# Semidefinite programming for permutation codes

###### Abstract.

We initiate study of the Terwilliger algebra and related semidefinite programming techniques for the conjugacy scheme of the symmetric group . In particular, we compute orbits of ordered pairs on acted upon by conjugation and inversion, explore a block diagonalization of the associated algebra, and obtain improved upper bounds on the size of permutation codes of lengths up to . For instance, these techniques detect the nonexistence of the projective plane of order six via and yield a new best bound for a challenging open case. Each of these represents an improvement on earlier Delsarte linear programming results.

## 1. Introduction and notation

Consider the symmetric group on . Let be
a collection of conjugacy classes of . A subset is an -*permutation code* if, for any two distinct
elements , their quotient belongs to
a class in . Note that the order of the terms is not important, since
permutations are conjugate with their inverses.

The *Hamming distance* between and in , denoted
, is the number of non-fixed points of .
Equivalently, if and are written in single-line notation,
counts the number of disagreements between corresponding
positions. It follows that is a metric for . To the best of
our knowledge, this metric was first considered by Farahat in [9].

In most investigations of permutation codes, the set of admissible
conjugacy classes reflects this metric and is instead taken as some subset . This interpretation for is that all conjugacy classes
with one-cycles, , are allowed for quotients . In other words, in this context, an -permutation code
is a subset of with all nonzero Hamming distances
belonging to . Based on the coding applications discussed below, the
standard choice for is the interval for some
*minimum distance* .

Let denote the maximum size of an -permutation code. When , this is simply written . Early work determining some bounds and values for began in the late 1970s in [7, 11], where the term ‘permutation array’ was used. Around 2000, permutation codes enjoyed a revival of interest with the discovery in [4, 10] of their applications to trellis codes. Then, the survey article [3] observed connections with permutation polynomials and also initiated the first serious computational attack on lower bounds on . A probabilistic lower bound appears in [15]. Meanwhile, linear programming (LP) upper bounds were investigated, first in [17], and then subsequently in [2, 8]. A growing body of related work has emerged, including constant composition codes, injection codes, and study of the packing and covering radii.

The LP bound rests on Delsarte’s theory of association schemes [6] applied to the conjugacy scheme of . More details follow in Section 2. In various combinatorial settings (block designs and binary codes, for instance) Delsarte LP bounds have been successfully improved using semidefinite programming (SDP). In general, the sizes of matrices required for this SDP tend to grow impractically large. So the most successful applications of SDP to designs and codes usually begin with an attack on the algebraic structure. In a little more detail, block diagonalizations of certain matrix algebras are desired in order to scale the computations. This is the content of Dion Gijswijt’s dissertation [12].

For a finite set , we use or to denote the
algebra of matrices with rows and columns indexed by .
(Some canonical ordering of is usually assumed, leading to the more
standard notation or , which we also use.) The
-entry of matrix is here denoted . All
linear spans and generated sub-algebras are
over , unless otherwise noted. The conjugate transpose
of is denoted and as usual is
*Hermitian* if . A sub--algebra of closed under
this Hermitian conjugate is called a - (matrix) algebra.

## 2. Bose-Mesner and Terwilliger algebras

An -*class association scheme* on a finite set is a list of
binary relations on such that is the identity
relation, the relations partition , and the following regularity
condition holds: given and with , the number of for which both and is a constant
depending only on . These values are
called the *structure constants* or *intersection numbers* of the
scheme. Here, we are mainly interested in *symmetric* schemes in which
each relation is symmetric and thus . Chris
Godsil’s notes [13] offer a readily available and comprehensive
reference on association schemes.

###### Example 2.1.

Consider , a finite group. The relations are indexed by the
conjugacy classes of , where corresponds with the trivial conjugacy
class. Declare in if and only if belongs to the
th conjugacy class. The fact that these partition is clear.
A character sum can compute the structure constants (see below). This is
the *conjugacy scheme* on .

Let . Define the zero-one matrices by

We have and , the all ones matrix. And from the definition of the structure constants,

(2.1) |

From this it follows that is a commutative
-matrix sub-algebra of of dimension .
This is the *Bose-Mesner algebra* of the scheme.

From spectral theory, also has a basis of orthogonal idempotents with . It is convenient to index , which is evidently one of the idempotents.

The *Hadamard* or *entrywise* product of matrices is with . For the
Bose-Mesner algebra, we have the dual relations

In particular, since is closed under , a parallel version of (2.1) exists; namely,

for some positive reals . These are called the *Krein
parameters* for the scheme.

The basis change matrices between and are denoted by and ; explicitly, the entries are given by

The being projections imply that the entries are the
eigenvalues of the matrices , with the column space of as the
associated eigenspace. For this reason, the matrices are called,
respectively, the *first* and *second eigenmatrix*. They are of
course related by .

For a subset , its *indicator vector* is , with if and only if . The *inner
distribution* of a nonempty is , where

The following observation leads to the famous Delsarte LP bound for cliques in association schemes.

