# Coding Theory and Cryptography

This was once posted on my USNA webpage. Since I’ve retired, I’m going to repost it here.

Coding Theory and Cryptography:
From Enigma and Geheimschreiber to Quantum Theory

(David Joyner, ed.) Springer-Verlag, 2000.
ISBN 3-540-66336-3

Summary: These are the proceedings of the “Cryptoday” Conference on Coding Theory, Cryptography, and Number Theory held at the U. S. Naval Academy during October 25-26, 1998. This book concerns elementary and advanced aspects of coding theory and cryptography. The coding theory contributions deal mostly with algebraic coding theory. Some of these papers are expository, whereas others are the result of original research. The emphasis is on geometric Goppa codes, but there is also a paper on codes arising from combinatorial constructions. There are both, historical and mathematical papers on cryptography.
Several of the contributions on cryptography describe the work done by the British and their allies during World War II to crack the German and Japanese ciphers. Some mathematical aspects of the Enigma rotor machine and more recent research on quantum cryptography are described. Moreover, there are two papers concerned with the RSA cryptosystem and related number-theoretic issues.

Chapters

• Reminiscences and Reflections of a Codebreaker, by Peter Hilton pdf
• FISH and I, by W. T. Tutte pdf
• Sturgeon, The FISH BP Never Really Caught, by Frode Weierud, pdf
• ENIGMA and PURPLE: How the Allies Broke German and Japanese Codes During the War, by David A. Hatch pdf
• The Geheimschreiber Secret, by Lars Ulfving, Frode Weierud pdf
• The RSA Public Key Cryptosystem, by William P. Wardlaw pdf
• Number Theory and Cryptography (using Maple), by John Cosgrave pdf
• A Talk on Quantum Cryptography or How Alice Outwits Eve, by Samuel J. Lomonaco, Jr. pdf
• The Rigidity Theorems of Hamada and Ohmori, Revisited, by T. S. Michael pdf
• Counting Prime Divisors on Elliptic Curves and Multiplication in Finite Fields, by M. Amin Shokrollahi pdf,
• On Cyclic MDS-Codes, by M. Amin Shokrollahi pdf
• Computing Roots of Polynomials over Function Fields of Curves, by Shuhong Gao, M. Amin Shokrollahi pdf
• Remarks on codes from modular curves: MAPLE applications, by David Joyner and Salahoddin Shokranian, pdf

For more cryptologic history, see the National Cryptologic Museum.

This is now out of print and will not be reprinted (as far as I know). The above pdf files are posted by written permission. I thank Springer-Verlag for this.

# The truncated tetrahedron covers the tetrahedron

At first, you might think this is obvious – just “clip” off each corner of the tetrahedron $\Gamma_1$ to create the truncated tetrahedron $\Gamma_2$ (by essentially creating a triangle from each of these clipped corners – see below for the associated graph). Then just map each such triangle to the corresponding vertex of the tetrahedron. No, it’s not obvious because the map just described is not a covering. This post describes one way to think about how to construct any covering.

First, color the vertices of the tetrahedron in some way.

The coloring below corresponds to a harmonic morphism $\phi : \Gamma_2\to \Gamma_1$:

All others are obtained from this by permuting the colors. They are all covers of $\Gamma_1 = K_4$ – with no vertical multiplicities and all horizontal multiplicities equal to 1. These 24 harmonic morphisms of $\Gamma_2\to\Gamma_1$ are all coverings and there are no other harmonic morphisms.

# Quartic graphs with 12 vertices

This is a continuation of the post A table of small quartic graphs. As with that post, it’s modeled on the handy wikipedia page Table of simple cubic graphs.

According to SageMath computations, there are 1544 connected, 4-regular graphs. Exactly 2 of these are symmetric (ie, arc transitive), also vertex-transitive and edge-transitive. Exactly 8 of these are vertex-transitive but not edge-transitive. None are distance regular.

Example 1: The first example of such a symmetric graph is the circulant graph with parameters (12, [1,5]), depicted below. It is bipartite, has girth 4, and its automorphism group has order 768, being generated by $(9,11), (5,6), (4,8), (2,10), (1,2)(5,9)(6,11)(7,10), (1,7), (0,1)(2,5)(3,7)(4,9)(6,10)(8,11)$.

Example 2: The second example of such a symmetric graph is the cuboctahedral graph, depicted below. It has girth 3, chromatic number 3, and its automorphism group has order 48, being generated by $(1,10)(2,7)(3,6)(4,8)(9,11), (1,11)(3,4)(6,8)(9,10), (0,1,9)(2,8,10)(3,7,11)(4,5,6)$.

# The Riemann-Hurwitz formula for regular graphs

A little over 10 years ago, M. Baker and S. Norine (I’ve also seen this name spelled Norin) wrote a terrific paper on harmonic morphisms between simple, connected graphs (see “Harmonic morphisms and hyperelliptic graphs” – you can find a downloadable pdf on the internet of you google for it). Roughly speaking, a harmonic function on a graph is a function in the kernel of the graph Laplacian. A harmonic morphism between graphs is, roughly speaking, a map from one graph to another that preserves harmonic functions.

They proved quite a few interesting results but one of the most interesting, I think, is their graph-theoretic analog of the Riemann-Hurwitz formula. We define the genus of a simple connected graph $\Gamma = (V,E)$ to be

${\rm genus}(\Gamma) = |E| - |V | + 1.$

It represents the minimum number of edges that must be removed from the graph to make it into a tree (so, a tree has genus 0).

Riemann-Hurwitz formula (Baker and Norine): Let $\phi:\Gamma_2\to \Gamma_1$ be a harmonic morphism from a graph $\Gamma_2 = (V_2,E_2)$ to a graph $\Gamma_1 = (V_1, E_1)$. Then

${\rm genus}(\Gamma_2)-1 = {\rm deg}(\phi)({\rm genus}(\Gamma_1)-1)+\sum_{x\in V_2} [m_\phi(x)+\frac{1}{2}\nu_\phi(x)-1].$

I’m not going to define them here but $m_\phi(x)$ denotes the horizontal multiplicity and $\nu_\phi(x)$ denotes the vertical multiplicity.

I simply want to record a very easy corollary to this, assuming $\Gamma_2 = (V_2,E_2)$ is $k_2$-regular and $\Gamma_1 = (V_1, E_1)$ is $k_1$-regular.

