N(m,n,d) answers the packing problem (with d = 2e+1).

Since our football pools have 13 matches, that explains the restriction to m+n ≤ 13 in the tables in [BHÖS]. However, the spanish Quiniela has 14+1 matches, so there is interest in extending these tables to m+n ≤ 14.

This wrong construction was used to show that N(13,2,6) ≥ 134, N(12,2,6) ≥ 68, N(12,1,5) ≥ 68, N(11,2,6) ≥ 38, N(11,1,5) ≥ 38, so that these five values are now in doubt. We can use N(12,1,5) ≥ N(13,0,5) = 64, N(11,1,5) ≥ N(12,0,5) = 32, N(10,3,6) ≥ N(11,2,6) ≥ N(12,1,6) = 32 instead. These smaller lower bounds are given in blue. Some of them were again improved later.

A large number of improved upper bounds was given in

Bart Litjens,These upper bounds are not quoted separately, but are given in the tables in purple.

Semidefinite bounds for mixed binary/ternary codes,

arXiv:1606.06930v1

m\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 1 | 3 | 9 | 18 | 38 | 99-111 |

1 | 1 | 1 | 2 | 6 | 12 | 33 | 71-76 | 198-222 |

2 | 1 | 2 | 4 | 9 | 22 | 52-65 | 134-152 | 396-444 |

3 | 2 | 3 | 6 | 18 | 42-44 | 99-125 | 266-304 | 711-888 |

4 | 2 | 6 | 12 | 28-30 | 72-88 | 186-238 | 504-608 | 1422-1749 |

5 | 4 | 8 | 22 | 54-60 | 144-165 | 348-457 | 1008-1216 | 2592-3259 |

6 | 8 | 16 | 38-44 | 108-118 | 288-317 | 672-855 | 1896-2332 | 5184-6362 |

7 | 16 | 26-30 | 72-83 | 192-225 | 576-604 | 1296-1612 | 3792-4443 | 10368-12171 |

8 | 20 | 50-59 | 144-154 | 384-414 | 1152 | 2560-3087 | 6912-8331 | |

9 | 40 | 96-108 | 288-292 | 768-796 | 1728-2130 | 4608-5924 | ||

10 | 72 | 192-212 | 512-552 | 1152-1492 | 3280-4081 | |||

11 | 144 | 384 | 848-1049 | 2304-2890 | ||||

12 | 256 | 768 | 1536-2011 | |||||

13 | 512 | 1120-1360 | ||||||

14 | 1024 |

m\n | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|

0 | 252-333 | 729-937 | 2187-2808 | 6561-7029 | 19683 | 59049 | 118098-153527 |

1 | 486-666 | 1458-1874 | 4374-4920 | 13122-14058 | 39366 | 78732-106288 | |

2 | 972-1284 | 2916-3514 | 8748-9840 | 26244-26790 | 59049-75920 | ||

3 | 1944-2464 | 5832-6846 | 17496-18589 | 40095-52488 | |||

4 | 3888-4764 | 11664-12887 | 34992-36337 | ||||

5 | 7776-9128 | 23328-25194 | |||||

6 | 15552-17485 |

The bound N(10,0,3) ≤ 72 (and hence N(11,0,3) ≤ 144) was obtained in

P.R.J. Östergård, T. Baicheva & E. Kolev,

Optimal binary one-error-correcting codes of length 10 have 72 codewords,

IEEE Trans. Inform. Theory45(1999) 1229-1231.

The bounds N(0,6,3) ≤ 38 (and hence N(a,6,3) ≤ 38.2^{a})
and N(0,7,3) ≤ 111 (and hence N(a,7+b,3) ≤ 111.2^{a}.3^{b})
were obtained in

P.R.J. Östergård,

Classification of binary/ternary one-error-correcting codes,

Discrete Math.223(2000) 253-262.

The bound N(0,10,3) ≤ 2808 is due to

W. Lang, J. Quistorff & E. Schneider,

New Results on Integer Programming for Codes,

Congr. Numer.188(2007) 97-107.

The bound N(3,5,3) ≥ 99 follows by
explicit
construction due to `Pedro fdez`, 2011-04-28.

The bound N(0,8,3) ≥ 252 follows by
explicit
construction due to `ehl555`, 2012-04-25.

