SLIDE 1
Rate–Diversity Optimal Multiblock Space–Time Codes via Sum-Rank Codes
Mohannad Shehadeh and Frank R. Kschischang
Department of Electrical and Computer Engineering University of Toronto {mshehadeh, frank}@ece.utoronto.ca
2020 IEEE International Symposium on Information Theory June 21–26, 2020
SLIDE 2 Channel Model
MIMO Rayleigh block-fading channel: ◮ nr receive antennas, nt transmit antennas ◮ L fading blocks of duration T Yi = ρHiXi + Wi for i = 1, 2, . . . , L ◮ Hi are nr × nt with iid CN(0, 1) entries ◮ Xi is nt × T transmission in the ith fading block ◮ Wi are nr × T with iid CN(0, 1) entries ◮ ρ related to SNR in the usual way E
L
ρXi2
F = L · T · SNR
SLIDE 3 Multiblock Space–Time Codes
◮ Fix a constellation which is a finite A ⊂ C ◮ An L-block nt × T code is a finite X ⊆ Ant×LT ◮ Any nt × LT codeword X ∈ X partitions as X =
X2 · · · XL
- ◮ Xi for i = 1, 2, . . . , L are nt × T sub-codewords
◮ Perfect channel knowledge at receiver with ML decoding ◮ T ≥ nt ◮ T = nt referred to as minimal-delay
SLIDE 4 Definition (Transmit Diversity Gain)
An L-block nt × T code X ⊆ Ant×LT will be said to achieve a transmit diversity gain of d if d = min
X,X ′∈X X=X ′ L
rank(Xi − X ′
i ).
[Guey et al., 1996], [Tarokh et al., 1998] ◮ If this is the case, then Pe(SNR) = O(SNR−nrd) as SNR → ∞ ◮ d ≤ Lnt ◮ d = Lnt referred to as full diversity
SLIDE 5
Rate–Diversity Tradeoff
Definition (Rate)
The rate R of an L-block nt × T code X ⊆ Ant×LT is R = 1 LT log|A| |X|.
Theorem (Rate–Diversity Tradeoff)
Let X ⊆ Ant×LT be an L-block nt × T code achieving transmit diversity gain d and rate R, then R ≤ nt − d − 1 L . [El Gamal and Hammons, 2003], [Lu and Kumar, 2005]
Proof.
True by a Singleton bound argument.
SLIDE 6 Codes over Extension Fields
◮ Fqm is an m-dimensional vector space over Fq ◮ Fix an ordered basis B = {β1, β2, . . . , βm} of Fqm over Fq ◮ Any c ∈ Fs
qm can be written as
c = m
i=1 βici1
m
i=1 βici2
· · · m
i=1 βicis
m
βi
ci2 · · · cis
- Definition (Matrix Representation)
The matrix representation of any c ∈ Fs
qm is given by the map
MB : Fs
qm −
→ Fm×s
q
defined by MB (c) = c11 c12 · · · c1s c21 c22 · · · c2s . . . . . . ... . . . cm1 cm2 · · · cms .
SLIDE 7 Sum-Rank Metric [N´
- brega and Uchˆ
- a-Filho, 2016]
◮ Fix a sum-rank length partition N = r1 + r2 + · · · + rL ◮ Any c ∈ FN
qm partitions as
c =
c(2) · · · c(L) where c(i) ∈ Fri
qm for i = 1, 2, . . . , L
◮ Define sum-rank distance between c, d ∈ FN
qm
dSR(c, d) =
L
rank(MB(c(i)) − MB(d(i))) ◮ r1 = r2 = · · · = rL = 1, L = N: Hamming distance ◮ r1 = N, L = 1: Rank distance [Gabidulin, 1985] ◮ Define minimum sum-rank distance of a code C ⊆ FN
qm
dSR(C) = min
c,d∈C c=d
dSR(c, d)
SLIDE 8
◮ If C is a k-dimensional linear code over Fqm, then k ≤ N − dSR(C) + 1 ◮ Maximum sum-rank distance (MSRD) codes meet above with equality ◮ L = N (Hamming distance): Classical Reed–Solomon codes ◮ L = 1 (Rank distance): Gabidulin codes [Gabidulin, 1985] Arbitrary sum-rank length partition: ◮ Gabidulin codes [Gabidulin, 1985] are MSRD for m ≥ N ◮ m ≥ N translates to T ≥ Lnt in construction of [Lu and Kumar, 2005] based on Gabidulin codes ◮ Linearized Reed–Solomon codes [Mart´ ınez-Pe˜ nas, 2018] are MSRD for m ≥ maxi ri ◮ m ≥ maxi ri translates to T ≥ nt
SLIDE 9 Definition (Rank-Metric-Preserving Map)
Let φ: Fq − → A be a bijection and ˜ φ: Fnt×T
q
− → Ant×T be an element-wise version. φ is rank-metric-preserving if, for all C, D ∈ Fnt×T
q
, C = D, rank(˜ φ(C) − ˜ φ(D)) ≥ rank(C − D) with the first rank over C and the second over Fq.
