Dihedral Permutation Channels

• Dihedral permutation channels are discrete memoryless channels whose transition matrices are convex combinations of permutation matrices from the dihedral group Dₙ.
• The channel model leverages polyhedral geometry, providing explicit facet descriptions, Ehrhart h*-vector insights, and Gorenstein structure for both odd and even n.
• The explicit geometric formulation enables uniform-input capacity optimization and efficient linear programming methods for channel design.

Searching arXiv for the specified paper and closely related context. Dihedral permutation channels are discrete memoryless channels whose transition matrices lie in the convex hull of the permutation matrices arising from the symmetries of a regular $n$-gon. In the formulation of the dihedral permutation polytope $P_n$, one begins with the dihedral group $D_n$ of order $2n$, maps each $\sigma \in D_n$ to its permutation matrix $P(\sigma)$, and sets

$$P_n := \operatorname{Conv}\{\,P(\sigma):\sigma\in D_n\} \subset \mathbb{R}^{n^2}.$$

This identifies $P_n$ simultaneously as a $0$–$1$-polytope inside the Birkhoff polytope $P_n$0, as the exact parameter space of all dihedral permutation channels, and as an object with an explicit facet description, dimension formula, Gorenstein structure, and Ehrhart $P_n$1-vector (Baumeister et al., 2012).

1. Definition and ambient representation

Let $P_n$2 be the symmetric group on $P_n$3 symbols, and for each $P_n$4 let

$P_n$5

with indices increasing from $P_n$6 to $P_n$7 and permutations acting on the right. The dihedral group admits the presentation

$P_n$8

where $P_n$9 is the $D_n$0-cycle $D_n$1 and $D_n$2 is a reflection, for example $D_n$3 (Baumeister et al., 2012).

The associated polytope is

$D_n$4

By construction, $D_n$5 is exactly the set of all convex combinations of the $D_n$6 matrices of $D_n$7. It sits inside the Birkhoff polytope $D_n$8, so it inherits the interpretation of a family of doubly stochastic matrices while imposing the additional restriction that only dihedral symmetries are allowed.

The vertices of $D_n$9 are exactly the $2n$0 permutation matrices

$2n$1

Thus the extreme points are in bijection with the rotations and reflections of the regular $2n$2-gon.

2. Odd-$2n$3 model and facet description

When $2n$4 is odd, $2n$5 is, by a coordinate permutation, affinely isomorphic to a polytope $2n$6 whose $2n$7 vertices are the rows of the $2n$8 matrix

$2n$9

where $\sigma \in D_n$0 is the $\sigma \in D_n$1 identity and $\sigma \in D_n$2 is the standard cyclic shift (Baumeister et al., 2012).

In this model, the affine hull of $\sigma \in D_n$3, and hence of $\sigma \in D_n$4, is cut out by two families of equations. The first is

$\sigma \in D_n$5

and the second is

$\sigma \in D_n$6

for $\sigma \in D_n$7 and $\sigma \in D_n$8, where $\sigma \in D_n$9 denotes $P(\sigma)$0.

Inside that affine space, the only facet-defining inequalities are

$P(\sigma)$1

Since there are $P(\sigma)$2 coordinates, there are $P(\sigma)$3 facets in total.

The geometric interpretation given for these constraints is specific. The equations

$P(\sigma)$4

enforce all row sums, and hence all column sums, to equal $P(\sigma)$5, implementing the doubly stochastic constraints “in bulk.” The relations denoted $P(\sigma)$6 force the dihedral commutation relations among blocks of powers of $P(\sigma)$7. The inequalities $P(\sigma)$8 are the $P(\sigma)$9–$$P_n := \operatorname{Conv}\{\,P(\sigma):\sigma\in D_n\} \subset \mathbb{R}^{n^2}.$$0-constraints that cut out the convex hull of permutation matrices.

3. Even-$$P_n := \operatorname{Conv}\{\,P(\sigma):\sigma\in D_n\} \subset \mathbb{R}^{n^2}.$$1 decomposition as a join

If $$P_n := \operatorname{Conv}\{\,P(\sigma):\sigma\in D_n\} \subset \mathbb{R}^{n^2}.$$2 is even, $$P_n := \operatorname{Conv}\{\,P(\sigma):\sigma\in D_n\} \subset \mathbb{R}^{n^2}.$$3 is, up to lattice-affine isomorphism, the join of two copies of $$P_n := \operatorname{Conv}\{\,P(\sigma):\sigma\in D_n\} \subset \mathbb{R}^{n^2}.$$4 (Baumeister et al., 2012). This decomposition determines the dimension, facets, and affine-hull equations of $$P_n := \operatorname{Conv}\{\,P(\sigma):\sigma\in D_n\} \subset \mathbb{R}^{n^2}.$$5 in the even case.