The bound N(3,6,3) ≥ 266 follows by
explicit
construction due to `PacoHH`, 2013-01-03.

The bound N(5,5,3) ≥ 348 follows by
explicit
construction due to `Pucho`, 2011-03-03.

The bound N(5,6,3) ≥ 1008 follows by
explicit
construction due to `Pucho`, 2012-04-23.

The bound N(11,2,3) ≥ 848 follows by
explicit
construction due to `Código`, 2011-04-10.

The bound N(13,1,3) ≥ 1120 follows by
explicit
construction due to `Código`, 2011-04-07.

The bound N(4,7,3) ≥ 1422 follows by
explicit
construction due to `PacoHH`, 2010-12-08.
It implies N(3,7,3) ≥ 711.

The bounds N(8,5,3) ≥ 2560 and N(10,4,3) ≥ 3280 follow by
explicit
construction due to `Código`, 2011-04-04.

The bounds N(11,3,3) ≥ 2304, N(9,5,3) ≥ 4608,
N(8,6,3) ≥ 6624, N(7,7,3) ≥ 9720, N(6,8,3) ≥ 14256 were
announced
by `PacoHH`, 2010-11-30. On
this page
one finds a zip file with codes confirming these.

The bounds N(8,6,3) ≥ 6912, N(7,7,3) ≥ 10368, N(6,8,3) ≥ 15552,
N(5,9,3) ≥ 23328, N(4,10,3) ≥ 34992, N(3,11,3) ≥ 40095 were
announced
by `Código`. All follow from the last two.
No explicit construction was given.

m\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 1 | 1 | 3 | 6 | 18 | 33 |

1 | 1 | 1 | 1 | 2 | 4 | 12 | 33 | 66 |

2 | 1 | 1 | 2 | 3 | 8 | 22 | 51-61 | 114-132 |

3 | 1 | 2 | 3 | 6 | 15 | 36-43 | 92-117 | 216-264 |

4 | 2 | 2 | 6 | 11 | 28-30 | 62-83 | 158-228 | 408-528 |

5 | 2 | 4 | 8 | 20 | 50-59 | 114-160 | 288-436 | 762-1056 |

6 | 4 | 8 | 16 | 34-40 | 96-114 | 216-308 | 576-825 | 1512-2112 |

7 | 8 | 16 | 26-30 | 64-80 | 192-220 | 408-585 | 1152-1576 | 3024-4224 |

8 | 16 | 20 | 50-59 | 128-153 | 384-407 | 768-1103 | 2304-3027 | |

9 | 20 | 40 | 96-108 | 256-288 | 548-771 | 1536-2105 | ||

10 | 40 | 72-76 | 192-212 | 420-548 | 1050-1480 | |||

11 | 72 | 144-152 | 384 | 784-1032 | ||||

12 | 144 | 256 | 768 | |||||

13 | 256 | 512 | ||||||

14 | 512 |

m\n | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|

0 | 99 | 243-297 | 729-891 | 1458-2561 | 4374-6839 | 8559-19270 | 24786-54774 |

1 | 162-198 | 486-594 | 972-1749 | 2916-4920 | 8019-13531 | 16767-37714 | |

2 | 324-396 | 729-1188 | 1944-3371 | 5589-9450 | 16038-27356 | ||

3 | 528-792 | 1296-2376 | 3726-6581 | 10692-18039 | |||

4 | 1056-1584 | 2484-4590 | 7128-13122 | ||||

5 | 1896-3168 | 4752-9180 | |||||

6 | 3660-6336 |

We have N(11,0,4) = N(10,0,3) and N(12,0,4) = N(11,0,3).

The bound N(0,7,4) ≤ 33 (with usual consequences for N(a,7+b,4)) is due to

P.R.J. Östergård,

On binary/ternary error-correcting codes with minimum distance 4,

in: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (M. Fossorier, H. Imai, S. Lin, and A. Poli, Eds.), LNCS 1719, Springer, Berlin 1999, pp. 472-481.

The bounds N(0,12,4) ≤ 6839 and N(0,13,4) ≤ 19270 are due to

Dion Gijswijt, Alexander Schrijver & Hajime Tanaka,

New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming,

JCT (A)113(2006) 1719-1731.

The bound N(8,2,4) ≥ 50 follows by
explicit
construction due to `PacoHH`, 2010-12-10.