Example (Gaussian Integer Map [Bossert et al., 2002])
◮ Z[ı] = {a + ıb | a, b ∈ Z} form a Euclidean domain ◮ If Π is prime in Z[ı], then Z[ı]/ΠZ[ı] is isomorphic to F|Π|2 ◮ The map φ: a → a − round a Π
with a interpreted as an integer is rank-metric-preserving
SLIDE 10 Rate–Diversity Optimal Multiblock Space–Time Codes
◮ Let r1 = r2 = · · · = rL = nt and N = Lnt ◮ Let m = T ◮ Let C ⊆ FLnt
qT be a linear code with generator matrix of the
form G =
G2 · · · GL
qT
with G1, G2, . . . , GL ∈ Fk×nt
qT
referred to as sub-codeword generators
SLIDE 11 Linearized Reed–Solomon Codes [Mart´ ınez-Pe˜ nas, 2018]
Define the FqT − → FqT functions: ◮ σ(a) = aq ◮ For all i ∈ N, define Ni(a) = σi−1(a)σi−2(a) · · · σ(a)a ◮ For all i ∈ N and a ∈ FqT , define Di
a(b) = σi(b)Ni(a)
Let q > L, T ≥ nt, x be a primitive element of FqT and let sub-codeword generators for C be Gi = β1 β2 · · · βnt Dxi−1(β1) Dxi−1(β2) · · · Dxi−1(βnt) D2
xi−1(β1)
D2
xi−1(β2)
· · · D2
xi−1(βnt)
. . . . . . ... . . . Dk−1
xi−1(β1)
Dk−1
xi−1(β2)
· · · Dk−1
xi−1(βnt)
for i = 1, 2, . . . , L.
Theorem ([Mart´ ınez-Pe˜ nas, 2018])
C is MSRD.
SLIDE 12 Rate–Diversity Optimal Multiblock Space–Time Codes
Let: ◮ T ≥ nt ◮ k = Lnt − d + 1 ◮ q = |Π|2 > L with φ the corresponding Gaussian integer map ◮ G =
G2 · · · GL
qT
be a linearized Reed–Solomon code generator X = ˜ φ(MB(uG1)⊺) ˜ φ(MB(uG2)⊺) · · · ˜ φ(MB(uGL)⊺)
qT
X is a rate–diversity optimal L-block nt × T code achieving transmit diversity gain d.
SLIDE 13
◮ Define bit rate Rb = 1 LT log2 |X| ◮ There is no tradeoff between Rb and d: Rb = R · log2 |A| ◮ There exist d = Lnt codes with arbitrarily large |A|
◮ Multiblock diversity–multiplexing optimal codes [Lu, 2008], [Yang and Belfiore, 2007] ◮ Codes from cyclic division algebras [Sethuraman et al., 2003] (repeat each codeword L times)
SLIDE 14 5 10 15 20 25 30 100 101 102 103 104 105 106 107
ε = 0 ε = . 5 ε = . 1 ε = 0.25 ε = . 7 5
Lnt |A| Rb/nt = 2 and d = ⌈(1 − ε)Lnt⌉
SLIDE 15 nr = nt = T = L = 2 and Rb ≈ 4
Reference code (Mult. CDA): 2-block diversity–multiplexing optimal code due to [Lu, 2008], [Yang and Belfiore, 2007]
5 10 15 20 25 10−5 10−4 10−3 10−2 10−1 100 SNR (dB) Codeword Error Rate
- Mult. CDA, d = 4, Rb = 4.00
Sum-Rank, d = 3, Rb = 4.09
−0.5 0.5 −0.5 0.5
−0.5 0.5 −0.5 0.5
Sum-Rank, d = 3
SLIDE 16 0 1 2 3 4 5 6 8 10 12 14 16 100 101 102 103 104 105 106 107
Sum-Rank
ε = ε = . 1 ε = 0.25 ε = 0.5
L |A| Rb = 4, nt = 2, and d = ⌈(1 − ε)Lnt⌉
L + 1
SLIDE 17
Concluding Remarks
◮ Linearized Reed–Solomon codes [Mart´ ınez-Pe˜ nas, 2018] translate to minimal-delay rate–diversity optimal multiblock space–time codes ◮ Such codes can perform well with small constellations compared to full diversity alternatives ◮ Future work:
◮ Further performance analysis, coding gain analysis ◮ Algebraic soft-decision decoding
