Self-dual binary codes from small covers and simple polytopes
We explore the connection between simple polytopes and self-dual binary codes via the theory of small covers. We first show that a small cover over a simple -polytope produces a self-dual code in the sense of Kreck–Puppe if and only if is -colorable and is odd. Then we show how to describe such a self-dual binary code in terms of the combinatorial information of . Moreover, we can define a family of binary codes , , from an arbitrary simple -polytope . We will give some necessary and sufficient conditions for to be a self-dual code. A spinoff of our study of such binary codes gives some new ways to judge whether a simple -polytope is -colorable in terms of the associated binary codes . In addition, we prove that the minimum distance of the self-dual binary code obtained from a -colorable simple -polytope is always .
Key words and phrases:Self-dual code, polytope, small cover.
2010 Mathematics Subject Classification:57S25, 94B05, 57M60, 57R91
A (linear) binary code of length is a linear subspace of the -dimensional linear space over (the binary field). The Hamming weight of an element , denoted by , is the number of nonzero coordinates in . Any element of is called a codeword. The Hamming distance of any two codewords is defined by:
The minimum of the Hamming distances for all , , is called the minimum distance of (which also equals the minimum Hamming weight of nonzero elements in ). A binary code is called type if and the minimum distance of is . We call two binary codes in equivalent if they differ only by a permutation of coordinates.
The standard bilinear form on is defined by
Note that for any , and
Then any linear binary code in has a dual code defined by
It is clear that . We call self-dual if . For a self-dual binary code , we can easily show the following
The length must be even;
For any , the Hamming weight is an even integer since ;
The minimum distance of is an even integer.
Self-dual binary codes play an important role in coding theory and have been studied extensively (see  for a detailed survey).
Puppe in  found an interesting connection between closed manifolds and self-dual binary codes. It was shown in  that an involution on an odd dimensional closed manifold with “maximal number of isolated fixed points” (i.e., with only isolated fixed points and the number of fixed points ) determines a self-dual binary code of length . Such an involution is called an -involution. Conversely, Kreck–Puppe  proved a somewhat surprising theorem that any self-dual binary code can be obtained from an -involution on some closed -manifold. Hence it is an interesting problem for us to search -involutions on closed manifolds. But in practice it is very difficult to construct all possible -involutions on a given manifold.
On the other hand, Davis and Januszkiewicz in  introduced a class of closed smooth manifolds with locally standard actions of elementary 2-group , called small covers, whose orbit space is an -dimensional simple convex polytope in . It was shown in  that many geometric and topological properties of can be explicitly described in terms of the combinatorics of and some characteristic function on determined by the -action. For example, the mod 2 Betti numbers of correspond to the -vector of . Any nonzero element determines a nontrivial involution on , denoted by . We call a regular involution on the small cover. So whenever is an -involution on where is odd, we obtain a self-dual binary code from .
Motivated by Kreck–Puppe and Davis–Januszkiewicz’s work, our purpose in this paper is to explore the connection between the theory of binary codes and the combinatorics of simple polytopes via the topology of small covers. We will show that a small cover over an -dimensional simple polytope admits a regular -involution only when is -colorable. A polytope is called -colorable if we can color all the facets (codimension-one faces) of the polytope by different colors so that any neighboring facets are assigned different colors. Moreover, we find that the self-dual binary code obtained from a regular -involution on depends only on the combinatorial structure of and the parity of . This motivates us to define a family of binary codes , , for any simple polytope (not necessarily -colorable).
The paper is organized as follows. In section 2, we explain the procedure of obtaining self-dual binary codes as described in  from -involutions on closed manifolds. In section 3, we first recall some basic facts of small covers and then investigate what kind of small covers can admit regular -involutions (see Theorem 3.2). In section 4, we spell out the self-dual binary code from a small cover with a regular -involution (see Corollary 4.5). It turns out that the self-dual binary code depends only on the combinatorial structure of the underlying simple polytope. In section 5, we study the properties of a family of binary codes , , associated to any simple -polytope . A spinoff of our study produces some new criteria to judge whether is -colorable in terms of the associated binary codes (see Proposition 5.6). In section 6, we will give some necessary and sufficient conditions for to be self-dual codes for general simple polytops (see Theorem 6.2). In section 7, we prove that the minimum distance of the self-dual binary code obtained from any -colorable simple -polytope is always (see Proposition 7.1). In section 8, we investigate some special properties of -colorable simple -polytopes. In section 9, we study what kind of doubly-even binary codes can be obtained from -colorable simple -polytopes. In particular, we show that the extended Golay code and the extended quadratic residue code cannot be obtained from any -colorable simple -poltyopes.
