fuchsian codes with arbitrary rates - ghent universitycage.ugent.be/cost/talks/blanco-chacon.pdf ·...

Fuchsian codes with arbitrary rates

Iván Blanco Chacón, Aalto University

20-09-2013

Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates

Summary

Basic concepts in wireless channels

Arithmetic Fuchsian groups: the rational case

Arithmetic Fuchsian groups: the general case

Fundamental domains and point reduction

Fuchsian codes with rate 3

Fuchsian codes with arbitrary rate


Basic concepts on Wireless channels

An Additive White Gaussian Noise (AWGN) is a sequence of

independent (in time) iidd complex Gaussian random variables. A

wireless channel a�ected only by AWGN has a channel equation

Y = X + N where X is a transmitted vector, N ∼ CN(0,Σ) and Y

the received vector.

We suppose that the AWGN originates at the receiving end.

A fading wireless channel is a channel subject to interference and

eventually to AWGN. It has a channel equation Y = HX + N,

where H is a fading matrix and Y ,X ,N as before.

We suppose that H is perfectly known to the receiver.



A QAM constellation is a set of the form

{(2n + 1, 2m + 1) : 0 ≤ |n|, |m| ≤ M}. It represents M2 di�erent

transmission states. It requires a labeling algorithm.

More generally, one has NUPAM and NUQAM alphabets,

non-uniformly distributed.

We are interested to send several NUPAM symbols simultaneously

by a single fading channel (SIMO), and to obtain a low complexity

decoding method.



How to send QAM/NUQAM symbols by a fading channel? Without

coding�>Need to solve: Let C be a constellation, suppose you

receive y , minimize

minx∈C ||y − Hx ||

Too complex by brute force if |C | >> 1!. Alternative: extra

structure on the codebook (lattices, matrix orders) . But how to

compare codes? Complexity and SNR/BEP.

SNR(Signal to noise ratio) = Energy/||Σ||

Diagrams BEP/SNR



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12 14 16 18 20

10-4

0.001

0.01

ô 4-G@15,1Dò 4-G@10,1Dì 4-G@6,1Dà 4-G@g=1Dæ 4-QAM

Linear codes coming from cyclic division algebras have complexity

O(|C |r ), with r ∈ Q, typically r = 1/2 (Alamouti) or r = 0,625

(Golden). Our approach is non-linear and yields complexity

O(log |C |).



a, b ∈ Q∗; B =(a,bQ

)= Q + QI + QJ + QK

I 2 = a; J2 = b, IJ = −JI = K

Reduced norm: N(x + yI + zJ + tK ) = x2 − ay2 − bz2 + abt2

Reduced Trace: Tr(x + yI + zJ + tK ) = 2x

ψ :

(a, b

Q

)↪→ M

(2,Q(

√a))

x + yI + zJ + tK 7→

(x + y

√a z + t

√a

b(z − t√a) x − y

√a

)

If B ⊗Qp is a division algebra, we say that B rami�es at p,

p prime or ∞; Q∞ = R.

D(B)=product of the rami�cation primes of B .

If B rami�es at p =∞, it is said to be de�nite; otherwise inde�nite.



An order O of B is a Z-lattice such that O ⊗Z Q ∼= B and such

that O is also a ring.

B =

(a, b

Q

)inde�nite; OB maximal order (up to conjugation)

O1

B = multiplicative group of elements reduced norm 1

Γ1B = ψ(O1

B)

If B is inde�nite of discriminant D, denote Γ(D, 1) := Γ1B

An arithmetic Fuchsian group of the �rst kind is a discrete group

Γ ⊆ GL (2,R) commensurable with Γ1B for some B .

Examples: Γ0(N), Takeuchi groups.


Arithmetic Fuchsian groups: the general case

F = Q(θ) totally real number �eld of degree n; RF its ring of

integers; a, b ∈ F ∗;

A quaternion F -algebra is B =(a,bQ

)= Q + QI + QJ + QK such

that I 2 = a; J2 = b, IJ = −JI = K .

An order O of B is a ring which is a rank 4 RF -lattice.

condition S: B rami�es EXACTLY at one absolute value in F

extending the usual one in Q.

Condition S allows us to assume that a > 0 so that we have again a

representation

ψ :

(a, b

Q

)↪→ M

(2,F (

√a))

x + yI + zJ + tK 7→

(x + y

√a z + t

√a

b(z − t√a) x − y

√a

)Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates


If Γ is an arithmetic Fuchsian group, then Γ acts on H. Afundamental domain is F such that

for any z ∈ H, there exists w ∈ F and g ∈ Γ such that

g · z = w ,

for any z ,w ∈ F and g ∈ Γ such that g · z = w , z ,w ∈ Fr(F).

Arithmetic Fuchsian groups have nice fundamental domains. They

tessellate H and we will use them for our coding purposes.

Condition S and rami�cation in prime ideals implies that F is

compact.

Let us see some examples:



Figura : Fundamental domain for SL (2,Z)

S =

(0 1

−1 0

), T =

(1 1

0 1

), SL (2,Z) = 〈S ,T 〉



Arithmetic Fuchsian group of signature (1, e):

Γ = 〈α, β|[α, β]e = ±1〉, α =

(λ 0

0 λ−1

), β =

(2 1

1 2

)(λ

algebraic number).

