Syndrome-Coupled Rate-Compatible Error-Correcting Codes for Flash - - PowerPoint PPT Presentation
Syndrome-Coupled Rate-Compatible Error-Correcting Codes for Flash Memories Pengfei Huang 1 , Yi Liu 1 , Xiaojie Zhang 2 , Paul H. Siegel 1 , Erich F. Haratsch 3 1 Center for Memory and Recording Research, UCSD 2 CNEX Labs, San Jose, CA 3 Seagate
Pengfei Huang1, Yi Liu1, Xiaojie Zhang2, Paul H. Siegel1, Erich F. Haratsch3
1 Center for Memory and Recording Research, UCSD 2 CNEX Labs, San Jose, CA 3 Seagate Technology, CA
NVMW, Mar. 12, 2018
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1
Introduction and Background
2
Syndrome-Coupled Code Construction
3
Correctable Patterns and Decoding
4
Capacity-Achieving Rate-Compatible Codes
5
Applications to MLC Flash Memories
6
Summary
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Flash memory channels degrade with use.
2000 4000 6000 8000 10000 Program/Erase (P/E) Cycle Count 10-6 10-5 10-4 10-3 10-2 Bit Error Rate (BER) Average RBER Lower Page RBER Upper Page RBER
Stronger error correction capability is needed as wear increases. We propose a general code construction to address this need: syndrome-coupled rate-compatible error-correcting codes
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Rate-compatible codes: a family of extended codes {Ci}i=1,...,M. Code Ci+1 obtained from code Ci by adding redundancy symbols. Decreasing rate, but increasing error-correcting capability. Example: a family of 3-level rate-compatible codes.
Codes: C1 ≺ C2 ≺ C3. Rates: R1 > R2 > R3.
c1 c2 c3 r2 r3
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Rate-compatible codes: a family of extended codes {Ci}i=1,...,M. Code Ci+1 obtained from code Ci by adding redundancy symbols. Decreasing rate, but increasing error-correcting capability. Example: a family of 3-level rate-compatible codes.
Codes: C1 ≺ C2 ≺ C3. Rates: R1 > R2 > R3.
c1 c2 c3 r2 r3
Challenge: How to systematically add these redundancy symbols.
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1
Introduction and Background
2
Syndrome-Coupled Code Construction
3
Correctable Patterns and Decoding
4
Capacity-Achieving Rate-Compatible Codes
5
Applications to MLC Flash Memories
6
Summary
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Example: a family of 3-level rate-compatible codes C1 ≺ C2 ≺ C3.
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Example: a family of 3-level rate-compatible codes C1 ≺ C2 ≺ C3. 1) Choose nested codes C3
1 ⊂ C2 1 ⊂ C1 1 = C1 = [n1, n1 − v1, d1]q.
Ci
1 = [n1, n1 − i m=1 vm, di]q for 1 ≤ i ≤ 3, d1 ≤ d2 ≤ d3.
Parity-check matrices of C1
1, C2 1, and C3 1.
HC1
1
HC2
1 =
1
HC2
1|C1 1
1 =
HC1
1
HC2
1|C1 1
HC3
1|C2 1
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Example: a family of 3-level rate-compatible codes C1 ≺ C2 ≺ C3. 1) Choose nested codes C3
1 ⊂ C2 1 ⊂ C1 1 = C1 = [n1, n1 − v1, d1]q.
Ci
1 = [n1, n1 − i m=1 vm, di]q for 1 ≤ i ≤ 3, d1 ≤ d2 ≤ d3.
Parity-check matrices of C1
1, C2 1, and C3 1.
HC1
1
HC2
1 =
1
HC2
1|C1 1
1 =
HC1
1
HC2
1|C1 1
HC3
1|C2 1
2) Choose two auxiliary nested codes A3
2 ⊂ A2 2.
A3
2 = [n2, v2 − λ3 2]q and A2 2 = [n2, v2]q.
Parity-check matrices of A2
2 and A3 2.
HA2
2
HA3
2 =
2
HA3
2|A2 2
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Example: a family of 3-level rate-compatible codes C1 ≺ C2 ≺ C3. 1) Choose nested codes C3
1 ⊂ C2 1 ⊂ C1 1 = C1 = [n1, n1 − v1, d1]q.