2. Binary codes from -involutions on manifolds
Let be an involution on a closed connected -dimensional manifold , which has only isolated fixed points. Let denote the binary group generated by . By Conner [8, p.82], the number of the fixed points of must be even. So we assume that , , in the following discussions. By [1, Proposition(1.3.14)], the following statements are equivalent.
(i.e. is an -involution);
is a free -module, so
The inclusion of the fixed point set, , induces a monomorphism
Next we assume that is an -involution on . So the image of in under the localization map is isomorphic to as graded algebras. It is shown in [7, 23] that the image can be described in the following way. For any vectors and in , define
It is clear that forms a commutative ring with respect to two operations and . Actually, is a boolean ring. Notice that for any . Let
Then is a -dimensional linear subspace of . Note that for any , the Hamming weight of is an even integer. The following lemma is immediate from our definitions.
Let be a binary code in with . Then the following statements are equivalent.
for any ;
for any .
By the localization theorem for equivariant cohomology (see ), we have isomorphisms
For any , we have
In addition, is isomorphic to the graded ring
where the ring structure of is given by
, where is generated by ;
For with each , , where
The operation on corresponds to the cup product in .
Each above can be thought of as a binary code in . Theorem 2.2 and the Poincaré duality of implies that
In addition, is perpendicular to with respect to . This is because
is self-dual if and only if .
3. Small covers with -involutions
3.1. Small covers
An -dimensional simple (convex) polytope is a polytope such that each vertex of the polytope is exactly the intersection of facets (-dimensional faces) of the polytope. Following , an -dimensional small cover is a closed smooth -manifold with a locally standard -action whose orbit space is homeomorphic to an -dimensional simple convex polytope , where a locally standard -action on means that this -action on is locally isomorphic to a faithful representation of on . Let denote the set of all vertices of and denote the set of all facets of . For any facet of , the isotropy subgroup of in with respect to the -action is a rank one subgroup of generated by an element of , denoted by . Then we obtain a map called the characteristic function associated to , which maps the facets meeting at each vertex of to linearly independent elements in . It is shown in  that up to equivariant homeomorphisms, can be recovered from in a canonical way (see (3.3)). Moreover, many algebraic topological invariants of a small cover can be easily computed from . Here is a list of facts on the cohomology rings of small covers proved in .
Let be the -th mod 2 Betti number of . Then
where is the -vector of .
Let denote the fixed point set of the -action on . Then
The equivariant cohomology is isomorphic as graded rings to the Stanley–Reisner ring of
where are all the facets of and are of degree , and is the ideal generated by all square free monomials of with in .
The mod- cohomology ring , where is an ideal determined by . In particular, is generated by degree elements.
3.2. Spaces constructed from simple polytopes with -colorings
Let be an -dimensional simple polytope in . For any , a -coloring on is a map . For any facet of , is called the color of . Let be a codimension- face of where . Define
Besides, let be the subgroup of generated by . The rank of is called the rank of , denoted by . It is clear that .
For any point , let denote the unique face of that contains in its relative interior. Then we define a space associated to by:
where if and only if and .
is a closed manifold if is non-degenerate (i.e. are linearly independent whenever ).
has connected components. So is connected if and only if .
There is a canonical -action on defined by:
let be the map sending any to .
For any face of with , let and
be the quotient homomorphism. Then induces a -coloring on by:
It is easy to see that is homeomorphic to .
Suppose is a small cover with characteristic function . Then is homeomorphic to . For any face of , is a closed connected submanifold of (called a facial submanifold of ), which is a small cover over .
3.3. Small covers with regular -involutions
Let be a small cover over an -dimensional simple polytope and be its characteristic function. Let us discuss under what condition there exists a regular -involution on .
The following statements are equivalent.
There exists a regular -involution on .
There exists a regular involution on with only isolated fixed points;
The image of consists of exactly elemnets (which implies that is -colorable) and so they form a basis of .
(a) implies (b) since by definition an -involution only has isolated fixed point.