###### thm 2.2 ((Delsarte, [6])).

The inner distribution of a nonempty subset of points in an association scheme with second eigenmatrix satisfies .

Now consider a pointed set with . Define diagonal zero-one matrices in by

Observe that, by analogy with the Bose-Mesner idempotents, these are projections with of rank one and .

The algebra obtained by extending
by these is
called the *subconstituent* or *Terwilliger algebra* of the scheme
with respect to . Unlike , the algebra is
not in general commutative.

Completing the duality, we have with

After this introduction, we are now interested exclusively in the conjugacy scheme on . Recall that the conjugacy classes (and irreducible characters) of are in correspondence with the integer partitions of . For a fixed , we will use to denote the number of partitions, and index the conjugacy classes as , with consisting only of the identity permutation. For any fixed , we have

Alternatively, as stated in [14],

where the sum is over all irreducible characters, and where is a representative in class , etc.

The Bose-Mesner algebra is a commutative sub-algebra of . Its eigenvalues are obtained nearly directly from the character table of .

###### Lemma 2.3 ((Tarnanen, [17])).

The second eigenvalue matrix for the conjugacy scheme on is given by

where index both the conjugacy classes and irreducible characters.

The idempotents and Krein parameters of this conjugacy scheme are also expressible via the irreducible characters, but we omit the details.

The linear programming bound of Theorem 2.2 now becomes

Given and , let denote this maximum value. Then

This was used with some success in [2, 17] to obtain some upper bounds on permutation codes with . Additionally, the LP led to a general upper bound on in [8].

When extending to the Terwilliger algebra in this setting, it is natural to take as the distinguished element. We have

The products are supported on the block. This offers a natural decomposition of the Terwilliger algebra in general. Note that is the zero matrix if and only if .

## 3. Isometries, orbits, and the centralizer algebra

Consider the automorphism group of the Bose-Mesner algebra of . On one hand, this is the intersection of the automorphism groups of the graphs whose adjacency matrices are the zero-one generators , . Alternatively, this is also the group of isometries of , when endowed with the Hamming distance metric . Let us also denote this group by .

Let and denote the group actions of left and right multiplication by , acting on itself. Let be the (involutoric) group action generated by inversion on . That these actions induce all isometries appears in [9], and is discussed further in [2].

###### Lemma 3.1.

Let be the subgroup of isometries which fixes the identity element. Its action on is generated by conjugations and inversion . These actions commute, and therefore .

For the extension to the Terwilliger algebra , we are concerned with how acts on pairs of permutations. In what follows, sums indexed by ‘’ are to be taken over all irreducible characters of .

###### Proposition 3.2.

The number of orbits of , acted on by conjugation and inversion is given by

###### Proof.

We use Burnside’s orbit counting lemma to obtain

Both terms in the sum are connected with interesting class functions on . The first term is just the sum of squared centralizers. Recall that orthonormality of the irreducible characters implies that for an element ,

the (size of the) centralizer of . So the first of our two summation terms can be rewritten

(3.1) |

For the second sum, observe that the condition is equivalent to . So for a fixed , the number of such is simply the number of square roots of in . As an aside, this counting problem for general groups was connected long ago (see [19]) with the ‘Frobenius-Schur indicator’

For our purposes, all irreducible representations of are real and therefore is always . The number of square roots of an element reduces in this case to the basic character sum For , the term we seek becomes

(3.2) | |||||

Now consider , acting on by conjugation, inversion, and the ‘coordinate swaps’ . A small extension of Proposition 3.2 also yields a character sum formula to count orbits for this action.

###### Proposition 3.3.

The number of orbits of , acted on by conjugation, inversion, and coordinate swaps, is given by

where is as in Proposition 3.2.

###### Proof.

New sums not already present in the proof of Proposition 3.2 count, for a fixed ,

Since conjugacy by is a bijection, it follows that each of these is counted simply by the centralizer of . Alternatively, we may count, over all , the number of square roots of times the size of the centralizer of . Using the expressions in the previous proof,

This, along with earlier terms, finishes the orbit count. ∎

Table 1 contains the first few interesting values of and .

Let denote the orbits of under . Define the corresponding zero-one matrices by

Put and observe that by Proposition 3.2 we have .

Alternatively, , where and . For , consider the zero-one matrix with

Then is the conjugacy representation. Define the action of inversion similarly

Let . The following rephrases the earlier definition of orbits under in terms of matrix algebras.

###### Proposition 3.4.

is the centralizer algebra of .

Since conjugacy classes of are left invariant by both conjugation and inversion, the orbits under admit a classification according to conjugacy classes. In other words, each matrix is supported on some block . The sum of all such supported on a given block is, when restricted to that block, the all-ones matrix .

With notation from Section 2, we recall that the ‘restricted’ Bose-Mesner algebra has conjugation-invariant generators supported on the block. So it follows that is a sub--algebra of . The reverse inclusion seems reasonable, but much more difficult to prove. Indeed, the dimensions in Table 1 grow much faster than the number of generators for .