Ci
1 = [n1, n1 − i m=1 vm, di]q for 1 ≤ i ≤ 3, d1 ≤ d2 ≤ d3.
Parity-check matrices of C1
1, C2 1, and C3 1.
HC1
1
HC2
1 =
1
HC2
1|C1 1
1 =
HC1
1
HC2
1|C1 1
HC3
1|C2 1
2) Choose two auxiliary nested codes A3
2 ⊂ A2 2.
A3
2 = [n2, v2 − λ3 2]q and A2 2 = [n2, v2]q.
Parity-check matrices of A2
2 and A3 2.
HA2
2
HA3
2 =
2
HA3
2|A2 2
3 = [n3, v3 + λ3 2]q.
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Given: Length-k information vector ✉ over Fq. ❝ ✉ s ❝ s ❝ ❛ s s ❝ ❝ ❛ ❝ ❝ s ❝ s ❝ ❛ ❛ ❛ s s ❝ ❝ ❛ ❛ ❝ ❝
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Given: Length-k information vector ✉ over Fq. 1) Compute codeword in C1 ❝1 = EC1(✉) ∈ C1. s ❝ s ❝ ❛ s s ❝ ❝ ❛ ❝ ❝ s ❝ s ❝ ❛ ❛ ❛ s s ❝ ❝ ❛ ❛ ❝ ❝
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Given: Length-k information vector ✉ over Fq. 1) Compute codeword in C1 ❝1 = EC1(✉) ∈ C1. 2) Compute extra redundancy for C2 s2 = ❝1HT
C2
1|C1 1
(compute syndrome s2 of ❝1) ❛2
2 = EA2
2(s2)
(encode syndrome s2) ❝2 = (❝1, ❛2
2)
(append to ❝1 to get ❝2 ∈ C2) s ❝ s ❝ ❛ ❛ ❛ s s ❝ ❝ ❛ ❛ ❝ ❝
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Given: Length-k information vector ✉ over Fq. 1) Compute codeword in C1 ❝1 = EC1(✉) ∈ C1. 2) Compute extra redundancy for C2 s2 = ❝1HT
C2
1|C1 1
(compute syndrome s2 of ❝1) ❛2
2 = EA2
2(s2)
(encode syndrome s2) ❝2 = (❝1, ❛2
2)
(append to ❝1 to get ❝2 ∈ C2) 3) Compute extra redundancy for C3 s3 = ❝1HT
C3
1|C2 1
(compute syndrome s3 of ❝1) Λ3
2 = ❛2 2HT A3
2|A2 2
(compute syndrome Λ3
2 of ❛2 2)
❛3
3 = EA3
3(s3, Λ3
2) (encode coupled syndrome s3, Λ3 2 )
❝3 = (❝1, ❛2
2, ❛3 3) (append to ❝2 to get ❝3 ∈ C3)
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Syndrome-coupled encoding: ❝1 = EC1(✉) ❝2 = (❝1, ❛2
2) =
2(s2)
= (❝1, ❛2
2, ❛3 3) =
2(s2), EA3 3(s3, Λ3
2)
c2 c3 r2 r3
Key idea: generate and encode syndromes progressively. Generalizes to any number of levels. Applicable to many codes, e.g., BCH, RS, LDPC, and Polar.
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Theorem
Assume the auxiliary codes in the construction have minimum distances: dmin(A2
2)
= d2 − d1 dmin(A3
2) = d3 − d1
dmin(A3
3)
= d3 − d2 Then, the syndrome-coupled codes Ci, 1 ≤ i ≤ 3, have length Ni = i
j=1 nj, dimension k = n1 − v1, and minimum distance di.
C1
1 = [n1, n1 − v1, d1]
C2
1 = [n1, n1 − 2 i=1 vi, d2]
A2
2 = [n2, v2, d2 − d1]
C3
1 = [n1, n1 − 3 i=1 vi, d3]
A3
2 = [n2, v2 − λ3 2, d3 − d1]
A3
3 = [n3, v3 + λ3 2, d3 − d2]
C1 = [n1, k, d1]q ≺ C2 = [n1 + n2, k, d2]q ≺ C3 = [n1 + n2 + n3, k, d3]q.