(b)(c) Suppose there exists so that the fixed points of on are all isolated. Let be an arbitrary vertex on and be the facets meeting at . By the construction of small covers, is a fixed point of the whole group . Let be a small neighborhood of . Since the action of on is locally standard, we observe that for , , the dimension of the fixed point set of in is equal to . Then since the fixed points of are all isolated, we must have .Next, take an edge of with two endpoints . Since is simple, there are facets such that and . Then , which implies . Since the 1-skeleton of is connected, we can deduce the image of consists of elements of which form a basis of .
(c)(a) Suppose is a basis of . Then by the construction of small covers, the fixed point set of the regular involution on is
So the number of fixed points of is equal to the number of vertices of , which is known to be . Then by the result (R1) in section 3.1, is an -involution on . ∎
It should be pointed out that for an -colorable simple -polytope , the image of a characteristic function might consist of more than elements of . In that case, the small cover defined by and admits no regular -involutions. So Theorem 3.2 only tells us that if an -dimensional small cover over admits a regular -involution, then is -colorable. But conversely, this is not true.
3.4. Descriptions of -colorable -dimensional simple polytopes
The following descriptions of -colorable simple -polytopes are due to Joswig .
Theorem 3.3 ([16, Theorem 16 and Corollary 21]).
Let be an -dimensional simple polytope, . The following statements are equivalent.
Each -face of has an even number of vertices.
Each face of with dimension greater than (including itself) has an even number of vertices.
Any proper -face of is -colorable.
Later we will give some new descriptions of -colorable simple -polytopes from our study of binary codes associated to general simple polytopes in section 5.
4. Self-dual binary codes from small covers
Let be an -dimensional small cover which admits a regular -involution. By Theorem 3.2, is an -dimensional -colorable simple polytope with an even number of vertices. Let be all the vertices of . The characteristic function of satisfies: is a basis of . By Theorem 3.2, is an -involution on . So by the discussion in section 2, we obtain a filtration
According to Theorem 2.2 and the property (R1) of small covers,
Then since for all , we have . Note that is self-dual in if and only if by (2.5). Then is self-dual if and only if (i.e. is odd and ). So we prove the following proposition.
Let be an -dimensional small cover which admits a regular -involution. Then is a self-dual code if and only if is odd and .
In the remaining part of this section, we will describe each , , explicitly in terms of the combinatorics of . First, any face of determines an element where the -th entry of is if and only if is a vertex of . In particular, and is a linear basis of . Note that for any faces of , we have
We define a sequence of binary codes as follows.
Changing the ordering of the vertices of only causes the coordinate changes in . So up to equivalences of binary codes, each is uniquely determined by .
For any -colorable simple -polytope with vertices, we have
By definition, can be colored by colors . Now choose an arbitrary color say , we observe that each vertex of is contained in exactly one facet of colored by . This implies that
where are all the facets of colored by . So . Moreover, by Theorem 3.3(d), the facets are -dimensional simple polytopes which are -colorable. So by repeating the above argument, we can show that and so on. Now it remains to show .
By definition, is spanned by . So it is obvious that . Let be all the vertices of . It is easy to see that is spanned by . Then since there exists an edge path on between any two vertices and of , belongs to . So . This finishes the proof. ∎
Let be an -dimensional small cover which admits a regular -involution. For any , the space coincides with .
Let be an -colorable simple -polytope with vertices. Then
If is odd, then is a self-dual code in if and only if . If is even, cannot be a self-dual code in for any .
Let be an -dimensional small cover which admits a regular -involution where is odd. Then the self-dual binary code is spanned by . So the minimum distance of is less or equal to .
Problem 1: For any -dimensional small cover which admits a regular -involution where is odd, determine the minimum distance of the self-dual binary code .
We will see in Proposition 7.1 that when , the minimum distance of is always equal to . For higher dimensions, it seems to us that the minimum distance of should be equal to . But the proof is not clear to us.
In the following, we are going to prove Theorem 4.3. For brevity, let
By the construction of , all the fixed points of on are where
4.1. Proof of Theorem 4.3
According to the result in (R4) of section 3.1, the cohomology ring of is generated as an algebra by . So as an algebra over , the equivariant cohomology ring is generated by elements of degree . In addition, the operation on corresponds to the cup product in . So we obtain from Theorem 2.2 that for any , . On the other hand, there is a similar structure on as well.
Claim-1: , .
Indeed, for any different facets