Corollary: Let $\Gamma_2 \rightarrow \Gamma_1$ be a non-trivial harmonic morphism from a connected $k_2$-regular graph
to a connected $k_1$-regular graph.
Then

$\sum_{x\in V_2}\nu_\phi(x) = k_2|V_2| - k_1|V_1|\deg (\phi).$

# The number-theoretic side of J. Barkley Rosser

By chance, I ran across a reference to a paper of J Barkey Rosser and it brought back fond memories of days long ago when I would browse the stacks in the math dept library at the University of Washington in Seattle. I remember finding papers describing number-theoretic computations of Rosser and Schoenfeld. I knew nothing about Rosser. Was he a number theorist?

J. Barkley Rosser, taken at Math meeting in Denver

Here’s all I could glean from different sources on the internet:
J. Barkley Rosser was born in Jacksonville, Florida in 1907. He earned both his Bachelor of Science (1929) and his Master of Science (1931) from the University of Florida. Both degrees were in physics. He obtained his Ph.D. in mathematics (in fact, mathematical logic) from Princeton University in 1934, under the supervision of Alonso Church. After getting his Ph.D., Rosser taught at Princeton, Harvard, and Cornell and spent the latter part of his career at the University of Wisconsin-Madison. As a logician, Rosser is known for his part in the Church-Rosser Theorem and the Kleene–Rosser Paradox in lambda calculus. Moreover, he served as president of the Association for Symbolic Logic. As an applied mathematician, he served as president of the Society of Industrial and Applied Mathematics (otherwise known as SIAM). While at the University of Wisconsin-Madison, he served as the director of the U.S. Army Mathematics Research Center. He continued to lecture well into his late 70s, and passed away at his home in Madison in 1989. He has a son, J. Barkley Rosser Jr, who’s an economist at James Madison University.

Lowell Schoenfeld spent his early years in New York City, graduating Cum Laude from the College of the City of New York in 1940. He went on to MIT to earn a Master’s. He received his Ph.D. in 1944 from the University of Pennsylvania under the direction of Hans Rademacher. (During his years in graduate school, he seems to have worked for the Philadelphia Navy Yard as well, writing reports on aircraft navigational computers.) After positions at Temple University and Harvard, he moved to the University of Illinois, where he met his future wife. He met Josephine M. Mitchell when she was a tenured Associate Professor and he was an untenured Assistant Professor. After they married, the University would no longer allow Mitchell to teach, so the couple both resigned their positions and eventually settled at Pennsylvania State University. They spent about 10 years there but in 1968 the couple moved to the University of Buffalo, where they remained until their retirements in the 1980s.

As far as I can tell, these are the papers they wrote together, all in analytic number theory:

[1] Rosser, J. Barkley; Schoenfeld, Lowell. “Approximate formulas for some functions of prime numbers”. Illinois J. Math. 6 (1962), no. 1, 64–94.
[2] Rosser, J. Barkley; Schoenfeld, Lowell; J.M. Yohe. “Rigorous Computation and the Zeros of the Riemann Zeta-Function,” 1969
[3] Rosser, J. Barkley; Schoenfeld, Lowell. “Sharper Bounds for the Chebyshev Functions $\theta (x)$ and $\psi (x)$” Mathematics of Computation Vol. 29, No. 129 (Jan., 1975), pp. 243-269
[4] Rosser, J. Barkley; Schoenfeld, Lowell. “Approximation of the Riemann Zeta-Function” 1971.

I haven’t seen a copy of the papers [2] and [4] in years but I’m guessing these are what I looked at as a teenager in Seattle, years ago, wandering through the stacks at the UW.

Rosser also wrote papers on topics in recreational mathematics, such as magic squares. Two such papers were co-written with R.J. Walker from Cornell University, who’s more well-known for his textbook Algebraic Curves:

Rosser, Barkley; Walker, R. J. “The algebraic theory of diabolic magic squares,” Duke Math. J. 5 (1939), no. 4, 705–728
Rosser, Barkley; Walker, R. J. “On the transformation group for diabolic magic squares of order four,” Bull. Amer. Math. Soc. 44 (1938), no. 6, 416–420.

Diabolic magic squares, also called pan-diagonal magic squares, are $n\times n$ squares of integers $1, 2, ..., n^2$ whose rows all add to a constant C, whose columns all add to C, whose diagonals both add to C, and whose “broken diagonals” all add to C. An example was given by the German artist Albrecht Durer in the 1514 engraving called Melencolia I: (where C=34):

# Harmonic morphisms from cubic graphs of order 8 to a graph of order 4

There are five simple cubic graphs of order 8 (listed here) and there are 6 connected graphs of order 4 (listed here). But before we get started, I have a conjecture.

Let $\Gamma_1$ be a simple graph on n1 vertices, $\Gamma_2$ a simple graph on n2 vertices, and assume there is a harmonic morphism $\phi:\Gamma_1 \to \Gamma_2$. Call an n1-tuple of “colors” $\{0,1,2,..., n2-1\}$ a harmonic color list (HCL) if it’s attached to a harmonic morphism in the usual way (the ith coordinate is j if $\phi$ sends vertex i of $\Gamma_1$ to vertex j of $\Gamma_2$). Let S be the set of all such HCLs. The automorphism group $G_1$ of $\Gamma_1$ acts on S (by permuting coordinates associated to the vertices of $\Gamma_1$, as does the automorphism group $G_2$ of $\Gamma_2$ (by permuting the “colors” associated to the vertices of $\Gamma_2$). These actions commute. Clearly S decomposes as a disjoint union of distinct $G_1\times G_2$ orbits. The conjecture is that there is only one such orbit.

Note: Caroline Melles has disproven this conjecture. Still, the question of the number of orbits is an interesting one, IMHO.

Onto the topic of the post! The 6 connected graphs of order 4 are called P4 (the path graph), D3 (the star graph, also $K_{3,1}$), C4 (the cycle graph), K4 (the complete graph), Paw (C3 with a “tail”), and Diamond (K4 but missing an edge). All these terms are used on graphclasses.org. The results below were obtained using SageMath.

