# The Assmus-Mattson Theorem, Golay codes, and Mathieu groups

A block design is a pair $(X,B)$, where $X$ is a non-empty finite set of $v>0$ elements called points, and $B$ is a non-empty finite multiset of size $b$ whose elements are called blocks, such that each block is a non-empty finite multiset of $k$ points. A design without repeated blocks is called a simple block design. If every subset of points of size $t$ is contained in exactly $\lambda$ blocks the the block design is called a $t$$(v,k,\lambda)$ design (or simply a $t$-design when the parameters are not specfied). When $\lambda = 1$ then the block design is called a $S(t,k,v)$ Steiner system.

Let $C$ be an [$n,k,d$] code and let $C_i = \{ c \in C\ |\ wt(c) = i\}$ denote the weight $i$ subset of codewords of weight $i$. For each codeword $c\in C$, let $supp(c)=\{i\ |\ c_i\not= 0\}$ denote the support of the codeword.

The first example below means that the binary [$24,12,8$]-code $C$ 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 $24$. This is a self-dual [$24,12,8$]-code. The set $X_8 = \{supp(c)\ |\ c \in C_8\}$ is a $5-(24, 8, 1)$ design; $X_{12} = \{supp(c)\ |\ c \in C_{12}\}$ is a $5-(24, 12, 48)$ design;and, $X_{16} = \{supp(c)\ |\ c \in C_{16}\}$ is a $5-(24, 16, 78)$ 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 $A_0, A_1, ..., A_n$ be the weight distribution of the codewords in a binary linear [$n , k, d$] code $C$, and let $A_0^\perp, A_1^\perp, ..., A_n^\perp$ be the weight distribution of the codewords in its dual [$n,n-k, d^\perp$] code $C^\perp$. Fix a $t$, $0, and let $s = |\{ i\ |\ A_i^\perp \not= 0, 0.
Assume $s\leq d-t$.

• If $A_i\not= 0$ and $d\leq i\leq n$ then $C_i = \{ c \in C\ |\ wt(c) = i\}$ holds a simple $t$-design.
• If $A_i^\perp\not= 0$ and $d^\perp\leq i\leq n-t$ then $C_i^\perp = \{ c \in C^\perp \ |\ wt(c) = i\}$ holds a simple $t$–design.
• If $A_i^\perp\not= 0$ and $d^\perp\leq i\leq n-t$ then $C_i^\perp = \{ c \in C^\perp \ |\ wt(c) = i\}$ holds a simple $t$–design.

In the Assmus and Mattson Theorem, $X$ is the set $\{1,2,...,n\}$ of coordinate locations and $B = \{supp(c)\ |\ c \in C_i\}$ is the set of supports of the codewords of $C$ of weight $i$. Therefore, the parameters of the $t$-design for $C_i$ are

• $t$ = given,
• $v = n$,
• $k = i$, (this k is not to be confused with dim(C)!),
• $b = A_i$,
• $\lambda = b*binomial(k,t)/binomial(v,t)$

(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 $M_{24}$. 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.

# Copyright law as it pertains to mathematics and mathematical software

Disclaimer: I am not a lawyer and this is not to be construed as legal advice. However, I find copyright law very interesting but complicated and wrote this only to try to explain some of the simpler aspects of copyright law which pertain to mathematical scholars.

This is a brief survey on some aspects of federal copyright law, as it pertains to mathematicians. (By mathematician, which we mean teachers or scholarly researchers at a non-profit educational institute; generally, the more commercial the enterprise, the more complicated the law is governing it. This small survey only discusses the simplest aspects.) It does not cover other aspects of intellectual property law, such as laws governing patents, trade secrets, and so on (see for example, [K]). We have used the excellent book [L] by Leaffer as a basis for this survey. Copyright law is very complex [US] but, I hope, this post shows that many parts of copyright law which pertain to mathematicians, is relatively uncomplicated.

U.S. copyright law applies to writings, or works ‘fixed in a tangible medium of expression’, produced by an author. For this article, we assume the author is a U. S. citizen and the work was produced on U. S. soil. However, a ‘writing’ is not assumed to be human-readable, so, for example, a software program in executable binary form, or ‘object code’, is included [L], section 3.06. The owner of the copyright of a work has the exclusive right for

• reproduce or copy the work,
• prepare derivative works,
• distribute the work,
• perform the work publicly,
• display the work publicly.

