This table gives, for all parameters *n* and *d*,
an interval for the maximum size *A*(*n*,*d*)
of an unrestricted binary block code. Unrestricted means in this context that no further constraint (such as linearity, constant weight, etc.) is imposed on
the binary codes.
A single value indicates that the lower and upper bounds
coincide: *A*(*n*,*d*)
is known exactly in such cases.
Superscripts denote the methods
by which the upper bounds were obtained, as detailed in the legend below.

The table gives the best known bounds on
*A*(*n*,*d*) for all *n* up to 28 and
all even *d* <= *n*.
For odd values of *d*, *A*(*n*,*d*) =
*A*(*n*+1,*d*+1).
The table can be extended to
larger and smaller values of *d* by noting that
*A*(*n*,*d*) = 2 for
2*n*/3 < *d* <= *n* and
*A*(*n*,2) = 2^{n-1} for all *n*.

Tables for *lower* bounds of binary codes are available on several
websites.
The lower bounds in the table below were copied from [LRS]
in January 2001 and are not regularly
updated. See [B] for updated lower bounds
and details on how they were obtained.

- Mar. 2, 2001: The web version of the table in [2] was published.
- June 1, 2001: 6 updates by Mounits, Etzion, and Litsyn using linear programming were included, from a preliminary version of [7].
- Oct. 28, 2003: 10 further updates by Mounits, Etzion, and Litsyn were included, from the published version of [7].
- July 8, 2004: 11 updates by Schrijver based on the Terwilliger algebra were included, from [8].

Please report further updates or corrections.

- 1-4: Theorem numbers in [2].
- S: Specific bounds described in the last paragraph of [2].
- M: Updates from [7].
- T: Updates from [8].

d=4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | ||
---|---|---|---|---|---|---|---|---|---|

n=4 | 2^{1} | n=4 | |||||||

5 | 2^{1} | 5 | |||||||

6 | 4^{1} | 2^{1} | 6 | ||||||

7 | 8^{1} | 2^{1} | 7 | ||||||

8 | 16^{1}
| 2^{1} | 2^{1} | 8 | |||||

9 | 20^{4}
| 4^{1} | 2^{1} | 9 | |||||

10 | 40^{1} | 6^{1} | 2^{1} | 2^{1} | 10 | ||||

11 | 72^{S} | 12^{1} | 2^{1}
| 2^{1} | 11 | ||||

12 | 144^{S}
| 24^{1} | 4^{1} | 2^{1} | 2^{1} | 12 | |||

13 | 256^{3} | 32^{S}
| 4^{1} | 2^{1} | 2^{1} | 13 | |||

14 | 512^{3} | 64^{3} | 8^{1} | 2^{1}
| 2^{1} | 2^{1} | 14 | ||

15 | 1024^{2} | 128^{3} | 16^{1} | 4^{1}
| 2^{1} | 2^{1} | 15 | ||

16 | 2048^{2} | 256^{2} | 32^{1} | 4^{1}
| 2^{1} | 2^{1} | 2^{1} | 16 | |

17 | 2720 - 3276^{3} | 256 - 340^{S} | 36 -
37^{S} | 6^{1} | 2^{1} | 2^{1}
| 2^{1} | 17 | |

18 | 5312 - 6552^{1}
| 512 - 680^{1} | 64 - 72^{S} | 10^{1}
| 4^{1} | 2^{1} | 2^{1} | 2^{1} | 18 |

19 | 10496 - 13104^{1} | 1024 - 1280^{T} | 128 -
142^{T} | 20^{1} | 4^{1} | 2^{1}
| 2^{1} | 2^{1} | 19 |

20 | 20480 - 26208^{1}
| 2048 - 2372^{4} | 256 - 274^{T} | 40^{1}
| 6^{1} | 2^{1} | 2^{1} | 2^{1} | 20 |

21 | 36864 - 43688^{M} | 2560 - 4096^{S}
| 512^{S} | 42 - 48^{S} | 8^{1} | 4^{1}
| 2^{1} | 2^{1} | 21 |

22 | 73728 - 87376^{M}
| 4096 - 6941^{4} | 1024^{3} | 50 - 87^{T}
| 12^{1} | 4^{1} | 2^{1} | 2^{1} | 22 |

23 | 147456 - 173015^{M} | 8192 - 13766^{T}
| 2048^{3} | 76 - 150^{4} | 24^{1} | 4^{1}
| 2^{1} | 2^{1} | 23 |

24 | 294912 -
344308^{2} | 16384 - 24106^{4} | 4096^{2} | 128 -
280^{3} | 48^{1} | 6^{1} | 4^{1}
| 2^{1} | 24 |

25 | 524288 - 599184^{M} | 16384 -
48008^{T} | 4096 - 5477^{T} | 176 - 503^{T} | 52 -
56^{S} | 8^{1} | 4^{1} | 2^{1} | 25 |

26 | 1048576 - 1198368^{M} | 32768 - 84260^{M} | 4096 -
9672^{M} | 270 - 859^{T} | 64 - 98^{S}
| 14^{1} | 4^{1} | 2^{1} | 26 |

27 | 2097152
- 2396736^{M} | 65536 - 157285^{M} | 8192 - 17768^{T}
| 512 - 1764^{3} | 128 - 169^{4} | 28^{1}
| 6^{1} | 4^{1} | 27 |

28 | 4194304 -
4793472^{M} | 131072 - 291269^{4} | 16384 - 32151^{T}
| 1024 - 3200^{3} | 178 - 288^{3} | 56^{1}
| 8^{1} | 4^{1} | 28 |

d=4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 |