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

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 = Z r p , any subgroup of p ) n is ( Z r ( Z r p ) k , for some k ≤ n . Then the encoding rate is R = kr log p n ◮ When the group is cyclic G = Z p r , any subgroup of ( Z p r ) n is p ⊕ Z k 2 p 2 ⊕ · · · ⊕ Z k r Z k 1 p r , where k 1 + k 2 + · · · + k r ≤ n . The encoding rate is R = log p � r i =1 ik i n

The 8PSK-AWGN channel x 2 x x 3 1 Is the channel ( X , Y , p ( y | x )) where j 2 π i 8 , i = 0 , . . . , 7 } , Y = R 2 X = { x i = e x 4 x 0 2 πσ e −� y − x i � 2 / 2 σ 2 1 and p ( y | x i ) = √ x x 5 7 x 6 ◮ When the code is over the Galois Field GF (2 3 ), 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 Z 8 , 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 Z m , for an m PSK-AWGN channel, achieves the channel capacity. ◮ The approach of Como has three steps: ◮ To derive a definition of group code capacity C G in such a way it is les or equal than the channel capacity, ◮ To show the existence of random group codes achieving C G . ◮ When C G = C it is declared that the group code capacity achieves the channel capacity.

Group Codes over Z 8 ◮ φ : U → X n means Group Codes over Z 3 2 = GF (8) φ : U → ( Z 8 ) n then ◮ φ : U → X n means φ : U → ( Z 3 2 ) n , then U ∼ = Z k 1 2 ⊕ Z k 2 4 ⊕ Z k 3 8 U ∼ = ( Z 3 2 ) k for some k 1 , k 2 , k 3 such that k 1 + k 2 + k 3 < n . for some k < n . ◮ The encoding rate is ◮ The encoding rate is R = 3 k R = k 1 +2 k 2 +3 k 3 n n ◮ It has one BI-AWGN ◮ In this case the sub-channels sub-channel with capacity are: the BI-AWGN with C 2 and the trivial capacity C 2 , the 8PSK-AWGN with capacity QPSK-AWGN with capacity C 8 = C , the capacity of the C 4 and the trivial channel. 8PSK-AWGN with capacity C 8 = 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, R l ≤ C l . Group Code Capacity for Z 8 ◮ The encoding capacity is Group Code Capacity for C G = min { 3 C 2 , 3 Z 3 2 = GF (8) 2 C 4 , C 8 } ◮ The encoding capacity is G. Como in [3] proves that C G = C the channel C G = min { 3 C 2 , C 8 } capacity by showing 3 C 2 ≥ 3 2 C 4 ≥ C 8 .

Group codes over the dihedral group D 4 ◮ In this case the sub-channels are BI-AWGN with capacity C 2 , QPSK-AWGN with capacity C 4 , VPSK-AWGN with capacity C V and the trivial sub-channel 8PSK-AWGN with capacity C 8 = C . ◮ We call the sub-channel VPSK because the matched group group is the Klein group Z 2 2 that in the algebraic literature is denoted by V . ◮ We show that the group code capacity is C G = min { 3 C 2 , 3 2 C 4 , 3 2 C V , C 8 } ◮ Since 3 C 2 ≥ 3 2 C 4 ≥ C 8 then C G = min { 3 2 C V , C 8 }

Group Code Capacity of group codes over D 4 ◮ A necessary condition for the equality C G = C 8 is that: 3 C V ≥ 2 C 8 ◮ If λ ( y ) and λ V are the output densities of C 8 and C V sub-channels, in terms of entropies the above inequality is equivalent 3 H ( λ ) ≥ 2 H ( λ V ) + H ( p 0 ) , where the H ( p 0 ) is the entropy of the conditional density p ( y | x 0 ).

Group Code Capacity of group codes over D 4 After some manipulation, the above inequality which is in terms of continuous entropies, is transformed in � � 2 λ ( y ) H ( ν ( y )) dy ≥ λ V ( y ) H ( ω V ( y )) dy , R 2 R 2 where ν ( y ) = ( ν 1 ( y ) , ν 2 ( y )) is a binary random variable and ω V ( y ) = ( ω V 1 ( y ) , ω V 2 ( y ) , ω V 3 ( y ) , ω V 4 ( y )) is a quaternary random variable.

Group Code Capacity of group codes over D 4 For a fixed � y � = ρ , in polar coordinates the above inequality becomes � 2 π � 2 π λ ( θ ) H ν ( θ ) d θ ≥ λ V ( θ ) H ω V ( θ ) d θ 0 0 were λ , λ V , H ν and H ω V have periods of else π/ 2 or π . We use these periodical properties of these functions to show this last integral. With this, C G = C 8 = C

Continuity of the work ◮ It is a remaining work to prove the existence of ensemble of group codes C n , over D 4 with rates R n whose supreme is C G . With this one would declare formally that the group capacity, over D 4 , 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 R 4 . ◮ 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.