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1
Introduction and Background
2
Syndrome-Coupled Code Construction
3
Correctable Patterns and Decoding
4
Capacity-Achieving Rate-Compatible Codes
5
Applications to MLC Flash Memories
6
Summary
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C1 = [n1, k, d1]q ≺ C2 = [n1 + n2, k, d2]q ≺ C3 = [n1 + n2 + n3, k, d3]q. ❝ ❝ ❛ ❛ ✉ s s ② ② ② ② ② ② ②
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C1 = [n1, k, d1]q ≺ C2 = [n1 + n2, k, d2]q ≺ C3 = [n1 + n2 + n3, k, d3]q. Transmit: ❝3 = (❝1, ❛2
2, ❛3 3) =
2(s2), EA3 3(s3, Λ3
2)
Receive: ② = (② 1, ② 2, ② 3), ② ∈ (Fq ∪ {?})n1+n2+n3. ② ②
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C1 = [n1, k, d1]q ≺ C2 = [n1 + n2, k, d2]q ≺ C3 = [n1 + n2 + n3, k, d3]q. Transmit: ❝3 = (❝1, ❛2
2, ❛3 3) =
2(s2), EA3 3(s3, Λ3
2)
Receive: ② = (② 1, ② 2, ② 3), ② ∈ (Fq ∪ {?})n1+n2+n3. For 1 ≤ i ≤ 3, let ti and τi denote the number of errors and erasures in the sub-block ② i of the received word ②.
Theorem
The code C3 can correct any combined error and erasure pattern that satisfies the following conditions: 2t1 + τ1 ≤d3 − 1, 2t2 + τ2 ≤d3 − d1 − 1, 2t3 + τ3 ≤d3 − d2 − 1.
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Transmit: ❝3 = (❝1, ❛2
2, ❛3 3) =
2(s2), EA3 3(s3, Λ3
2)
Receive: ② = (② 1, ② 2, ② 3). ❛ ② s ❛ ❛ ❛ ② s ❛ s s ❝ ❝ ② s s ✉ ❝ ✉
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Transmit: ❝3 = (❝1, ❛2
2, ❛3 3) =
2(s2), EA3 3(s3, Λ3
2)
Receive: ② = (② 1, ② 2, ② 3). 1) ❛3
3 = DA3
3(② 3)
( s3, Λ3
2) = E−1 A3
3 (
❛3
3)
2 determines the coset of A3 2 ⊂ A2 2 that ❛2 2 lies in.
❛ ② s ❛ s s ❝ ❝ ② s s ✉ ❝ ✉
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Transmit: ❝3 = (❝1, ❛2
2, ❛3 3) =
2(s2), EA3 3(s3, Λ3
2)
Receive: ② = (② 1, ② 2, ② 3). 1) ❛3
3 = DA3
3(② 3)
( s3, Λ3
2) = E−1 A3
3 (
❛3
3)
2 determines the coset of A3 2 ⊂ A2 2 that ❛2 2 lies in.
2) ❛2
2 = DA3
2(② 2|
Λ3
2)
A2
2 (
❛2
2)
( s2, s3) determines the coset of C3
1 ⊂ C1 1 = C1 that ❝1 lies in.
❝ ② s s ✉ ❝ ✉
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Transmit: ❝3 = (❝1, ❛2
2, ❛3 3) =
2(s2), EA3 3(s3, Λ3
2)
Receive: ② = (② 1, ② 2, ② 3). 1) ❛3
3 = DA3
3(② 3)
( s3, Λ3
2) = E−1 A3
3 (
❛3
3)
2 determines the coset of A3 2 ⊂ A2 2 that ❛2 2 lies in.
2) ❛2
2 = DA3
2(② 2|
Λ3
2)
A2
2 (
❛2
2)
( s2, s3) determines the coset of C3
1 ⊂ C1 1 = C1 that ❝1 lies in.