1. We start with the graph $\Gamma_1$ listed 1st on wikipedia’s Table of simple cubic graphs and defined using the sage code sage: Gamma1 = graphs.LCFGraph(8, [2, 2, -2, -2], 2). This graph $\Gamma_1$ has diameter 3, girth 3, and its automorphism group G is generated by (5,6), (1,2), (0,3)(4,7), (0,4)(1,5)(2,6)(3,7), $|G|=16$. This graph is not vertex transitive. Its characteristic polynomial is $x^8 - 12x^6 - 8x^5 + 38x^4 + 48x^3 - 12x^2 - 40x - 15$. Its edge connectivity and vertex connectivity are both 2. This graph has no non-trivial harmonic morphisms to D3 or P4 or C4 or Paw. However, there are 48 non-trivial harmonic morphisms to $\Gamma_2=K4$. For example,
(the automorphism group of K4, ie the symmetric group of degree 4, acts on the colors {0,1,2,3} and produces 24 total plots), and (again, the automorphism group of K4, ie the symmetric group of degree 4, acts on the colors {0,1,2,3} and produces 24 total plots). There are 8 non-trivial harmonic morphisms to $\Gamma_2={\rm Diamond}$. For example, and Here the automorphism group of K4, ie the symmetric group of degree 4, acts on the colors {0,1,2,3}, while the automorphism group of the graph $\Gamma_1$ acts by permuting some of the coordinates, for example, it can swap the 5th and 6th coordinates.Next, we take for $\Gamma_1$ the graph listed 2nd on wikipedia’s Table of simple cubic graphs and defined using the sage code sage: Gamma1 = graphs.LCFGraph(8, [4, -2, 4, 2], 2). This graph $\Gamma_1$ has diameter 3, girth 3, and its automorphism group G is generated by (1,7)(2,6)(3,5), (0,4)(1,3)(5,7), $|G|=4$ (obviously too small to act transitively on the vertices). Its characteristic polynomial is $x^8 - 12x^6 - 4x^5 + 38x^4 + 16x^3 - 36x^2 - 12x + 9$, its edge connectivity and vertex connectivity are both 3. This graph has no non-trivial harmonic morphisms to D3 or P4 or C4 or Paw or K4. However, it has 4 non-trivial harmonic morphisms to Diamond. They are:
Let $\Gamma_1$ denote the graph listed 3rd on wikipedia’s Table of simple cubic graphs and defined using the sage code sage: Gamma1 = graphs.LCFGraph(8, [2, 4, -2, 3, 3, 4, -3, -3], 1). This graph $\Gamma_1$ has diameter 2, girth 3, and its automorphism group G is generated by (4,6), (1,2)(3,5), (0,1)(5,7), $|G|=12$. It does not act transitively on the vertices. Its characteristic polynomial is $x^8 - 12x^6 - 2x^5 + 36x^4 - 31x^2 + 12x$ and its edge connectivity and vertex connectivity are both 3.
This graph has no non-trivial harmonic morphisms to P4 or C4 or Paw or K4 or Diamond. However, it has 6 non-trivial harmonic morphisms to D3, for example,

The automorphism group of D3 (the symmetric group of degree 3) acts by permuting the colors {0,1,2,3} and so yields a total of 6=3! such harmonic color plots.Let $\Gamma_1$ denote the graph listed 4th on wikipedia’s Table of simple cubic graphs and defined using the sage code sage: Gamma1 = graphs.LCFGraph(8, [4, -3, 3, 4], 2). This example is especially interesting. Otherwise known as the “cube graph” $Q_3$, this graph $\Gamma_1$ has diameter 3, girth 4, and its automorphism group G is generated by ((2,4)(5,7), (1,7)(4,6), (0,1,4,5)(2,3,6,7), $|G|=48$. It is vertex transitive. Its characteristic polynomial is $x^8 - 12x^6 + 30x^4 - 28x^2 + 9$ and its edge connectivity and vertex connectivity are both 3.
This graph has no non-trivial harmonic morphisms to D3 or P4 or Paw. However, it has 24 non-trivial harmonic morphisms to C4, 24 non-trivial harmonic morphisms to K4, and 24 non-trivial harmonic morphisms to Diamond. An example of a non-trivial harmonic morphism to K4:

A few examples of a non-trivial harmonic morphism to Diamond:
and
A few examples of a non-trivial harmonic morphism to C4:

The automorphism group of C4 acts by permuting the colors {0,1,2,3} cyclically, while the automorphism group G acts by permuting coordinates. These yield more harmonic color plots.

# Duursma zeta function of a graph

I’m going to start off with two big caveats:

1. This is not Duursma‘s definition, it’s mine.
2. I’m not convinced (yet?) that it’s a useful idea to examine such a zeta function.

The Duursma zeta function of a linear block (error-correcting) code is due to Iwan Duursma and is a fascinating object of study. (More precisely, it’s defined for “formal” linear block codes, ie, defined via a weight enumerator polynomial with a suitable MacWilliams-like functional equation.) Sometimes it satisfies an analog of the Riemann hypothesis and sometimes it doesn’t. And sometimes that analog is still an open question.

Duursma zeta function of a code

Before we define the Duursma zeta function of a graph, we introduce the Duursma zeta function of a code.

Let $C$ be an $[n,k,d]_q$ code, ie a linear code over $GF(q)$ of length $n$, dimension $k$, and minimum distance $d$. In general, if $C$ is an $[n,k,d]_q$-code then we use $[n,k^\perp,d^\perp]_q$ for the parameters of the dual code, $C^\perp$. It is a consequence of Singleton’s bound that $n+2-d-d^\perp\geq$, with equality when $C$ is an MDS code. Motivated by analogies with local class field theory, in [Du] Iwan Duursma introduced the (Duursma) zeta function $\zeta=\zeta_C$:

$\zeta_C(T)=\frac{P(T)}{(1-T)(1-qT)},$
where $P(T)=P_C(T)$ is a polynomial of degree $n+2-d-d^\perp$, called the zeta polynomial of $C$. We next sketch the definition of the zeta polynomial.

If $C^\perp$ denotes the dual code of $C$, with parameters $[n,n-k,d^\perp]$ then the MacWilliams identity relates the weight enumerator $A_{C^\perp}$ of $C^\perp$ to the weight enumerator $A_{C}$ of $C$:

$A_{C^\perp}(x,y) = |C|^{-1}A_C(x+(q-1)y,x-y).$
A polynomial $P(T)=P_C(T)$ for which

$\frac{(xT+(1-T)y)^n}{(1-T)(1-qT)}P(T)=\dots +\frac{A_C(x,y)-x^n}{q-1}T^{n-d}+\dots \ .$
is a (Duursma) zeta polynomial of $C$.

Theorem (Duursma): If $C$ be an $[n,k,d]_q$ code with $d\geq 2$ and $d^\perp\geq 2$ then the Duursma zeta polynomial $P=P_C$ exists and is unique.

See the papers of Duursma for interesting properties and conjectures.

Duursma zeta function of a graph

Let $\Gamma=(V,E)$ be a graph having $|V(\Gamma)|$ vertices and $|E(\Gamma)|$ edges. We define the zeta function of $\Gamma$ via the Duursma zeta function of the binary linear code defined by the cycle space of $\Gamma$.