The bound N(3,5,4) ≥ 36 follows by
explicit
construction due to `PacoHH`, 2010-11-28.

The bound N(4,5,4) ≥ 62 follows by
explicit
construction due to `PacoHH`, 2010-12-02.

The bound N(5,4,4) ≥ 50 follows by
explicit
construction due to `PFPGU`, 2012-05-08.

The bound N(5,5,4) ≥ 114 follows by
explicit
construction due to `Pedro fdez`, 2011-03-01.

The bound N(7,5,4) ≥ 408 follows by
explicit
construction due to `Joan`, 2011-02-27.

The bound N(3,6,4) ≥ 92 follows by
explicit
construction due to `PacoHH`, 2010-12-16.

The bound N(4,6,4) ≥ 158 follows by
explicit
construction due to `Código`, 2011-04-02.

The bound N(5,6,4) ≥ 216 follows by
explicit
construction due to `PacoHH`, 2011-03-16.

The bound N(2,7,4) ≥ 114 follows by
explicit
construction due to `PacoHH`, 2011-03-16.

The bound N(4,7,4) ≥ 408 follows by
explicit
construction due to `Código`, 2011-04-02.

The bounds N(3,8,4) ≥ 528, N(4,8,4) ≥ 1056 follow by
explicit
construction due to `spaik` and `Código`, 2011-04-25.

The bound N(5,8,4) ≥ 1896 follows by
explicit
construction due to `spaik`, 2012-04-25.

The bound N(6,8,4) ≥ 3660 follows by
explicit
construction due to `PacoHH`, 2013-01-02.

The bound N(5,7,4) ≥ 762 follows by
explicit
construction due to `Pucho`, 2012-04-25.

The bound N(6,7,4) ≥ 1512 follows by
explicit
construction due to `Pucho`, 2011-03-02.

The bound N(7,7,4) ≥ 3024 follows by
explicit
construction due to `Pucho`, 2012-04-25.

The bound N(10,3,4) ≥ 420 follows by
explicit
construction due to `PacoHH`, 2012-12-21.

The bound N(11,3,4) ≥ 784 follows by
explicit
construction due to `Pucho`, 2012-05-21.

The bound N(9,4,4) ≥ 548 follows by
explicit
construction due to `PFPGU`, 2012-05-21.

The bound N(10,4,4) ≥ 1050 follows by
explicit
construction due to `PacoHH`, 2012-11-08.

The bound N(0,13,4) ≥ 8559 follows by
explicit
construction due to `Código`, 2011-03-26.

The bound N(1,13,4) ≥ 16767 follows by
explicit
construction due to `Código`, 2011-03-27.

The bound N(0,14,4) ≥ 24786 follows by
explicit
construction due to `Código`, 2011-03-19.

All unexplained lower bounds can be found in the
file
provided
by `PacoHH`, 2011-03-03.

m\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 10 |

1 | 1 | 1 | 1 | 1 | 2 | 3 | 8 | 18 |

2 | 1 | 1 | 1 | 2 | 3 | 6 | 15 | 36 |

3 | 1 | 1 | 2 | 3 | 6 | 12 | 24-27 | 72 |

4 | 1 | 2 | 2 | 4 | 9 | 18 | 48-54 | 144 |

5 | 2 | 2 | 4 | 6 | 14 | 33-36 | 96-108 | 216-288 |

6 | 2 | 3 | 6 | 12 | 28 | 66-72 | 144-216 | 378-563 |

7 | 2 | 4 | 9 | 20-24 | 44-56 | 99-144 | 255-407 | 648-1047 |

8 | 4 | 7 | 16 | 34-44 | 84-112 | 180-288 | 453-755 | |

9 | 6 | 12 | 26-31 | 64-85 | 136-216 | 318-534 | ||

10 | 12 | 24 | 48-61 | 128-158 | 234-390 | |||

11 | 24 | 38-43 | 96-115 | 192-292 | ||||

12 | 32 | 64-83 | 192-213 | |||||

13 | 64 | 128-156 | ||||||

14 | 128 |

m\n | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|

0 | 27 | 81 | 243 | 729 | 729-1557 | 2187-4078 | 6561-10624 |

1 | 54 | 162 | 486 | 729-1138 | 1458-2927 | 4374-7598 | |

2 | 108 | 324 | 729-849 | 972-2105 | 2916-5512 | ||

3 | 216 | 486-601 | 729-1519 | 1944-3964 | |||

4 | 324-420 | 729-1099 | 1458-2801 | ||||

5 | 486-791 | 1458-2000 | |||||

6 | 972-1437 |

N(0,12,5) ≤ 1557 and N(0,13,5) ≤ 4078 is due to Gijswijt et al., see above.