-2 -1 0 1 2

-2

-1

0

1

2

Figura : Fundamental domain for signature (1, 2)Iván Blanco Chacón, Aalto University Fuchsian codes with arbitrary rates


-2 -1 0 1 2

1

2

3

4

Figura : Fundamental domain for signature Γ(6, 1) and tessellation



Problem 1: How to �nd presentations for an arithmetic Fuchsian

group? This is equivalent to the problem of how to produce

fundamental domains.(Bayer, Alsina, Voight)

Problem 2: Given a presentation of an arithmetic Fuchsian group,

decompose matrices as products of the generators. Equivalent to

the problem of given a fundamental domain F and a point outside

it, to �nd a transformation which brings the point inside F . (Bayer,B., Remón).

The following result implies the low decoding complexity of our

codes:

Theorem (Bayer-B. 2012, Bayer-Remón 2013)

Given a fundamental domain for an arithmetic Fuchsian group,

there exists an explicit point reduction algorithm doing at most as

many matrix products as the minimal length of the input matrix.


Fuchsian codes of rate 3

Alsina and Bayer have shown:

Γ(6, 1) ={γ =

1

2

(α β

−β′ α′

)|α, β ∈ Z[

√3], det(γ) = 1, α ≡ β mod 2

}.

Alternatively Γ(6, 1) = 〈γ1, γ2, S〉, γ1 = 1

2

(1 +√3 −3 +

√3

3 +√3 1−

√3

)

and γ2 = 1

2

(1 +√3 3−

√3

−3−√3 1−

√3

).

Consider in Γ(6, 1) the subgroup given by the group of units of

norm 1 in the natural order Z[1, I , J,K ]. Since α, β are restricted to

the determinant condition, they carry 3 degrees of freedom over Z.



Suppose one Tx -transmission antenna- and one or more Rx

-receiving antennas-

Suppose a �nite codebook C and a �nite collection of 4-tuples of

integers {(xi , yi , zi , ti )}|C |i=1such that x2i − ay2i − bzi + abti = 1.

Want to summarize each 4-tuple into a suitable signal (coding),

send this signal by the antenna, and decode it. Each 4-tuple will

give a matrix belonging to an arithmetic Fuchsian group of the �rst

kind attached to B =(a,bQ

)(x , y , z , t) γ =

(x + y

√a z + t

√b

−z + t√b x − y

√a

)



Fix a fundamental domain F for Γ. We will send γ(τ), where

τ ∈ F is an interior point.

Interior points are Γ-inequivalent, so we will have as many symbols

as codewords.

Design Problem number 1: How to choose τ?

Want to transmit over a fading channel. But assume for simplicity

just AWGN.

KEY IDEA: Transmit γ(τ). It belongs to γ(F). It is an interior

point. For any interior point w ∈ γ(F), the reduction algorithm

returns a representative z ∈ F and the unique transformation that

brings z into w . Suppose that the fundamental domain, the

euclidean center τ of it and the AWGN is such that the BEP is very

very small. Then, the reduction algorithm returns ±γ for γ(τ) + N.



Some simulations:

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12 14 16 18 20

10-4

0.001

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Figura : 4NUF constellations



Some simulations:

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15 20 25 30

10-5

10-4

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0.1





Some simulations:

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1





F = Q(θ)/Q totally real of degree n; B =(a,bF

)meeting condition

S ; �x an RF -order O (for simplicity, the natural one). We can

regard Γ := ψ(O)1 as matrices(x +√ay z +

√bt

b(z −√at x −

√ay)

); x , y , z , t ∈ RF .

Since x =∑n−1

k=0mkθ

k , with mk ∈ Z and analogously y , z , t, the

matrices in M(2,RF [

√a])carry 4n items of information.

New proposal: Fix F for Γ. Send a 4-tuple (x , y , z , t) satisfying

x2 − ay2 − bz2 + abt2 = 1 as a matrix acting on τ ∈ F , γ(τ).

Send γ(τ) by an AWGN channel and use the reduction point

algorithm to decode.



Proposition

The code rate of the proposed code (using N channels) is 3n/N.

Example 1: Take B =(

3,−1Q(√7)

)and the natura order

Z[√7][1, I , J,K ]. The corresponding Fuchsian code has rate 6. The

matrix

( √7 +√3

√3√

3√7−√3

)is identi�ed with the 8-tuple

(0, 1, 1, 0, 0, 0, 1, 0).

Example 2: Take F the maximal totally real sub�eld of the p-th

cyclotomic �eld. And a quaternion F -algebra meeting condition S

(p = 13 works, we think that in�nitely many). Then, the Fuchsian

code has rate 3(p−1)2

.


Bibliography

M. Alsina, P. Bayer: Quaternion orders, quadratic forms and

Shimura curves. CRM Monograph Series, 22. American

Mathematical Society, Providence, RI 2004.

M. Alsina, I. Blanco-Chacón, D. Remón, C. Hollanti: Fuchsian

codes for AWGN channels (extended journal version).

Submited.

I. Blanco-Chacón, D. Remón, C. Hollanti: Fuchsian codes for

AWGN channels. Proceedings of the International Workshop in

Cryptography and Coding WCC2013, 496-507.

F. E. Oggier, J.C. Bel�ore, E. Viterbo: Cyclic division algebras:

A tool for space-time coding. Foundations and Trends in

Communications and Information Theory, vol. 4, no. 1 (2007).


Thank you for your attention

Contact: ivan.blancochacon@aalto.�


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