Theorem (see [DKR], [HB], [JV]): The binary code $B=B_\Gamma$ generated by the rows of the incidence matrix of $\Gamma$ is the cocycle space of $\Gamma$ over $GF(2)$, and the dual code $B^\perp$ is the cycle space $Z=Z_\Gamma$ of $\Gamma$:

$B_\Gamma^\perp = Z_\Gamma.$
Moreover,
(a) the length of $B_\Gamma$ is $|E|$, dimension of $B_\Gamma$ is $|V|-1$, and the minimum distance of $B_\Gamma$ is the edge-connectivity of $\Gamma$,
(b) length of $Z_\Gamma$ is $|E|$, dimension of $Z_\Gamma$ is $|E|-|V|+1$, and the minimum distance of $Z_\Gamma$ is the girth of $\Gamma$.

Call $Z_\Gamma$ the cycle code and $B_\Gamma$ the cocycle code.

Finally, we can introduce the (Duursma) zeta function $\Gamma$:

$\zeta_\Gamma(T)=\zeta_{Z_\Gamma} =\frac{P(T)}{(1-T)(1-qT)},$
where $P=P_\Gamma=P_{Z_\Gamma}$ is the Duursma polynomial of $\Gamma$.

Example: Using SageMath, when $\Gamma = W_5$, the wheel graph on 5 vertices, we have

$P_\Gamma(T) = \frac{2}{7}T^4 + \frac{2}{7}T^3 + \frac{3}{14}T^2 + \frac{1}{7}T + \frac{1}{14}.$
All its zeros are of absolute value $1/\sqrt{2}$.

References

[Du] I. Duursma, Combinatorics of the two-variable zeta function, in Finite fields and applications, 109–136, Lecture Notes in Comput. Sci., 2948, Springer, Berlin, 2004.

[DKR] P. Dankelmann, J. Key, B. Rodrigues, Codes from incidence matrices of graphs, Designs, Codes and Cryptography 68 (2013) 373-393.

[HB] S. Hakimi and J. Bredeson, Graph theoretic error-correcting codes, IEEE Trans. Info. Theory 14(1968)584-591.

[JV] D. Jungnickel and S. Vanstone, Graphical codes revisited, IEEE Trans. Info. Theory 43(1997)136-146.

# Harmonic morphisms to P_3 – examples

This post expands on a previous post and gives more examples of harmonic morphisms to the path graph $\Gamma_2=P_3$.

If $\Gamma_1 = (V_1, E_1)$ and $\Gamma_2 = (V_2, E_2)$ are graphs then a map $\phi:\Gamma_1\to \Gamma_2$ (that is, $\phi: V_1\cup E_1\to V_2\cup E_2$) is a morphism provided

1. if $\phi$ sends an edge to an edge then the edges vertices must also map to each other: $e=(v,w)\in E_1$ and $\phi(e)\in E_2$ then $\phi(e)$ is an edge in $\Gamma_2$ having vertices $\phi(v)\in V_2$ and $\phi(w)\in V_2$, where $\phi(v)\not= \phi(w)$, and
2. if $\phi$ sends an edge to a vertex then the edges vertices must also map to that vertex: if $e=(v,w)\in E_1$ and $\phi(e)\in V_2$ then $\phi(e) = \phi(v) = \phi(w)$.

As a non-example, if $\Gamma_1$ is a planar graph, if $\Gamma_2$ is its dual graph, and if $\phi:\Gamma_1\to\Gamma_2$ is the dual map $V_1\to E_2$ and $E_1\to V_2$, then $\phi$ is not a morphism.

Given a map $\phi_E : E_1 \rightarrow E_2 \cup V_2$, an edge $e_1$ is called horizontal if $\phi_E(e_1) \in E_2$ and is called vertical if $\phi_E(e_1) \in V_2$. We say that a graph morphism $\phi: \Gamma_1 \rightarrow \Gamma_2$ is a graph homomorphism if $\phi_E (E_1) \subset E_2$. Thus, a graph morphism is a homomorphism if it has no vertical edges.

Suppose that $\Gamma_2$ has at least one edge. Let $Star_{\Gamma_1}(v)$ denote the star subgraph centered at the vertex v. A graph morphism $\phi : \Gamma_1 \to \Gamma_2$ is called harmonic if for all vertices $v \in V(\Gamma_1)$, the quantity
$\mu_\phi(v,f)= |\phi^{-1}(f) \cap Star_{\Gamma_1}(v)|$
(the number of edges in $\Gamma_1$ adjacent to $v$ and mapping to the edge $f$ in $\Gamma_2$) is independent of the choice of edge $f$ in $Star_{\Gamma_2}(\phi(v))$.

An example of a harmonic morphism can be described in the plot below as follows: $\phi:\Gamma_1\to \Gamma_2=P_3$ sends the red vertices in $\Gamma_1$ to the red vertex of $\Gamma_2=P_3$, the green vertices in $\Gamma_1$ to the green vertex of $\Gamma_2=P_3$, and the white vertices in $\Gamma_1$ to the white vertex of $\Gamma_2=P_3$.

Example 1:

Example 2:

Example 3:

# John Cleese on creativity

Creativity is not a talent. It is a way of operating. John Cleese has written and lectured on creativity and listed the 5 factors that you can arrange to be more creative:

1. Space                                                                                                                                                       You need to escape your usual pressures.
2. Time                                                                                                                                                         You need your space for a specific block of time.
3. Time                                                                                                                                                         Giving your mind as long as possible to come up with something  original, be  patient with yourself.
4. Confidence                                                                                                                                              Nothing will stop you being creative so effectively as the fear of  making a mistake.
5. Humor                                                                                                                                                      Have fun. Humor gets us from closed mode to the open mind quicker than anything else.

# Dodecahedral Faces of M12

### by Ann Luers Casey

This post constitutes part of the math honors thesis written in spring 1997 at the USNA, advised by David Joyner. It is in the public domain.

Groups are objects in mathematics that measure symmetry in nature. A group is a set with a binary operation that has an inverse, an identity and is associative. For example, a clock has 12-fold symmetry. A more unusual group is a sporadic, non-abelian simple group. It can be very interesting to look more closely at such a group that arises naturally. One such group is M12. This post explores two different ways of creating M12 and then looks at twelve different ways M12 appears in mathematics, hence the pun the “dodecahedral faces” in the title. Specifically, this post relates M12 to the Mongean shuffle, hexads of a Steiner system, Golay codes, the Hadamard matrix of order 12, 5-transitivity, presentations, crossing the Rubicon, the minimog, the kitten, mathematical blackjack, sporadic groups, and the stabilizer in M24 of a dodecad.

