reflection group codes and their decodingmath.hawaii.edu/~jb/gpcodes-hand.pdf · marc fossorier,...

Reflection Group Codes and their Decoding

Marc Fossorier, J.B. Nation and Wes Peterson

University of Hawai‘i at Manoa

Ashikaga, August 2010

Fossorier, Nation, Peterson Reflection Group Codes

Background

History.zipBasic idea due to David Slepian in 1950s and 1960s:choose a group of orthogonal matrices and a point on asphere, and use the orbit of that point by that group as aset of signals for communication.Refinement by Mittelholzer and Lahtonen in 1996.Further refinement by F, N, P in 2004–2009.Extension to complex groups by Kim, Shepler, N in2008-2010.

For comparisonModulation via nPSK or QAM

First taskRecall a lot of group theory you never knew!


Linear Algebra Review

Inner products and transposes

(x,y) = xT y(Mx,y) = (x,MT y) = xT MT y‖x‖ =

√(x,x) and d(a,b) = ‖a− b‖

Orthogonal matrices

M is orthogonal if MMT = I, i.e., MT = M−1

If M is orthogonal, then

(Mx,y) = (x,M−1y)

(Mx,My) = (x,y)

‖Mx− y‖ = ‖x−M−1y‖


Groups of orthogonal matrices

All n × n orthogonal matrices form a group O(n).

I is orthogonal.The inverse of an orthogonal matrix is orthogonal.Products of orthogonal matrices are orthogonal.

Examples

Dihedral groups Hk2 generated by the matrices[

c −ss c

]and

[1 00 −1

]where c = cos 2π

k and s = sin 2πk

The group An−1 of all n × n permutation matricesThe group Bn of all n × n permutation matrices with entries±1


Basic idea

Let G be a finite group of isometries on Rn.G acts on the sphere of radius 1.Choose an initial vector x0 ∈ V with ‖x0‖ = 1.Associate a group element g with each message m.m −→ x = g−1x0 −→ r = x + n −→ g′r ≈ x0 −→ m′

G acts faithfully and transitively on Gx0 = {gx0 : g ∈ G}(or if not, factor out the isotropy group).


Points vs. bitstrings as signals

2n-PSK: With dihedral groups, use points on the circle totransmit bit-strings of length blog 2nc.With the group E6, use points on the sphere in R6 torepresent bit-strings of length 15.With the group B32, use points on the sphere in R32 torepresent bit-strings of length 148.More generally, we can identify the group elementsthemselves with the (not necessarily binary) message.


Reflection groups

Reflection group

...a group G generated by a set of reflections on Rn.

ReflectionFor α a unit vector, Sα(x) = x− 2(x, α)α.

Reflection planes

Nα = {x : (x, α) = 0} is the set of vectors fixed by Sα.

Roots∆(G) = {α : Sα ∈ G}


Example - A2

Notation on whiteboard

A = Sα

B = Sβ

G = {I,A,B,AB,BA,ABA}


Regions

FundamentalsThe reflection planes divide the space into regions.Pick one region to be the fundamental region FR(G).Choose an initial vector x0 in the fundamental region.A root β is positive if (β,x0) > 0.The positive roots whose reflection planes form the walls ofthe fundamental region are the fundamental roots.FR(G) = {y : (α,y) > 0 for every fundamental root α}


Reflection groups (cont.)

Generators and length

G is generated by the reflections Sα with α a fundamentalroot.The length `(g) is the minimum number k such that g is aproduct of k fundamental reflections.Roughly speaking, ‖gx0 − x0‖ increases with `(g): if`(Sαg) = `(g) + 1, then ‖Sαgx0 − x0‖ > ‖gx0 − x0‖.

Classification of irreducible reflection groups

An, Bn, Dn, E6, E7, E8, F4, Hn2 , I3, I4


The action of G on roots

Technical definitionLet ∆G(g) denote the set of positive roots for which gα is anegative root.

TheoremIf α is a positive root and v is not on the reflecting plane Nα,then Nα is between v and gv if and only if α ∈ ∆G(g−1).

TheoremIf Sα is a fundamental reflection, then Sαα = −α and Sα

permutes all the other positive roots.

TheoremIf α is any root and u and v are on the same side of thereflection plane Nα, then v is closer than Sαv to u. If u and vare on opposite sides of Nα, then Sαv is closer than v to u.