The dimension is

$$P_n := \operatorname{Conv}\{\,P(\sigma):\sigma\in D_n\} \subset \mathbb{R}^{n^2}.$$6

Each copy of $$P_n := \operatorname{Conv}\{\,P(\sigma):\sigma\in D_n\} \subset \mathbb{R}^{n^2}.$$7 contributes $$P_n := \operatorname{Conv}\{\,P(\sigma):\sigma\in D_n\} \subset \mathbb{R}^{n^2}.$$8 non-negativity facets $$P_n := \operatorname{Conv}\{\,P(\sigma):\sigma\in D_n\} \subset \mathbb{R}^{n^2}.$$9, so $P_n$0 has $P_n$1 facets in total, namely

$P_n$2

together with the corresponding inequalities in the second copy’s coordinates.

The affine-hull equations are those of each copy of $P_n$3 on its coordinates, exactly as in the odd case but with $P_n$4 replaced by $P_n$5. Accordingly, the even-$P_n$6 geometry is governed by the same structural pattern as the odd-$P_n$7 model, but assembled through a join construction rather than a single block model.

4. Structural invariants

The dimension of $P_n$8 is

$P_n$9

This parity dependence is one of the principal structural distinctions in the theory (Baumeister et al., 2012).

The polytope $0$0 is a lattice polytope, since its vertices lie in $0$1. For odd $0$2, $0$3 is Gorenstein of codegree $0$4, reflexive after an $0$5-fold dilation and translation, and has normalized volume $0$6. For even $0$7, $0$8 is the join $0$9, is Gorenstein of codegree $1$0, and has normalized volume $1$1.

Its Ehrhart series is written as

$1$2

The corresponding $1$3-vector is explicit: $1$4 and

$1$5

if $1$6 is even, that is,

$1$7

In particular, the $1$8-vector is symmetric, unimodal, and sums to the normalized volume.

These invariants place $1$9 within the interaction of permutation polytopes, lattice polytope theory, and Ehrhart theory. The Gorenstein and reflexive features are not auxiliary: they are extracted directly from the explicit polyhedral description.

5. Characterization as dihedral permutation channels

A discrete memoryless channel with input and output alphabets of size $P_n$00 is specified by an $P_n$01 transition matrix $P_n$02 with nonnegative entries summing to $P_n$03 in each row. Such a channel is called a dihedral permutation channel if

$P_n$04

Equivalently,

$P_n$05

Thus the polytope $P_n$06 is the exact parameter space of all dihedral permutation channels (Baumeister et al., 2012).

There is also an intrinsic characterization: $P_n$07 This expresses the same class of channels without explicit reference to the coefficients $P_n$08. In that form, the admissible transition matrices are precisely the doubly stochastic matrices that respect all dihedral symmetries.

This equivalence links the channel model to graph symmetry. The adjacency matrix of the $P_n$09-cycle encodes the regular $P_n$10-gon combinatorially, while commutation with $P_n$11 imposes compatibility with the dihedral action. As a result, the geometric object $P_n$12 and the channel class defined by dihedral symmetry coincide exactly.

6. Information-theoretic consequences

Since $P_n$13 acts transitively on the input alphabet, every $P_n$14 is a group-symmetric channel. By standard symmetry arguments, the capacity-achieving input distribution is uniform: $P_n$15 This gives a closed structural statement for the entire class, rather than a channel-by-channel optimization result (Baumeister et al., 2012).

For any

$P_n$16

the row-conditional entropy $P_n$17 is independent of $P_n$18 and equals the Shannon entropy $P_n$19 of the mixing weights. Hence

$P_n$20

In particular, $P_n$21 is noiseless, with capacity $P_n$22, exactly when one $P_n$23.

The polyhedral description also has algorithmic consequences. Because $P_n$24 is cut out by linear constraints and $P_n$25 facets, any linear-objective optimization over this class can be carried out by linear or convex programming over $P_n$26 with size $P_n$27 in the number of variables. The complete facet description guarantees polynomial-time separation oracles and hence efficient algorithms for channel design within the dihedral class.

Taken together, these consequences show that the explicit geometry of $P_n$28 translates directly into information-theoretic structure: uniform-input optimality, a closed-form capacity formula, and tractable optimization over the admissible family of channels.