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An example of a non-Abelian group code achieving channel capacity

Jorge P. Arpasi Universidade Federal do Pampa - UNIPAMPA September 2018

Group Codes

◮ Given a channel (X, Y, p(y|x)) and a group G matched to X,

a group code C is a subgroup of G n.

◮ Equivalently a group code is the image Im(φ) of some

injective homomorphism φ : U → X n, where U is the uncoded information group. From this U ∼ = C

◮ The rate of the code is R = log U n . ◮ When the group is the finite field G = Zr p, any subgroup of

(Zr

p)n is (Zr p)k, for some k ≤ n. Then the encoding rate is

R = kr log p

n ◮ When the group is cyclic G = Zpr , any subgroup of (Zpr )n is

Zk1

p ⊕ Zk2 p2 ⊕ · · · ⊕ Zkr pr , where k1 + k2 + · · · + kr ≤ n. The

encoding rate is R = log p r

i=1 iki

n

The 8PSK-AWGN channel

x x x x x x x x

2 3 4 5 6 7 1

Is the channel (X, Y, p(y|x)) where X = {xi = e

j2πi 8 , i = 0, . . . , 7}, Y = R2

and p(y|xi) =

1 √ 2πσe−y−xi2/2σ2 ◮ When the code is over the Galois Field GF(23), it is linear. In

this case, Gallager in [1] shows that there is an ensemble of random linear codes achieving the channel capacity.

◮ When the code is over the the group Z8, it was conjectured

by Loeliger in [2] if there were ensemble of group codes achieving the channel capacity.

◮ G. Como in [3] shows that a group code over a cyclic groups

Zm, for an mPSK-AWGN channel, achieves the channel capacity.

◮ The approach of Como has three steps:

◮ To derive a definition of group code capacity CG in such a way

it is les or equal than the channel capacity,

◮ To show the existence of random group codes achieving CG. ◮ When CG = C it is declared that the group code capacity

achieves the channel capacity.

Group Codes over Z3

2 = GF(8) ◮ φ : U → X n means

φ : U → (Z3

2)n, then

U ∼ = (Z3

2)k

for some k < n.

◮ The encoding rate is R = 3k n ◮ It has one BI-AWGN

sub-channel with capacity C2 and the trivial 8PSK-AWGN with capacity C8 = C, the capacity of the channel. Group Codes over Z8

◮ φ : U → X n means

φ : U → (Z8)n then U ∼ = Zk1

2 ⊕ Zk2 4 ⊕ Zk3 8

for some k1, k2, k3 such that k1 + k2 + k3 < n.

◮ The encoding rate is

R = k1+2k2+3k3

n ◮ In this case the sub-channels

are: the BI-AWGN with capacity C2, the QPSK-AWGN with capacity C4 and the trivial 8PSK-AWGN with capacity C8 = C.

Imposing the converse of the Shannon’s coding theorem the rate of each sub-channel must be inferior than the respective capacity, that is, Rl ≤ Cl

. Group Code Capacity for Z3

2 = GF(8) ◮ The encoding capacity is

CG = min{3C2, C8} Group Code Capacity for Z8

◮ The encoding capacity is

CG = min{3C2, 3 2C4, C8}

• G. Como in [3] proves that

CG = C the channel capacity by showing 3C2 ≥ 3

2C4 ≥ C8.

Group codes over the dihedral group D4

◮ In this case the sub-channels are BI-AWGN with capacity C2,

QPSK-AWGN with capacity C4, VPSK-AWGN with capacity CV and the trivial sub-channel 8PSK-AWGN with capacity C8 = C.

◮ We call the sub-channel VPSK because the matched group

group is the Klein group Z2

2 that in the algebraic literature is

denoted by V .

◮ We show that the group code capacity is

CG = min{3C2, 3 2C4, 3 2CV , C8}

◮ Since 3C2 ≥ 3 2C4 ≥ C8 then

CG = min{3 2CV , C8}

Group Code Capacity of group codes over D4

◮ A necessary condition for the equality CG = C8 is that:

3CV ≥ 2C8

◮ If λ(y) and λV are the output densities of C8 and CV

sub-channels, in terms of entropies the above inequality is equivalent 3H(λ) ≥ 2H(λV ) + H(p0), where the H(p0) is the entropy of the conditional density p(y|x0).

Group Code Capacity of group codes over D4 After some manipulation, the above inequality which is in terms of continuous entropies, is transformed in 2

• R2

λ(y)H(ν(y))dy ≥

• R2

λV (y)H(ωV (y))dy, where ν(y) = (ν1(y), ν2(y)) is a binary random variable and ωV (y) = (ωV 1(y), ωV 2(y), ωV 3(y), ωV 4(y)) is a quaternary random variable.

Group Code Capacity of group codes over D4 For a fixed y = ρ, in polar coordinates the above inequality becomes 2π λ(θ)Hν(θ)dθ ≥ 2π λV (θ)HωV (θ)dθ were λ, λV , Hν and HωV have periods of else π/2 or π. We use these periodical properties of these functions to show this last

• integral. With this,

CG = C8 = C

Continuity of the work

◮ It is a remaining work to prove the existence of ensemble of

group codes Cn, over D4 with rates Rn whose supreme is CG. With this one would declare formally that the group capacity,

• ver D4, achieves the channel capacity.

◮ Another important non-Abelian group is the group of

quaternions which is naturally matched to a a constellation of the unitary sphere of R4.

◮ In this case the group capacity equals the channel capacity?, if

so the group capacity achieves the channel capacity?

• R. G. Gallager, Information Theory and Reliable

Communication. Wiley and Sons, 1968.

• H. A. Loeliger, “Signal sets matched to groups,” IEEE Trans.
• Inform. Theory, vol. IT 37, pp. 1675–1682, November 1991.
• G. Como and F. Fagnani, “The capacity of abelian group

codes over symmetric channels,” IEEE Trans. Inform. Theory,

• vol. IT 45, no. 01, pp. 3–31, 2009.