Before explaining these terms, exceptions to these rights, and how these rights relate especially to mathematical works, we discuss works for which copyright law cannot be applied. The law is designed to protect creative written works.

• Ideas are generally not subject to copyright. From section 102 of [US]:In no case does copyright protection for an original work of authorship extend to any idea, procedure, process, system, method of operation, concept, principle, or discovery, regardless of the form in which it is described, explained, illustrated, or embodied in such work.
• An unoriginal work, or a work ‘mechanically produced’, say by a computer program whose use requires no originality, are not copyrightable (more precisely, are not subject to a separate copyright – the program could, for example, output copyrighted elements). For example, the output of an automatic theorem proving program is not copyrightable. On the other hand, the output of an image processing program which takes an image and applies a de-noising algorithm is a “mechanical” derivation of the original image, so the copyright is the same as that of the original.
• Facts are is not copyrightable. It doesn’t matter how much money or man power it took to discover, collect, or obtain it. (However, there are various laws which can be used to protect such intellectual property, such as trade secret laws.) For example, you cannot copyright a theorem, such Fermat’s Last Theorem, as it is a fact.
Indeed, section 102(b) of the copyright law (chapter 1, section 102 in [US]) says:

In no case does copyright protection for an original work of authorship extend to any idea, procedure, process, system, method of operation, concept, principle, or discovery, regardless of the form in which it is described, explained, illustrated, or embodied in such work.

I have put `discovery’ in bold for emphasis.

In some cases, a creative arrangement of data is copyrightable, for example, statistical data displayed in an unusual way, even if the data itself is not.

• Works in the public domain (in particular most ‘official’ works by the U. S. government), are not copyrightable. All written works eventually pass into the public domain. Due to the variety of copyright laws which have been passed in the United States over the years, the duration of copyright depends on when the work was written, if it is a joint work (or a ‘work for hire’) or not, and various other factors. In fact, all of chapter 3 or the copyright code [US] is devoted to to duration, so it is complicated. However, life plus 70 years should apply in most cases.
From section 302(a) of [US]:
• In General. — Copyright in a work created on or after January 1, 1978, subsists from its creation and, except as provided by the following subsections, endures for a term consisting of the life of the author and 70 years after the author’s death.

For the owner of a creative mathematical work, whether it is an article or a piece of software, we explain next what these rights mean.

• Reproduction: A reproduction is to fix a copy in a tangible and relatively permanent form, such as a xerox copy or a file on a computer (though a copy stored in your cache is exempted). Aside from non-profit, educational, government, or ‘fair use’, the copyright holder has the sole right to make unlimited copies of the work. For example, if you publish a paper or book, you often sign over your copyright to a publisher. If anyone could make a copy of your article freely, the commercial interest of the publisher would disappear. Similarly, if you wrote a mathematical software program which you wanted to market, you would want to restrict the copies of the program to those who paid for it. A research paper downloaded from the internet and then emailed to a colleague is an example of a reproduction.However, there is a ‘fair use’ exception to copyright law regarding copying for personal use if you are a scholar (at a non-profit institute) or the educational use of your students if you are a teacher (at a non-profit institute). These do not apply to commercial think-tanks or to commercial training centers. The guidelines are different for research than for educational use, but the basic idea is to copy no more than is necessary. The guidelines for education are more strict. Generally, 1000 words or 10% of the material (the minimum of the two) are recommended limits [L], section 10.12.
• Distribution: A work is distributed if it is made available to the ‘public’ in some form. For example, a copy in a public library or a file posted on a world-accessible internet site are publicly distributed. However, defining the term ‘public’ precisely in this context is a technical legal matter, for which we refer to [L], section 8.13.

For more details, we refer to Leaffer, [L], or Joyce et al [J]. You are encouraged to consider placing on your works one of the Creative Commons licenses [C] or one of the FSF licenses [F], whatever you feel is appropriate. These licenses allow others to distribute your work legally, enabling more people can learn from your mathematical efforts.

Bibiliography (I’ve included [Le] and [V] for related and, I think, interesting reading.)

[F] Free Software Foundation, http://www.fsf.org/

[J] C. Joyce, M. Leaffer, P. Jaszi, and T. Ochoa, Copyright law, 7th edition, LexisNexis, 2006.

[K]} B. Klemens, Math you can’t use, Brookings Institute Press, Washington DC, 2006.

[L] M. Leaffer, Understanding copyright law, 4th edition, LexisNexis, 2005.

[Le] L. Lessig, Code 2.0, http://codev2.cc/