A **block design** is a pair , where is a non-empty finite set of elements called **points**, and is a non-empty finite multiset of size whose elements are called **blocks**, such that each block is a non-empty finite multiset of points. A design without repeated blocks is called a **simple block design**. If every subset of points of size is contained in exactly blocks the the block design is called a —** design** (or simply a **-design** when the parameters are not specfied). When then the block design is called a ** Steiner system**.

Let be an [] code and let denote the **weight subset** of codewords of weight . For each codeword , let denote the **support** of the codeword.

The first example below means that the binary []-code has the property that the (support of the) codewords of weight 8 (resp, 12, 16) form a 5-design.

Example: Let $C$ denote the extended binary Golay code of length . This is a self-dual []-code. The set is a design; is a design;and, is a design.

This is a consequence of the following theorem of Assmus and Mattson.

**Assmus and Mattson Theorem** (section 8.4, page 303 of [HP]):

* Let be the weight distribution of the codewords in a binary linear [] code , and let be the weight distribution of the codewords in its dual [] code . Fix a , , and let .
Assume .
*

*If and then holds a simple -design.**If and then holds a simple –design.**If and then holds a simple –design.*

In the Assmus and Mattson Theorem, is the set of coordinate locations and is the set of supports of the codewords of of weight . Therefore, the parameters of the -design for are

- = given,
- ,
- , (this k is not to be confused with dim(C)!),
- ,

(by Theorem 8.1.6, p. 294, in \cite{HP}). Here is a SAGE example.

sage: C = ExtendedBinaryGolayCode()

sage: C.assmus_mattson_designs(5)

['weights from C: ',

[8, 12, 16, 24],

'designs from C: ',

[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)], [5, (24, 24, 1)]],

'weights from C*: ',

[8, 12, 16],

'designs from C*: ',

[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)]]]

sage: C.assmus_mattson_designs(6)

0

The automorphism group of the extended binary Golay code is the Mathieu group . Moreover, the code is spanned by the codewords of weight 8.

References:

[HP] W. C. Huffman, V. Pless, **Fundamentals of error-correcting codes**, Cambridge Univ. Press, 2003.

[CvL] P. Cameron, J. van Lint, **Graphs, codes and designs**, Cambridge Univ. Press, 1980.