The bound N(6,4,5) ≥ 28 follows by
explicit
construction due to `Joan`, 2010-11-21.

The bound N(8,3,5) ≥ 34 follows by
explicit
construction due to `Pucho`, 2011-03-19.

The bound N(8,4,5) ≥ 84 follows by
explicit
construction due to `Código`, 2011-03-19.

The bound N(8,5,5) ≥ 180 follows by explicit construction
due to `PacoHH`, 2011-06-07.

The bounds N(11,1,5) ≥ 38 and N(7,6,5) ≥ 240 follow by
explicit
construction due to `PacoHH`, 2011-02-18 or earlier.
This link also reveals that `PacoHH` is Francisco Hernández Hernández.

The bound N(6,7,5) ≥ 378 follows by
explicit
construction due to `spaik`, 2011-06-02.

The bound N(8,6,5) ≥ 453 follows by
explicit
construction due to `PacoHH`, 2011-06-05.

The bound N(7,6,5) ≥ 255 was
announced
by `Código`. The bound N(5,9,5) ≥ 1458 was
announced
by `Pucho`.
Constructions are given
here.

m\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |

1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 6 |

2 | 1 | 1 | 1 | 1 | 2 | 3 | 6 | 12 |

3 | 1 | 1 | 1 | 2 | 3 | 4 | 12 | 24 |

4 | 1 | 1 | 2 | 2 | 4 | 8 | 18 | 48 |

5 | 1 | 2 | 2 | 3 | 6 | 12 | 33-36 | 96 |

6 | 2 | 2 | 3 | 6 | 12 | 24 | 66-72 | 144-192 |

7 | 2 | 2 | 4 | 8 | 18-22 | 44-48 | 99-142 | 216-375 |

8 | 2 | 4 | 7 | 16 | 32-39 | 66-96 | 168-273 | |

9 | 4 | 6 | 12 | 26-30 | 56-75 | 112-192 | ||

10 | 6 | 12 | 24 | 44-56 | 88-144 | |||

11 | 12 | 24 | 32-43 | 88-107 | ||||

12 | 24 | 32 | 64-83 | |||||

13 | 32 | 64 | ||||||

14 | 64 |

m\n | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|

0 | 9 | 27 | 81 | 243 | 729 | 729-1449 | 2187-3660 |

1 | 18 | 54 | 162 | 486 | 729-1073 | 1458-2657 | |

2 | 36 | 108 | 324 | 729-803 | 972-1935 | ||

3 | 72 | 216 | 486-574 | 729-1414 | |||

4 | 144 | 324-425 | 729-1036 | ||||

5 | 216-276 | 486-744 | |||||

6 | 324-527 |

N(0,13,6) ≤ 1449 is due to Gijswijt et al., see above.

N(10,3,6) ≥ 44 follows by
explicit
construction due to `Pucho`, 2011-03-15.

N(6,5,6) ≥ 24 follows by
explicit
construction due to `Pucho`, 2011-03-15.

N(9,4,6) ≥ 56 follows by
explicit
construction due to `Joan`, 2011-03-15.

N(10,4,6) ≥ 88 follows by
explicit
construction due to `PacoHH`, 2011-03-15.

N(11,3,6) ≥ 88 follows by
explicit
construction due to `Pedro fdez`, 2011-05-05.

N(8,6,6) ≥ 168 follows by
explicit
construction due to `spaik`, 2012-04-18.