3) ❝1 = DC3
1(② 1|
s2, s3) ˆ ✉ = E−1
C1
1 (
❝1) ✉
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Transmit: ❝3 = (❝1, ❛2
2, ❛3 3) =
2(s2), EA3 3(s3, Λ3
2)
Receive: ② = (② 1, ② 2, ② 3). 1) ❛3
3 = DA3
3(② 3)
( s3, Λ3
2) = E−1 A3
3 (
❛3
3)
2 determines the coset of A3 2 ⊂ A2 2 that ❛2 2 lies in.
2) ❛2
2 = DA3
2(② 2|
Λ3
2)
A2
2 (
❛2
2)
( s2, s3) determines the coset of C3
1 ⊂ C1 1 = C1 that ❝1 lies in.
3) ❝1 = DC3
1(② 1|
s2, s3) ˆ ✉ = E−1
C1
1 (
❝1) Output: ˆ ✉.
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1
Introduction and Background
2
Syndrome-Coupled Code Construction
3
Correctable Patterns and Decoding
4
Capacity-Achieving Rate-Compatible Codes
5
Applications to MLC Flash Memories
6
Summary
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Goal: A family of rate-compatible codes that achieve the capacities of a set of degraded q-ary symmetric channels simultaneously.
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Goal: A family of rate-compatible codes that achieve the capacities of a set of degraded q-ary symmetric channels simultaneously.
Channels W1 ≻ W2 ≻ W3, capacities C(W1) > C(W2) > C(W3). Any rates R1 > R2 > R3 such that Ri < C(Wi). Rate-compatible codes C1 ≺ C2 ≺ C3, where Ci has rate Ri. For code Ci over Wi, error probability P(Ni)
e
(Ci) → 0, as Ni → ∞.
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Goal: A family of rate-compatible codes that achieve the capacities of a set of degraded q-ary symmetric channels simultaneously.
Channels W1 ≻ W2 ≻ W3, capacities C(W1) > C(W2) > C(W3). Any rates R1 > R2 > R3 such that Ri < C(Wi). Rate-compatible codes C1 ≺ C2 ≺ C3, where Ci has rate Ri. For code Ci over Wi, error probability P(Ni)
e
(Ci) → 0, as Ni → ∞.
Approach: Design syndrome-coupled rate-compatible codes using nested capacity-achieving codes as component codes.
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Goal: A family of rate-compatible codes that achieve the capacities of a set of degraded q-ary symmetric channels simultaneously.
Channels W1 ≻ W2 ≻ W3, capacities C(W1) > C(W2) > C(W3). Any rates R1 > R2 > R3 such that Ri < C(Wi). Rate-compatible codes C1 ≺ C2 ≺ C3, where Ci has rate Ri. For code Ci over Wi, error probability P(Ni)
e
(Ci) → 0, as Ni → ∞.
Approach: Design syndrome-coupled rate-compatible codes using nested capacity-achieving codes as component codes.
Lemma
Consider a set of M degraded q-ary symmetric channels W1 ≻ W2 ≻ · · · ≻ WM. For any rates R1 > R2 > · · · > RM such that Ri < C(Wi), there exists a sequence
1 = [n, kM = RMn]q ⊂ CM−1 1
= [n, kM−1 = RM−1n]q ⊂ · · · ⊂ C1
1 = [n, k1 = R1n]q such that the error probability of Ci 1 over Wi, under
nearest-codeword (ML) decoding, satisfies P(n)
e (Ci 1) → 0, as n goes to infinity.
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Goal: A family of rate-compatible codes that achieve the capacities of a set of degraded q-ary symmetric channels simultaneously.
Channels W1 ≻ W2 ≻ W3, capacities C(W1) > C(W2) > C(W3). Any rates R1 > R2 > R3 such that Ri < C(Wi). Rate-compatible codes C1 ≺ C2 ≺ C3, where Ci has rate Ri. For code Ci over Wi, error probability P(Ni)
e
(Ci) → 0, as Ni → ∞.
Approach: Design syndrome-coupled rate-compatible codes using nested capacity-achieving codes as component codes.