Definitions:

Homomorphism: Let G1, G2 be groups with *1 denoting the group operation for G1 and *2 the group operation for G2. A function f : G1–>G2 is a homomorphism if and only if for all a,b, in Gwe have

f(a *1 b) = f(a) *2 f(b).

Isomorphism: If a homomorphism is bijective, then it is called an
isomorphism.

Automorphism: An isomorphism from a group G to itself is an automorphism.

Notation:

• Let Fq denote the finite field with q elements, q is a power of a prime.
• Z = the invertible scalar 2×2 matrices with entries in Fqx.
• Let PGL2(Fq) = GL2(Fq)/Z = {A*Z | A is in GL2(Fq)}, with multiplication given by
(A*Z)(B*Z) = (A*B)Z. This is the projective linear group over Fq.
• LF(Fq) is the group of linear fractional transformations x–>(ax+b)/(cx+d).

Claim: There is a group theoretic isomorphism between PGL2(Fq) and LF(Fq). (See [11], Theorem 9.47 for a proof.)

Claim: LF(Fq) acts 3-transitively on the set P1(Fq) (q>3). I.e., one can send any triple to any other triple in P1(Fq) by using a suitable linear fractional transformation. (See [11], Theorem 9.48 for a proof.)

Theorem

PSL2(Fq) = < x–>x+1, x–>kx, x–>-1/x>, where k is any element in Fqthat generates the multiplicative group of squares.

For a proof, see [12], ch 10, section 1.

One way to construct the Mathieu group M12 is the following, accredited to Conway.

M12 = < PSL2(F11), (2 10)(3 4)(5 9)(6 7) >.More explicitly, let

• f1 be a cyclic permutation = x–> x+1 = (0,1,2,…,10)(inf).
• f2 = x–>kx = (0)(1 3 9 5 4)(2 6 7 10 8)(inf) when k=3.
• f3 = x–>-1/x = (0 inf)(1 10)(2 5)(3 7)(4 8)(6 9).
• f4 = (2 10)(3 4)(5 9)(6 7).

Then M12 = < f1, f2, f3, f4 >. Therefore, M12 is a subgroup of the symmetric group on 12
symbols, namely inf, 0, 1, …, 10.

Another way to construct M12 is given later under 5-transitivity.

There are many occurrences of M12 in mathematics, but here I will list and explain twelve of them:

1. Mongean Shuffle
3. Golay Code
5. 5- Transitivity
6. Presentations
7. Crossing the Rubicon
8. <a href="m_12.htm#M12 and the Minimog”>M12 and the Minimog
9. Kitten
10. Mathematical Blackjack or Mathieu’s 21
12. <a href="m_12.htm#Stabilizer in M24 of a dodecad”>Stabilizer in M24 of a dodecad

## 1. Mongean Shuffle

The Mongean shuffle concerns a deck of twelve cards, labeled 0 through 11. The permutation

r(t) = 11-t

reverses the cards around. The permutation

s(t) = min(2t,23-2t)

is called the Mongean Shuffle. The permutation group M12 is generated by r and s: M12 = < r,s >, as a subgroup of S12. (See [12], Chap. 11, Sec. 17 or [18])

 Jacob Steiner (1796-1863) was a Swiss mathematician specializing in projective goemetry. (It is said that he did not learn to read or write until the age of 14 and only started attending school at the age of 18.) The origins of “Steiner systems” are rooted in problems of plane geometry.

Let T be a given set with n elements. Then the Steiner system S(k,m,n) is a collection S = {S1, … ,Sr} of subsets of T such that

• |Si| = m,
• For any subset R in T with |R| = k there is a unique i, 1<=i<=n such that R is contained in Si. |S(k,m,n)| = (n choose k)/(m choose k).

If any set H has cardinality 6 (respectively 8, 12) then H is called a hexad, (respectively octad, dodecad.)

Let’s look at the Steiner System S(5,6,12) and M12. We want to construct the Steiner system S(5,6,12) using the projective line P1(F11). To define the hexads in the Steiner system, denote

• the projective line over F11 by P1(F11)={inf,0,1,…,10}.
• Q = {0,1,3,4,5,9}=the quadratic residues union 0
• G = PSL2(F11)
• S = set of all images of Q under G. (Each element g in G will send Q to a subset of P1(F11). )

There are always six elements in such a hexad. There are 132 such hexads. If I know five of the elements in a hexad of S, then the sixth element is uniquely determined. Therefore S is a Steiner system of type (5,6,12).

Theorem:
M12 sends a hexad in a Steiner system to another hexad in a Steiner system. In fact, the automorphism group of a Steiner system of type (5,6,12) is isomorphic to M12.

(For a proof, see [11], Theorem 9.78.)

The hexads of S form a Steiner system of type (5,6,12), so

M12 = < g in S12 | g(s) belongs to S, for all s in S > .

In other words, M12 is the subgroup stabilizing S. The hexads support the weight six words of the Golay code, defined next. (For a proof, see  [6].)

## 3. Golay Code

 ” The Golay code is probably the most important of all codes for both practical and theoretical reasons.” ([17], pg. 64) M. J. E. Golay (1902-1989) was a Swiss physicist known for his work in infrared spectroscopy among other things. He was one of the founding fathers of coding theory, discovering GC24 in 1949 and GC12 in 1954.

A code C is a vector subspace of (Fq)for some n >=1 and some prime power q =pk.
An automorphism of C is a vector space isomorphism, f:C–>C.

If w is a code word in Fqn, n>1, then the number of non-zero coordinates of w is called the weight w, denoted by wt(w). A cyclic code is a code which has the property that whenever (c0,c1,…,cn-1) is a code word then so is (cn-1,c0,…,cn-2).
If c=(c0,c1,…,cn-1) is a code word in a cyclic code C then we can associate to it a polynomial g_c(x)=c0 + c1x + … + cn-1xn-1. It turns out that there is a unique monic polynomial with coefficients in Fq

of degree >1 which divides all such polynomials g_c(x). This polynomial is called
a generator polynomial for C, denoted g(x).

Let n be a positive integer relatively prime to q and let alpha be a primitive n-th root of unity. Each generator g of a cyclic code C of length n has a factorization of the form g(x) = (x-alphak1)… (x-alphakr), where {k1,…,kr} are in {0,…,n-1} [17]. The numbers alphaki, 1≤ i≤ r, are called the zeros of the code C.

If p and n are distinct primes and p is a square mod n, then the quadratic residue code of length n over Fp is the cyclic code whose generator polynomial has zeros
{alphak | k is a square mod n} [17]. The ternary Golary code GC11 is the quadratic
residue code of length 11 over F3.

