Staggered Diagonal Embedding Based Linear Field Size Streaming Codes - - PowerPoint PPT Presentation
Staggered Diagonal Embedding Based Linear Field Size Streaming Codes Vinayak Ramkumar * , Myna Vajha * , M. Nikhil Krishnan , P. Vijay Kumar * * Department of Electrical Communication Engineering, IISc Bangalore Department of Electrical and
Vinayak Ramkumar*, Myna Vajha*, M. Nikhil Krishnan†,
* Department of Electrical Communication Engineering, IISc Bangalore † Department of Electrical and Computer Engineering, University of Toronto
IEEE International Symposium on Information Theory 2020 Los Angeles, California, USA 21-26 June 2020
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◮ A stream of packets sent over a network ◮ Some packets are dropped or lost (packet erasure) ◮ congestion, deep fade in wireless link, arriving after deadline ◮ Atmost τ future packets (along with all past packets) can be used for recovering any erased packet
1 2 3 4 5 6 7 8 9
Source 𝜐 = 4
1 2 3 4 5 6 7 8 9
Destination
i erased packet Network
Packet 2 decoded
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Multiplayer Gaming
Interactive Voice & Video Latency
Ultra reliability and low latency
Enhanced Broadband Massive IoT
5G
Industrial Automation Traffic Control & Safety Tele-surgery
AR Multiplayer VR
◮ underlying stream of packets could correspond to voice, video or data
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◮ Gilbert Elliot (GE) Channel is a commonly-accepted model ◮ Not tractable, difficult to design codes
BEC(𝜗) BEC(1)
G B
𝛽 𝛾 (1 − 𝛽) (1 − 𝛾)
◮ Delay Constrained Sliding Window (DCSW) Channel ◮ A deterministic approximation of GE channel under decoding delay constraint ◮ Admissible erasure patterns (AEP) of {a, b, w, τ} DCSW channel ◮ within every sliding window of size w: either ≤ a random erasures or a burst of ≤ b erasures ◮ decoding delay constraint τ
9 10 a = 2 random erasures 1 2 3 4 6 7 8 5 w = 5 burst of b = 4 erasures w = 5 i erased pkt
(a = 2, b = 4, w = 5, τ = 4)
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◮ Streaming code is a packet-level code that can correct from all AEP of DCSW channel within the decoding delay constraint τ. ◮ Non-trivial only if a ≤ b ≤ τ. ◮ Turns out WOLOG we can set w = τ + 1. ◮ This reduces parameter set from {a, b, w, τ} to {a, b, τ}. ◮ The optimal rate of an (a, b, τ) streaming code is given by 1: Ropt = τ + 1 − a τ + 1 − a + b .
1 Badr et al., “Layered Constructions for Low-Delay Streaming Codes”, Trans. IT, 2017. 5/28
Martinian- Sundberg [2004] Martinian- Trott [ISIT 2007] Badr et al. [2017] Krishnan- Shukla-Kumar [Jan 2019]
Burst-only Burst and Random Erasures
Near Rate-Optimal Rate-Optimal Rate-Optimal & Quadratic Field-size Domanovitz- Fong-Khisti [April 2019] Fong et al. [ISIT 2018] Krishnan- Kumar [ISIT 2018] Rate-Optimal, Explicit & Quadratic Field-size Krishnan et al. [Sept 2019] Simple Streaming Codes 6/28
◮ Decoding delay constraint τ = ⇒ for decoding i-th code symbol only till (i + τ)-th code symbol is accessible. ◮ Short code: block length n ≤ τ + 1
◮ All non-erased symbols can be used for decoding of any erased symbol.
◮ Long code: n > τ + 1
◮ Partial-knowledge decoding is needed for some symbols.
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
known inaccessible
𝜐 +1 = 5
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Encoder
time
message packets coded packet
. . . . . (1) ()
() ()
()
◮ Coded packet stream obtained by embedding code symbols of a scalar block code.
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◮ Codewords of [n, k] scalar block code are diagonally placed in the packet stream. ◮ This approach needs n − k ≥ b.
m1 m1 m1 m2 m2 m2 m3 m3 m3 m4 m4 m4 m5 m5 m5 m6 m6 m6 p1 p1 p1 p2 p2 p2 p3 p3 p3 p4 p4 p4 p5 p5 p5 p6 p6 p6 (0) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)(11)(12)(13)
DE of [12, 6] scalar code to get (a = 4, b = 6, τ = 9) streaming code ◮ Optimal rate code needs n > τ + 1, for a < b = ⇒ Long code ◮ Best known explicit construction requires field size > τ 2 and n = τ + 1 − a + b.
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◮ Codewords are embedded diagonally with gaps in the packet stream. ◮ This approach reduces the impact
◮ In the example, burst of 6 erasures results in erasure of atmost 4 consecutive code symbols.
◮ Dispersion span N ◮ N ≤ τ + 1 = ⇒ short code, N > τ + 1 = ⇒ long code
1 2 3 4 5 6 7 8 9 c0 c1 c2 c3 c4 c5 c6 c7 N=10 c0 c0 c0 c1 c1 c1 c2 c2 c2 c3 c3 c3 c4 c4 c4 c5 c5 c6 c6 c6 c6 c7 c7 c7
𝑦(0) 𝑦(1) 𝑦(2) 𝑦(3) 𝑦(4) 𝑦(5) 𝑦(6) 𝑦(7) 𝑦(8) 𝑦(9) 𝑦(10) 𝑦(11)
SDE of [n = 8, k = 4] MDS code for (a = 4, b = 6, τ = 9) streaming code Krishnan et al., “Simple streaming codes for reliable, low-latency communication”, Comm Letters, 2020.
