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

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2 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

3 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 aected 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.

4 Basic concepts on Wireless channels A QAM constellation is a set of the form {(2n + 1, 2m + 1) : 0 n, m M}. It represents M 2 dierent 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.

5 Basic concepts on Wireless channels 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 min x 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. Diagrams BEP/SNR SNR(Signal to noise ratio) = Energy/ Σ

6 Basic concepts on Wireless channels QAM 4g 1 46,1 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 ).

7 Arithmetic Fuchsian groups: the rational case ( ) a, b Q a,b ; B = Q = Q + QI + QJ + QK I 2 = a; J 2 = b, IJ = JI = K Reduced norm: N(x + yi + zj + tk) = x 2 ay 2 bz 2 + abt 2 Reduced Trace: Tr(x + yi + zj + tk) = 2x ( ) a, b ψ : M ( 2, Q( a) ) Q ( x + y a z + ) t a x + yi + zj + tk b(z t a) x y a If B Q p is a division algebra, we say that B ramies at p, p prime or ; Q = R. D(B)=product of the ramication primes of B. If B ramies at p =, it is said to be denite; otherwise indenite.

8 Arithmetic Fuchsian groups: the rational case An order O of B is a Z-lattice such that O Z Q = B and such that O is also a ring. ( ) a, b B = indenite; O B maximal order (up to conjugation) Q O 1 = multiplicative group of elements reduced norm 1 B Γ 1 = B ψ(o1 ) B If B is indenite of discriminant D, denote Γ(D, 1) := Γ 1 B An arithmetic Fuchsian group of the rst kind is a discrete group Γ GL (2, R) commensurable with Γ 1 for some B. B Examples: Γ 0 (N), Takeuchi groups.

9 Arithmetic Fuchsian groups: the general case F = Q(θ) totally real number eld of degree n; R F its ring of integers; a, b F ; ( ) a,b A quaternion F -algebra is B = Q = Q + QI + QJ + QK such that I 2 = a; J 2 = b, IJ = JI = K. An order O of B is a ring which is a rank 4 R F -lattice. condition S: B ramies 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 ( ) a, b ψ : Q x + yi + zj + tk M ( 2, F ( a) ) ( x + y a z + t a b(z t a) x y a )

10 Fundamental domains and point reduction If Γ is an arithmetic Fuchsian group, then Γ acts on H. A fundamental 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 ramication in prime ideals implies that F is compact. Let us see some examples:

11 Fundamental domains and point reduction S = ( Figura : Fundamental domain for SL (2, Z) ), T = ( ), SL (2, Z) = S, T

12 Fundamental domains and point reduction Arithmetic Fuchsian group of signature ( (1, ) e): ( λ 0 Γ = α, β [α, β] e = ±1, α =, β = 0 λ 1 algebraic number) ) (λ Figura : Fundamental domain for signature (1, 2)

13 Fundamental domains and point reduction Figura : Fundamental domain for signature Γ(6, 1) and tessellation

14 Fundamental domains and point reduction 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.

15 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) = γ 1, γ 2, S, γ 1 = 1 2 ( and γ 2 = ) 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.

16 Fuchsian codes of rate 3 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 {(x i, y i, z i, t i )} C such that i=1 x 2 ay 2 bz i i i + abt i = 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 a,b kind attached to B = Q ( x + y a (x, y, z, t) γ = z + t b z + t b x y a )

17 Fuchsian codes of rate 3 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.

18 Fuchsian codes of rate 3 Some simulations: QAM 4g 1 46,1 Figura : 4NUF constellations

19 Fuchsian codes of rate 3 Some simulations: QAM 8g 1 86,1 Figura : 8NUF constellations

20 Fuchsian codes of rate 3 Some simulations: QAM Figura : 16NUF constellations

21 Fuchsian codes with arbitrary rate ( F = Q(θ)/Q totally real of degree n; B = a,b F ) meeting condition S; x an R F -order O (for simplicity, the natural one). We can regard Γ := ψ(o) 1 as matrices ( x + ay z + ) bt b(z at x ; x, y, z, t R F. ay) Since x = n 1 k=0 m kθ k, with m k Z and analogously y, z, t, the matrices in M ( 2, R F [ a] ) carry 4n items of information. New proposal: Fix F for Γ. Send a 4-tuple (x, y, z, t) satisfying x 2 ay 2 bz 2 + abt 2 = 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.

22 Fuchsian codes with arbitrary rate Proposition The code rate of the proposed code (using N channels) is 3n/N. ( Example 1: Take B = 3, 1 Q( 7) ) and the natura order Z[ 7][1, ( I, J, K]. The corresponding Fuchsian code has rate 6. The ) matrix is identied with the 8-tuple (0, 1, 1, 0, 0, 0, 1, 0). Example 2: Take F the maximal totally real subeld of the p-th cyclotomic eld. And a quaternion F -algebra meeting condition S (p = 13 works, we think that innitely many). Then, the Fuchsian code has rate 3(p 1) 2.

23 Bibliography M. Alsina, P. Bayer: Quaternion orders, quadratic forms and Shimura curves. CRM Monograph Series, 22. American Mathematical Society, Providence, RI 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, F. E. Oggier, J.C. Belore, E. Viterbo: Cyclic division algebras: A tool for space-time coding. Foundations and Trends in Communications and Information Theory, vol. 4, no. 1 (2007).

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