The ternary Golay code GC12 is the quadratic residue code of length 12 over F3 obtained by appending onto GC11 a zero-sum check digit [12].

Theorem:
There is a normal subgroup N of Aut(GC12) of order 2 such that Aut(GC12)/N is isomorphic to M12. M12 is a quotient of Aut(GC12) by a subgroup or order 2. In other words, M12 fits into the following short exact sequence:

1–>N–>Aut(GC12)–>M12–>1

Where i is the embedding and N in Aut(GC12) is a subgroup of order 2. See [6].

 Jacques Hadamard (1865-1963) was a French mathematician who did important work in analytic number theory. He also wrote a popular book “The psychology in invention in the mathematical field” (1945).

A Hadamard matrix is any n x n matrix with a +1 or -1 in every entry such that the absolute value of the determinant is equal to nn/2.

An example of a Hadamard matrix is the Paley-Hadamard matrix. Let p be a prime of the form 4N-1, p > 3. A Paley-Hadamard matrix has order p+1 and has only +1’s and -1’s as entries. The columns and rows are indexed as (inf,0,1,2,…,p-1). The infinity row and the infinity column are all +1’s. The zero row is -1 at the 0th column and at the columns that are quadratic non-residues mod p; the zero row is +1 elsewhere. The remaining p-1 rows are cyclic shifts of the finite part of the second row. For further details, see for example [14].

When p = 11 this construction yields a 12×12 Hadamard matrix.

Given two Hadamard Matrices A, B we call them left-equivalent if there is an nxn signed permutation matrix P such that PA = B.

The set {P nxn signed permutation matrix| AP is left equivalent to A} is called the automorphism group of A. In other words, a matrix is an automorphism of the Hadamard matrix, if it is a nxn monomial matrix with entries in {0,+1,-1} and when it is multiplies the Hadamard matrix on the right, only the rows may be permuted, with a sign change in some rows allowed.

Two nxn Hadamard matrices A, B are called equivalent if there are nxn signed permutation matrices P1, P2 such that A = P1 *B *P2.

All 12×12 Hadamard matrices are equivalent ([13][16] pg. 24). The group of automorphisms of any 12×12 Hadamard matrix is isomorphic to the Mathieu group M12 ([14] pg 99).

## 5. 5-Transitivity

 Emile Mathieu (1835-1890) was a mathematical physicist known for his solution to the vibrations of an elliptical membrane.

The fact that M12 acts 5-transitively on a set with 12 elements is due to E. Mathieu who proved the result in 1861. (Some history may be found in [15].)

There are only a finite number of types of 5-transitive groups, namely Sn (n>=5), An (n>=7), M12 and M24. (For a proof, see [11])

Let G act on a set X via phi : G–>SX. G is k-transitive if for each pair of ordered k-tuples (x1, x2,…,xk), (y1,y2,…,yk), all xi and yi elements belonging to X, there exists a g in G such that yi = phi(g)(xi) for each i in {1,2,…,k}.

M12 can also be constructed as in Rotman [11], using transitive extensions, as follows (this construction appears to be due originally to Witt). Let fa,b,c,d(x)=(ax+b)/(cx+d), let

M10 = < fa,b,c,d, fa’,b’,c’,d’ |ad-bc is in Fqx, a’d’-b’c’ is not in Fqx >,

q = 9.

pi = generator of F9x, so that F9x = < pi> = C8.

Using Thm. 9.51 from Rotman, we can create a transitive extension of M10. Let omega be a new symbol and define

M11 = < M10, h| h = (inf, omega)(pi, pi2)(pi3,pi7) (pi5,pi6)>.

Let P1(F9) = {inf, 0, 1, pi, pi2, … , pi7}. Then M11 is four transitive on Y0 = P1(F9) union {omega}, by Thm 9.51.

Again using Thm. 9.51, we can create a transitive extension of M11. Let sigma be a new symbol and define

M12 = < M11, k>, where k = (omega, sig)(pi,pi3) (pi2,pi6)(pi5,pi7). M12 is 5-transitive on Y1 = Y0 union {sig}, by Th. 9.51.

Now that we constructed a particular group that is 5-transitive on a particular set with 12 elements, what happens if we have a group that is isomorphic to that group? Is this new group also 5-transitive?

Let G be a subgroup of S12 isomorphic to the Mathieu group M12. Such a group was constructed in Section 1.

Lemma: There is an action of G on the set {1,2,…,12} which is 5-transitive.

proof: Let Sig : G –> M12 be an isomorphism. Define g(i) = Sig(g)(i), where i = {1,2,…,12}, g is in G. This is an action since Sig is an isomorphism. Sig-1(h)(i) = h(i) for all g in M12, i in Y1. Using some h in M12, any i1,…,i5

in Y1 can be sent to any j1,…,j5 in Y1. That is, there exists an h in M12 such that h(ik) = jk, k= 1,…,5 since M12 is 5-transitive. Therefore, Sig-1(h)(ik) = jk = g(ik). This action is 5-transitive. QED

In fact, the following uniqueness result holds.

Theorem: If G and G’ are subgroups of S12 isomorphic to M12 then they are conjugate in S12.

(This may be found in [7], pg 211.)

## 6. Presentations

The presentation of M12 will be shown later, but first I will define a presentation.

Let G = < x1,…,xn | R1(x1,…xn) = 1, …, Rm(x1,…,xn) = 1> be the smallest group generated by x1,…,xn satisfying the relations R1,…Rm. Then we say G has presentation with generators x1,…,xn and relations R1(x1,…xn) = 1, …, Rm(x1,…,xn) = 1.

Example: Let a = (1,2,…,n), so a is an n-cycle. Let Cn be the cyclic group, Cn = < a > =
{1,a,…,an-1}. Then Cn has presentation < x | xn=1 > = all words in x, where x satisfies xn.=1 In fact, < x | xn = 1 > is isomorphic to < a >. Indeed, the isomorphism
< x | xn = 1 > –> < a > is denoted by xk –> ak, 0 <= k <= n-1. Two things are needed for a presentation:

• generators, in this case x, and
• relations, in this case xn = 1.

Example: Let G be a group generated by a,b with the following relations; a2 = 1, b2 = 1, (ab)2 = 1:

G = < a,b | a2 = 1, b2 = 1, (ab)2 = 1 > = {1,a,b,ab}.

This is a non-cyclic group of order 4.