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◮ Here N ≤ τ + 1 = ⇒ No partial-knowledge decoding needed ◮ Simple streaming codes are rate-optimal when gcd(b, τ + 1 − a
) = b. ◮ Pick [n = (m + 1)a, k = ma] MDS code
a b-a
a b-a a
◮ a code symbols are placed in a consecutive packets followed by (b − a) gaps.
◮ Burst of b erasure = ⇒ Erasure of a consecutive code symbols
◮ For SDE with N ≤ τ + 1, these codes are the only rate-optimal codes.
Krishnan et al., “Simple streaming codes for reliable, low-latency communication”, Comm Letters, 2020.
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◮ Enlarging the set of O(τ) field size rate-optimal streaming codes ◮ SDE Based Construction
◮ O(τ) field size rate-optimal (a, b, τ) streaming codes when gcd(b, τ + 1 − a) ∈ {a, a + 1, . . . , b − 1, b}
◮ DE based Constructions
◮ O(τ) field size rate-optimal (a, b, τ) streaming codes when
i) τ + 1 ≥ a + b and b(mod a) = 0 ii) τ + 1 ≥ a + b and b(mod a) = a − 1
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◮ N > τ + 1 = ⇒ partial-knowledge decoding is required
x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7) x(8) x(9) x(10) x(11) x(12) x(13) x(14) x(15) x(16)
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
1 2 3 4 5 6 7 8 9 10 11 12 13
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
SDE of [10, 6] scalar code with N = 14 to construct (a = 2, b = 6, τ = 10) streaming code
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c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
1 2 3 4 5 6 7 8 9 10 11 12 13
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
inaccessible
◮ Recover c0 in the presence of an additional random erasure or a burst of 4 erasures starting at c0, by accessing only till c7.
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
1 2 3 4 5 6 7 8 9 10 11 12 13
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
known
◮ Recover c2 in the presence of an additional random erasure or a burst of 4 erasures starting at c2, with c0, c1 known. Conditions for other ci are similar
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gcd( b
, τ + 1 − a
) = g ∈ {a, a + 1, . . . , b − 1, b} Ropt = τ + 1 − a τ + 1 − a + b = m m + ℓ ◮ We first construct an [n = (m + ℓ)a, k = ma] scalar code.
a g-a
a g-a a
◮ SDE of this code is done by putting a code symbols consecutively followed by (g − a) gaps until all code symbols are put together.
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Ia Z1 Z1 Z1 Z2 Z2 Z2 C (m-ℓ+1)a (ℓ-1)a ℓa ℓa a a
p-c matrix of the scalar code ◮ C is a Cauchy matrix. ◮ Every square sub-matrix of C is invertible. ◮ Z = [Z1 Z2] is a Zero-band (ZB) generator matrix of [2a, a] MDS code.
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* * * * * * * * * * * *
3 × 6 ZB matrix
◮ Field size q ≥ (m + 1)a = a(τ+1−a+g)
g
= O(τ). ◮ When g = b, we get simple streaming codes.
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◮ g = gcd(6, 9) = 3, ℓ = 2, m = 3
1 2 3 4 5 6 7 8 9 10 11 12 13
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
Placement pattern
◮ Burst erasure of length 6 = ⇒ Erasure of 4 consecutive code symbols
4 X 4 Cauchy matrix 2 X 4 ZB MDS gen matrix 20/28
◮ For recovery of c0 non-erased symbols only till c7 are accessible.
1 2 3 4 5 6 7 8 9 10 11 12 13
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
ˆ H = 1 2 3 4 5 6 7
c00 c01 c02 c03 1 c10 c11 c12 c13 ◮ Removing all-zero columns from ˆ H gives generator matrix of a [6, 2] MDS code. = ⇒ Random erasure recovery of c0. ◮ Column 0 of ˆ H is not in linear span of columns 1, 2 and 3. = ⇒ Burst erasure recovery of c0. ◮ Similar arguments work for c1.
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1 2 3 4 5 6 7 8 9 10 11 12 13
c0 c1 c2 c3 c4 c5 c6 c7 c8 c9
◮ For recovery of ci, i ∈ [2 : 9], all non-erased symbols can be accessed.
0 1 2 3 4 5 6 7 8 9
◮ Any two columns among last 8 columns of H are linearly independent = ⇒ Random erasure recovery of ci, i ∈ [2 : 9].
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◮ Any 4 consecutive columns among last 8 columns of H forms an invertible matrix. = ⇒ Burst erasure recovery of ci, i ∈ [2 : 9]. invertible invertible
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invertible invertible invertible
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Ia Z1 Z1 Z1 Z2 Z2 Z2 C
𝜐+1-b
b-a b b a a
p-c matrix ◮ If τ + 1 ≥ a + b and a|b, the DE of this scalar code results in rate-optimal streaming code, even when gcd(b, τ + 1 − a) < a. ◮ Field size q ≥ τ + 1 = O(τ)
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Z2
*
Ia-1 Z1 Z1 Z1 Z2 Z2 C
𝜐+1-b
b-a b b a-1 a a-1 a a-1 a-1
* p-c matrix of modified code Z ∗
2 = Z2(0 : a − 2, 0 : a − 2)
◮ When τ + 1 ≥ a + b and b = ℓa + a − 1, DE of this modified scalar code leads to rate-optimal streaming code. ◮ Field size q ≥ τ + 1 = O(τ)
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