Action and length

TheoremTFAE.

`(g) = k .For any vector v that is not in any reflecting plane, thenumber of reflecting planes that separate v and gv is k .|∆G(g)| = k .


Subgroups

Types of subgroups

parabolic - generated by fundamental reflectionsreflection - generated by reflectionsother

DefinitionIf H is a reflection subgroup, the fundamental region FR(H) isthe region containing x0 and bounded by the reflection planesof H.

TheoremIf H is a reflection subgroup of G, then FR(G) ⊆ FR(H).


Coset leaders

Let H be a reflection subgroup of G.

LemmaEach left coset gH contains a unique element c of minimallength.Choose c as the coset leader.

TheoremIf x ∈ FR(H), then there is a unique coset leader c such thatcx ∈ FR(G) and we have an efficient algorithm to find it.


Summary of group theory background

group G of orthogonal matricesreflection, reflection group, reflection subgroupinitial vector x0

fundamental regions FR(H) ⊇ FR(G)

coset leader


The algorithm

SetupChoose a chain of reflection subgroups

{I} = G0 < G1 < · · · < Gk = G

The fundamental regions are nested in reverse order:

Rn = FR(I) ⊇ FR(G1) ⊇ · · · ⊇ FR(G)


The algorithm

EncodeGiven a message m,take the corresponding group element g.Write it as a product of coset leaders g = ck . . . c1.Transmit x = g−1x0 = c−1

1 . . . c−1k x0.


The algorithm

Noiser = x + n


The algorithm

Decoder ∈ FR(G0)

Find the coset leader d1 with d1r ∈ FR(G1)

Find the coset leader d2 with d2d1r ∈ FR(G2)

etc.to obtain dk . . . d1r ∈ FR(G)

Decode g′ = dk . . . d1 −→ m′.


Notation for the examples

Coset leadersThe coset leaders are arranged into graphs.

Matrices

τ12 =

0 1 01 0 00 0 1

σ1 =

−1 0 00 1 00 0 1

etc.


Decoding A4

τ12

τ12

τ12

τ12

τ23

τ23

τ23

τ34

τ34

τ45

x0 = 〈−2,−1, 0, 1, 2〉

1


Decoding A6 bent

τ17 τ12 τ27 τ26

τ12

τ67

τ23

τ12

τ36

τ67

τ35

τ23

τ12

τ56

τ67

τ34

τ23

τ12

τ45

τ56

τ67

1


Decoding B4

σ1 σ4 τ14 σ2 τ12 τ24 σ3 τ34τ23

τ12

x0 = 〈0.707, 1.707, 2.707, 3.707〉

1


Average number of comparisons to decode An

n an a′n log2(n + 1)!

4 7.7 7.1 6.98 24.2 19.5 18.4

16 81.6 57.4 48.332 292.9 182.2 122.7

wherean is the average number of comparisons needed todecode An using straight coset leader graphs (parabolicsubgroups)a′n is the average number of comparisons needed todecode An using bent coset leader graphs (a differentsubgroup sequence)log2(n + 1)! is the theoretical minimum average number ofcomparisons


Average number of comparisons to decode Bn

n bn b′n b

′′n n + log2 n!

4 11.0 8.9 8.7 8.68 38.6 27.3 24.0 23.3

16 142.3 88.6 67.7 60.332 542.0 307.9 204.5 149.7

wherebn is the average number of comparisons to decode Bnusing parabolic subgroups

b′n is the average number of comparisons to decode Bn

using intermediate subgroups with straight CL graphs

b′′n is the average number of comparisons to decode Bn

using intermediate subgroups with bent CL graphsn + log2 n! is the theoretical minimum average number ofcomparisons


Refinements

Error controlReceived vectors from neighboring regions decode togroup elements with only one coset leader different.A linear block code can be super-imposed.

EfficiencyChoosing the right subgroup sequence can doubledecoding efficiency.Complex permutation groups G(r , k ,n) work similarly withlarge minimum distance.


Conclusions

Use of subgroup decoding may make group codingpractical.Group codes with nontrivial isotropy subgroups can bedecoded with this algorithm (though some care isrequired).Further testing and analysis is required.Other groups and other decoding methods should beconsidered.Thank you!


reflection group codes and their decodingmath.hawaii.edu/~jb/gpcodes-hand.pdf · marc fossorier,...

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