Two presentations of M12 are as follows:

M12 = < A,B,C,D | A11 = B5 = C2 = D2 = (BC)2 = (BD)2 = (AC)= (AD)3 = (DCB)2 = 1, AB =A3 >

= < A,C,D | A11 = C2 = D2 = (AC)3 = (AD)3 = (CD)10 = 1, A2(CD)2A = (CD)2 >.

In the first presentation above, AB = B-1AB. These are found in [6] and Chap. 10 Sec. 1.6 [12].

## 7. Crossing the Rubicon

The Rubicon is the nick-name for the Rubik icosahedron, made by slicing the icosahedron in half for each pair of antipodal vertices. Each vertex can be rotated by 2*pi/5 radians, affecting the vertices in that half of the Rubicon, creating a shape with 12 vertices, and six slices. The Rubicon and M12 are closely related by specific moves on the Rubicon.

Let f1, f2, …,f12 denote the basic moves of the Rubicon, or a 2*pi/5 radians turn of the sub-pentagon about each vertex. Then according to Conway,

M12 = < x*y-1 | x,y are elements of {f1, f2, …,f12 } >.

Actually, if a twist-untwist move, x*y-1, as above, is called a cross of the Rubicon, then M12 is generated by the crosses of the Rubicon! ([1], Chap. 11 Sec. 19 of [12])

<a name="M12 and the Minimog”>

## 8. M12 and the Minimog

Using the Minimog and C4 (defined below), I want to construct the Golay code GC12.

The tetracode C4 consists of 9 words over F3:

  0 000,     0 +++,    0 ---,         where 0=0, +=1, and -=2 all mod 3.
+ 0+-,     + +-0,    + -0+,
- 0-+,     - +0-,    - -+0.


Each (a,b,c,d) in C4 defines a linear function f : F3 –> F3, where f(x) = ax+b, f(0) = b, f(1) = f(+) = c, f(2) = f(-) = d, and a is the “slope” of f. This implies a + b = c (mod 3), b – a = d (mod 3).

Minimog: A 4×3 array whose rows are labeled 0,+,-, that construct the Golay code in such a way that both signed and unsigned hexads are easily recognized.

A col is a word of length 12, weight 3 with a “+” in all entries of any one column and a “0” everywhere else. A tet is a word of length 12, weight 4 who has “+” entries in a pattern such that the row names form a tetracode word, and “0” entires elsewhere. For example,


_________          _________
| |+    |          | |+    |
| |+    |          |+|  +  |
| |+    |          | |    +|
---------          ---------
"col"              "tet"


The above “col” has “+” entries in all entries of column 2, and “0” entries elsewhere.
The above “tet” has a “+” entry in each column. The row names of each “+” entry are +, 1, +, – respectively. When put together, + 0+- is one of the nine tetracode words.

Lemma: Each word belongs to the ternary Golay Code GC12 if and only if

• sum of each column = -(sum row0)
• row+ – row is one of the tetracode words.

This may be found in [4].

Example:

|+|+ + +|      col sums: ----      row+ - row-: --+0
|0|0 + -|      row0 sum: + = -(sum of each col)
|+|+ 0 -|



How do I construct a Golay code word using cols and tets? By the Lemma above, there are four such combinations of cols and tets that are Golay code words. These are: col – col, col + tet, tet – tet, col + col – tet.

Example:

  col-col         col+tet      tet-tet       col+col-tet

| |+   -|       | |+ +  |    |+|0 + +|      | |- + +|
| |+   -|       |+|  -  |    |-|  -  |      |-|  0 +|
| |+   -|       | |  + +|    | |    -|      | |  + 0|
? ? ? ?         + 0 ? -      - ? - +        + 0 + -


“Odd-Man-Out”: The rows are labeled 0,+,-, resp.. If there is only one entry in a column then the label of the corresponding row is the Odd Man Out. (The name of the odd man out is that of the corresponding row.) If there is no entry or more than one entry in the column then the odd man out is denoted by “?”.

For example, in the arrays above, the Odd-Men-Out are written below the individual arrays.

For the Steiner system S(5,6,12), the minimog is labeled as such:

                              ______________
|0  3 inf  2 |
|5  9  8  10 |
|4  1  6  7  |
--------------


The four combinations of cols and tets above that construct a Golay code word yield all signed hexads. From these signed hexads, if you ignore the sign, there are 132 hexads of the Steiner system S(5,6,12) using the (o, inf, 1) labeling discussed in Section 9 below. There are a total of 265 words of this form, but there are 729 Golay code words total. So, although the above combinations yield all signed hexads, they do not yield all hexads of the Golay code ([12] pg. 321).

The hexad for the tet-tet according to the S(5,6,12) Minimog above would be (0, inf, 2, 5, 8, 7).

The rules to obtain each hexad in this Steiner system is discussed in Section 9 below.

A Steiner system of type (5,6,12) and the Conway-Curtis notation can be obtained from the Minimog. S12 sends the 3×4 minimog array to another 3×4 array. The group M12 is a subgroup of S12 which sends the Minimog array to another array also yielding S(5,6,12) in Conway-Curtis notation.

## 9. Kitten

The kitten is also an interesting facet of the Minimog. Created by R.T. Curtis,
kittens come from the construction of the Miracle Octal Generator, or MOG, also created by R.T. Curtis. (A description of the MOG would be too far afield for this post, but further information on the MOG can be gotten from [3] or [6].)

Suppose we want to construct a Steiner system from the set T = {0, 1, …, 10, inf}.
The kitten places 0, 1, and inf at the corners of a triangle, and then creates a rotational symmetry of triples inside the triangle according to R(y) = 1/(1-y) (as in [2], section 3.1). A kitten looks like:

                                infinity

6

2     10

5     7      3

6     9      4     6

2    10     8      2     10

0                                    1

Curtis' kitten


where 0, 1, inf are the points at infinity.

Another kitten, used to construct a Steiner system from the set T = {0, 1, …, 10, 11} is

                                   6

9

10     8

7     2      5

9     4     11     9

10     8     3      10     8

1                                    0

Conway-Curtis' kitten


The corresponding minimog is

                  _________________________
|  6  |  3  |  0  |  9  |
|-----|-----|-----|-----|
|  5  |  2  |  7  | 10  |
|-----|-----|-----|-----|
|  4  |  1  |  8  | 11  |
|_____|_____|_____|_____|


(see Conway [3]).