###### Conjecture 3.5.

The Terwilliger algebra of the conjugacy scheme of has as a basis the zero-one matrices , , defined via orbits of under . That is, as -algebras, we have .

Conjecture 3.5 was verified up to in [1]. We have extended this verification to but leave the general problem to future work.

Fortunately, the center of is reasonably easy to describe. The following appears in [16] for general groups; here (and from now on) we are interested in .

###### Proposition 3.6.

The primitive central idempotents of are given by

(3.3) |

where indexes the irreducible characters of .

Since the commute, they have a common basis of eigenvectors. Consider a unitary matrix which has orthonormal eigenvectors of the , organized by columns. Then is block-diagonalized by simple blocks. Our goal later, in Section 5, is to decompose these simple blocks into basic blocks. For now, though, we have sufficient background to introduce semidefinite programming for permutation codes.

## 4. Semidefinite programming

Let be a subset of , and recall that is the zero-one indicator vector of .

Define now the subgroup of consisting of all isometries such that . Let denote the complement of this subgroup, consisting of all with . It is clear that , while . This defines two matrices

which are roughly analogous to the inner distribution vector of Section 2.

By construction, both and are positive semidefinite (symmetric) matrices. And the trace of gives the cardinality of . The -entry of is always , since appears in all copies under . The the -entry equals the -entry in , and, being symmetric, it is also equal to the -entry. Since is transitive on , one has the following formula and lemma.

(4.1) |

(In particular, the matrix has a diagonal of .)

###### Lemma 4.1.

Consider an entry . If no quotient in belongs to the same conjugacy class as , then .

Next, we have the key connection with the centralizer algebra discussed in Section 3.

###### Proposition 4.2.

The two matrices and belong to the algebra .

###### Proof.

Recall that are the zero-one matrices representing the orbits of under the action of . With the convention that , one has , where the coefficients are explicitly determined by the formula

Observe that .

Let denote the subset of indices such that contains elements of type , or , and let be the complementary set .

With some rearranging and (4.1), it follows easily that

where the matrices are defined by if and by the following expression if :

In both cases, we have the required matrices expressed as linear combinations of the . ∎

We now summarize the above facts concerning and .

###### Proposition 4.3.

Let . For any -permutation code , we have

From this, the following SDP system is obtained, considering the coefficients as variables in :

(4.2) |

Given and , let denote the maximum value of this program. Then

Recall that the matrices have size . So presently, we can only directly consider (4.2) for . Since permutation codes are well understood for , our main contribution at this stage is a table of SDP bounds for . For , it is necessary to consider an equivalent SDP obtained via block-diagonalization of the algebra . That is the topic of the next section.

We conclude with Table 2, indicating the SDP bounds from (4.2) for and various distance sets . For comparison, the Delsarte LP bounds are given, along with the quantity , which is an upper bound either for or (see [17]).

## 5. Block diagonalization and bounds for

First, let us recall the notion of -isomorphism. A bounded linear map
such that for all , and
is called a -*homomorphism* of
algebras. Such a map which is also bijective is naturally called a -*isomorphism* and the two
algebras are -*isomorphic*. A -isomorphism
of matrix algebras preserves positive semidefiniteness.

Next, we follow Gijswijt in [12] and Vallentin in [18] by considering the -isomorphism which ‘block-diagonalizes’ our algebra .

In more detail, we can regard as a semisimple -module, , acted upon by . Therefore, decomposes as a sum of orthogonal submodules , where has columns forming a basis of the range of the matrix defined in (3.3), for . Each is a direct sum of, say, submodules: , with for all . Let denote the common dimension of . Incidentally, this parameter is also given by where is the -th irreducible character of .

For the centralizer algebra of , we correspondingly have a decomposition of as an orthogonal direct sum of, say, , each of which is in turn decomposes into submodules of dimension .

The action of elements in can be naturally decomposed along each orbit (i.e. conjugacy class). Let denote the restriction of the character on the conjugacy class . So is the number of permutations fixed by . Depending on the conjugacy class of , it can be written or for all . In the first case, is the number of permutations in commuting with a fixed permutation . In the latter case, is the number of permutations in such that, for a fixed , . So far, to the best of our knowledge, no closed form formula is known for these two expressions.

For each , splits as a sum of irreducible characters , , of as

The coefficients are given by

So each irreducible representation of appears times in the action of and the dimension of is equal to . The non-zero values of for are given in Table 3.

It remains to explicitly decompose each submodule of into smaller ‘basic blocks’.

The method begins as follows. For each , compute an element . Then define , a subspace of . Note that the vector can be obtained as for any vector since is idempotent. So, for instance, any non-zero row of affords a choice of .

By Schur’s lemma, the subspace meets each in a -dimensional subspace. Then any acts on by multiplication. If is an orthogonal basis of , then the basic block of can be computed as

(5.1) |

Each basic block is an matrix. Recall the matrix has nonzero entry in the position iff . So the inner product in the numerator of (5.1) is