N(2,12,6) ≥ 972 follows by
explicit
construction due to `PacoHH`, 2011-07-04.

m\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 6 | 14 | 36 | 60-108 | 162-324 | 243-805 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 6 | 12 | 24-28 | 48-72 | 108-216 | 243-591 | |

2 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 9 | 18-24 | 36-56 | 75-144 | 162-432 | ||

3 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 7 | 16-18 | 28-48 | 57-112 | 108-288 | |||

4 | 1 | 1 | 1 | 2 | 2 | 3 | 6 | 14 | 24-36 | 48-96 | 84-224 | ||||

5 | 1 | 1 | 2 | 2 | 3 | 6 | 12 | 21-28 | 36-72 | 69-174 | |||||

6 | 1 | 2 | 2 | 3 | 4 | 9 | 18-23 | 33-53 | 61-130 | ||||||

7 | 2 | 2 | 2 | 4 | 8 | 16 | 24-41 | 58-99 | |||||||

8 | 2 | 2 | 4 | 6 | 16 | 22-31 | 44-74 | ||||||||

9 | 2 | 3 | 4 | 12 | 18-23 | 36-53 | |||||||||

10 | 2 | 4 | 8 | 16-18 | 28-41 | ||||||||||

11 | 4 | 6 | 12 | 24-31 | |||||||||||

12 | 4 | 12 | 20-24 | ||||||||||||

13 | 8 | 16-19 | |||||||||||||

14 | 16 |

N(0,10,7) ≤ 14 is due to

K.S. Kapralov,

The nonexistence of ternary (10,15,7) codes,

Proc. 7th international workshop on algebraic and combinatorial coding theory (ACCT'2000), Bansko, Bulgaria, 18-24 June, 2000, pp. 189-192.

N(0,11,7) ≤ 36 (and hence N(0,12,7) ≤ 108, N(0,13,7) ≤ 324) is due to

M.J. Letourneau & S.K. Houghten,

Optimal Ternary (11,7) and (14,10) Codes,

Journal of Combinat. Math. and Combinat. Computing51(2004) 159-164.

The bound N(0,12,7) ≥ 60 follows by
explicit
construction due to `spaik`, 2011-06-22.

The bound N(0,13,7) ≥ 162 follows by
explicit
construction (18 cosets of <1011100011122,0100011122111>)
due to `spaik`, 2011-06-11.

The bound N(1,11,7) ≥ 48 follows by
explicit
construction (16 cosets of <0^{3}1^{9}>)
due to `Código`, 2011-03-13.

The bounds N(1,12,7) ≥ 108 and N(2,12,7) ≥ 162 and N(3,11,7) ≥ 108 follow from N(0,13,7) ≥ 162.

The bound N(1,13,7) ≥ 243 follows by
explicit
construction due to `PacoHH`, 2011-06-12.

The bound N(2,11,7) ≥ 75 follows from N(2,12,8) ≥ 75.

The bound N(3,9,7) ≥ 28 follows by
explicit
construction due to `spaik`, 2012-04-08.

The bound N(3,10,7) ≥ 57 follows by
explicit
construction due to `spaik`, 2012-05-09.

The bounds N(4,8,7) ≥ 24, N(5,8,7) ≥ 36 follow by
explicit
construction due to `Pucho`, 2011-03-13.

The bound N(4,9,7) ≥ 48 follows by
explicit
construction due to `Código`, 2011-04-02.

The bound N(4,10,7) ≥ 84 follows by
explicit
construction due to `spaik`, 2012-04-17.

The bound N(5,7,7) ≥ 21 follows by
explicit
construction due to `PacoHH`, 2010-12-05.

The bound N(5,9,7) ≥ 69 follows by
explicit
construction due to `spaik`, 2011-06-11.

The bound N(6,6,7) ≥ 18 follows by
explicit
construction due to `PacoHH`, 2010-12-17.

The bound N(6,7,7) ≥ 33 follows by
explicit
construction due to `spaik`, 2011-07-18.

The bound N(6,8,7) ≥ 61 follows by
explicit
construction due to `spaik`, 2012-04-02.

The bounds N(7,7,7) ≥ 58 and N(6,8,7) ≥ 60 follow by
explicit
construction due to `spaik`, 2011-07-17.

The bound N(8,5,7) ≥ 22 follows by
explicit
construction due to `PacoHH`, 2012-03-26.

The bound N(8,6,7) ≥ 44 follows by
explicit
construction due to `spaik`, 2012-12-19.

The bound N(9,5,7) ≥ 36 follows by
explicit
construction due to `PacoHH`, 2012-12-16.