The first kitten shown consists of the three points at 0, inf, 1 with an arrangement of points of the plane corresponding to each of them. This correspondence is:

         6 |10 | 3              5 | 7 |3               5 | 7 | 3
2 | 7 | 4              6 | 9 |4               9 | 4 | 6
5 | 9 | 8              2 |10 |8               8 | 2 |10

inf-picture             0-picture              1-picture


A union of two perpendicular lines is called a cross. There are 18 crosses of the kitten:

                ___________________________________________
|* * * |* * * |* * * |*     |  *   |    * |
|*     |  *   |    * |* * * |* * * |* * * |
|*     |  *   |    * |*     |  *   |    * |
-----------------------------------------
_________________________________________
|*     |  *   |* *   |*     |*   * |    * |
|*     |  *   |* *   |  * * |  *   |    * |
|* * * |* * * |    * |  * * |*   * |* * * |
-----------------------------------------
_________________________________________
|*   * |    * |  * * |  *   |  * * |* *   |
|*   * |* *   |*     |*   * |  * * |    * |
|  *   |* *   |  * * |*   * |*     |* *   |
------------------------------------------



A square is a complement of a cross. The 18 squares of a kitten are:

                ___________________________________________
|      |      |      |  * * |*   * |* *  |
|  * * |*   * |* *   |      |      |     |
|  * * |*   * |* *   |* *   |*   * |* *  |
-----------------------------------------
_________________________________________
|  * * |*   * |    * |  * * |  *   |* *   |
|  * * |*   * |    * |*     | *  * |* *   |
|      |      |* *   |*     |  *   |      |
-----------------------------------------
_________________________________________
|  *   |* *   |*     |*   * |*     |    * |
|  *   |    * |  * * |  *   |*     |* *   |
|*   * |    * |*     |  *   |  * * |    * |
-----------------------------------------


The rules to obtain a hexad in the {0,1,inf} notation are the following:

• A union of parallel lines in any picture,
• {0, 1, inf} union any line,
• {Two points at infinity} union {square in a picture corresponding to omitted point at infinity},
• {One point at infinity} union {cross in the corresponding picture at infinity}.

(See [2])

M12 is isomorphic to the group of automorphisms of the Steiner system S(5,6,12) in the Conway-Curtis notation.

## 10. Mathematical Blackjack or Mathieu’s 21

Mathematical Blackjack is a card game where six cards from the group {0,1,…,11} are laid out face up on a table. The rules are:

• each player must swap a card with a card from the remaining six, that is lower than the card on the table;
• the first player to make the sum of all six cards less than 21 loses.

According to Conway and Ryba [8, section V, part (d)], the winning strategy of this game is to choose a move which leaves a Steiner hexad from S(5,6,12) in the shuffle
notation, whose sum is greater than or equal to 21, on the table.

The shuffle notation for the hexad, used in the Mathematical Blackjack game, is shown below (see also the description in the hexad/blackjack page):

              8 |10 |3            5 |11 |3            5 |11 |3
9 |11 |4            2 | 4 |8            8 | 2 |4
5 | 2 |7            7 | 9 |10           9 |10 |7

0-picture          1-picture          6-picture


Riddle: Assuming the strategy, player A just made a winning hexad move that will force player B to make the sum under 21 on his next turn. Joe Smith walks up to player B and offers to shuffle all 12 cards while player A isn’t looking, for a fee. Player B grabs at his chance thinking that a random shuffle will let him back in the game. How is it that player B still loses?

Joe is actually working for Player A. Joe does not shuffle the cards randomly, but instead uses the M12 group generated by r, s (see section 1) to shuffle the cards. Since the M12 group preserves hexads, player A still has a winning game. (He and Joe split the money.)

A simple group is a group with no normal subgroups except itself and {1}. Most simple groups are from a family such as PSL2(Fp), p>3 or An, n >= 5. However there exist some simple groups outside of such well known families. These are called sporadic simple groups. M12 is a sporadic simple group of order 95,040. The only smaller sporadic group is M11 of order 7,920. (See [10] pg. 211)

<a name="Stabilizer in M24 of a dodecad”>

## 12. Stabilizer in M24 of a dodecad.

M24 is a sporadic simple group of order 244,823,040 containing M12 as a subgroup. The Steiner system S(5,8,24) is a collection of 8 element subsets, called octads, from a 24 element set X, with the property that any five elements in X determine a unique octad in the system. There are (24 choose 5)/(8 choose 5) = 759 of these octads. M24 is the subgroup of SX which sends the set of octads to itself. Two octads, O1, O2, intersect in a subset of X of order 0,2,4,6 or 8 [14]. If |O1 intersect O2| = 2 then O1 + O2 is order 12. Such a subset of X is called a dodecad. M12 is isomorphic to

{g in M24 | g(O1 + O2) = (O1 + O2)} = the stablizer of the dodecad O1 + O2.
(See [6] for details)

## References

1. W. D. Joyner, Mathematics of the Rubik’s Cube (USNA Course notes), 1997.
2. R. T. Curtis, “The Steiner System S(5,6,12), the Mathieu Group M12 and the ‘Kitten’ ,” Computational Group Theory, Academic Press, London, 1984.
3. J. H. Conway, “hexacode and Tetracode- MOG and MINIMOG,” Computational Group Theory (ed. Atkinson), Academic Press, London, 1984.
4. Vera Pless, “Decoding the Golay Code,” Transactions of Information Theory, IEEE, 1986, (pgs 561-567).
5. R. T. Curtis, “A new Combinatorial approach to M24“, Mathematical Proceeding of the Cambridge Philosophical Society, Vol. 79, 1974.
6. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, “M12,”,
Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
7. Robinson, A Course in the Theory of Groups, Springer, 1996.
8. J. H. Conway, N. Sloane, “Lexicographic Codes: Error-Correcting Codes
from Game Theory,” Transactions on Information Theory, IEEE, 1986.
9. A .Adler, “The modular Curve X(11) and the Mathieu group M11“,
Proc. London Math Society 74(1997)1-28.
10. T. Thompson, From Error-Correcting Codes Through Sphere
Packings to Simple Groups
, The Mathematical Association of
America, 1983.
11. Rotman, J, Introduction to the Theory of Groups, 4th ed.
Springer-Verlag, 1995.
12. J. Conway, N. Sloane, Sphere Packings, Lattices, and Groups,
Springer-Verlag, 3rd ed., 1999.
13. B. Kostant, “The Graph of the truncated icosahedron and the
last letter of Galois.” Notices of the A.M.S. 42(1995)959-
968.
14. E. Assmus, “On the Automorphism Groups of Paley-Hadamard
Matrices.” Combinatorial Mathematics and its Applications.
University of North Carolina Press, 1969, (pgs 98-103).
15. P. Greenberg, Mathieu Groups, Courant Institute of Math and
Science, New York University, 1973.
16. P. Cameron, J. Van Lint, Designs, Graphs, Codes, and Their