The bound N(10,4,7) ≥ 28 follows by
explicit
construction due to `Pucho`, 2011-03-13.

m\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 6 | 12 | 36 | 54-95 | 108-237 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 9 | 24 | 39-67 | 81-179 | |

2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 7 | 18 | 36-48 | 75-134 | ||

3 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 13 | 24-36 | 54-96 | |||

4 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 6 | 12 | 24-26 | 42-72 | ||||

5 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 8 | 18-24 | 30-52 | |||||

6 | 1 | 1 | 2 | 2 | 3 | 4 | 7 | 14-16 | 28-44 | ||||||

7 | 1 | 2 | 2 | 2 | 4 | 6 | 12 | 23-32 | |||||||

8 | 2 | 2 | 2 | 3 | 6 | 12 | 20-24 | ||||||||

9 | 2 | 2 | 3 | 4 | 9 | 18 | |||||||||

10 | 2 | 2 | 4 | 6 | 16 | ||||||||||

11 | 2 | 4 | 6 | 12 | |||||||||||

12 | 4 | 4 | 12 | ||||||||||||

13 | 4 | 8 | |||||||||||||

14 | 8 |

N(0,13,8) ≤ 95 is due to Gijswijt et al., see above.

N(0,13,8) ≥ 54 follows by
explicit
construction due to `PacoHH`, 2011-03-14.

N(1,12,8) ≥ 39 follows by
explicit
construction due to `Código`, 2011-04-04.

N(4,10,8) ≥ 42 follows by
explicit
construction due to `Joan`, 2011-07-23.

N(3,11,8) ≥ 45 follows by
explicit
construction due to `PacoHH`, 2011-04-26.

N(2,12,8) ≥ 75 follows by
explicit
construction due to `spaik`, 2011-06-26.

N(1,13,8) ≥ 81 follows by
explicit
construction due to `spaik`, 2011-06-27.

N(0,14,8) ≥ 108 follows by
explicit
construction due to `Joan`, 2011-07-24.

N(7,7,8) ≥ 23 follows by
explicit
construction due to `spaik`, 2011-07-23.

N(6,8,8) ≥ 28 follows by
explicit
construction due to `spaik`, 2012-04-10.

N(4,9,8) ≥ 24 follows by
explicit
construction given by `Pedro fdez`, 2011-04-12.

N(5,9,8) ≥ 30 follows by
explicit
construction given by `PacoHH`, 2011-03-18.

N(8,6,8) ≥ 20 follows by
explicit
construction given by `Pucho`, 2012-04-22.

m\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 9 | 27 | 36-62 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 4 | 6 | 18 | 30-50 | |

2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 12 | 27-36 | ||

3 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 10 | 20-24 | |||

4 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 8 | 18-20 | ||||

5 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 6 | 14-16 | |||||

6 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 6 | 12 | ||||||

7 | 1 | 1 | 2 | 2 | 2 | 3 | 6 | 10 | |||||||

8 | 1 | 2 | 2 | 2 | 3 | 4 | 8 | ||||||||

9 | 2 | 2 | 2 | 3 | 4 | 6 | |||||||||

10 | 2 | 2 | 2 | 4 | 6 | ||||||||||

11 | 2 | 2 | 4 | 6 | |||||||||||

12 | 2 | 3 | 4 | ||||||||||||

13 | 2 | 4 | |||||||||||||

14 | 4 |

N(0,14,9) ≥ 36 follows by
explicit
construction due to `spaik`, 2011-04-09.

N(1,13,9) ≥ 30 follows by
explicit
construction due to `spaik`, 2011-06-05.

N(3,11,9) ≥ 20 follows by
explicit
construction due to `spaik`, 2012-04-30.

m\n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 6 | 13 |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 10-12 | |

2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 9 | ||

3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 4 | 7 | |||

4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 6 | ||||

5 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 6 | |||||

6 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 | ||||||

7 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 4 | |||||||

8 | 1 | 1 | 2 | 2 | 2 | 3 | 4 | ||||||||

9 | 1 | 2 | 2 | 2 | 3 | 4 | |||||||||

10 | 2 | 2 | 2 | 2 | 4 | ||||||||||

11 | 2 | 2 | 2 | 3 | |||||||||||

12 | 2 | 2 | 3 | ||||||||||||

13 | 2 | 2 | |||||||||||||

14 | 2 |