Lemma
Consider a set of M degraded q-ary symmetric channels W1 ≻ W2 ≻ · · · ≻ WM. For any rates R1 > R2 > · · · > RM such that Ri < C(Wi), there exists a sequence
1 = [n, kM = RMn]q ⊂ CM−1 1
= [n, kM−1 = RM−1n]q ⊂ · · · ⊂ C1
1 = [n, k1 = R1n]q such that the error probability of Ci 1 over Wi, under
nearest-codeword (ML) decoding, satisfies P(n)
e (Ci 1) → 0, as n goes to infinity.
Examples: Polar codes on DMC; BCH, Reed-Muller codes on BEC.
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Rates of component codes must satisfy specified relations ❝ ❝ ❛ ❛ ✉ s s ② ② ② ② ② ② ②
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Rates of component codes must satisfy specified relations Component codes (capacity-achieving with specified rates)
C1
1 = [n1, n1 − v1], R1
C2
1 = [n1, n1 − 2 i=1 vi], R2
A2
2 = [n2, v2], R2
C3
1 = [n1, n1 − 3 i=1 vi], R3
A3
2 = [n2, v2 − λ3 2], R3
A3
3 = [n3, v3 + λ3 2], R3
❝ ❝ ❛ ❛ ✉ s s ② ② ② ② ② ② ②
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Rates of component codes must satisfy specified relations Component codes (capacity-achieving with specified rates)
C1
1 = [n1, n1 − v1], R1
C2
1 = [n1, n1 − 2 i=1 vi], R2
A2
2 = [n2, v2], R2
C3
1 = [n1, n1 − 3 i=1 vi], R3
A3
2 = [n2, v2 − λ3 2], R3
A3
3 = [n3, v3 + λ3 2], R3
Rate-compatible code rates
C1 = [n1, k]q (R1) ≺ C2 = [n1 + n2, k]q (R2) ≺ C3 = [n1 + n2 + n3, k]q (R3).
❝ ❝ ❛ ❛ ✉ s s ② ② ② ② ② ② ②
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Rates of component codes must satisfy specified relations Component codes (capacity-achieving with specified rates)
C1
1 = [n1, n1 − v1], R1
C2
1 = [n1, n1 − 2 i=1 vi], R2
A2
2 = [n2, v2], R2
C3
1 = [n1, n1 − 3 i=1 vi], R3
A3
2 = [n2, v2 − λ3 2], R3
A3
3 = [n3, v3 + λ3 2], R3
Rate-compatible code rates
C1 = [n1, k]q (R1) ≺ C2 = [n1 + n2, k]q (R2) ≺ C3 = [n1 + n2 + n3, k]q (R3).
Decoding error probability over W3 Transmit: ❝3 = (❝1, ❛2
2, ❛3 3) =
2(s2), EA3 3(s3, Λ3
2)
Receive: ② = (② 1, ② 2, ② 3). Decoding: ② 3 → ② 2 → ② 1.
P(N3)
e
(C3) ≤1 −
e
(C3
1)
e
(A3
2)
e
(A3
3)
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1
Introduction and Background
2
Syndrome-Coupled Code Construction
3
Correctable Patterns and Decoding
4
Capacity-Achieving Rate-Compatible Codes
5
Applications to MLC Flash Memories
6
Summary
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BCH component codes C2
1 = [8191, 7398]2 (t = 61) ⊂ C1 1 = [8191, 7697]2 (t = 38)
A2
2 = [398, 299]2 (t = 11)
❝ ❝ ❝ ❛
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BCH component codes C2
1 = [8191, 7398]2 (t = 61) ⊂ C1 1 = [8191, 7697]2 (t = 38)
A2
2 = [398, 299]2 (t = 11)
2-level rate-compatible codes C1 = [8191, k = 7697]2 ≺ C2 = [8589, k = 7697]2 R1 = 0.9397 > R2 = 0.8961
❝ ❝ ❝ ❛
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BCH component codes C2
1 = [8191, 7398]2 (t = 61) ⊂ C1 1 = [8191, 7697]2 (t = 38)
A2
2 = [398, 299]2 (t = 11)
2-level rate-compatible codes C1 = [8191, k = 7697]2 ≺ C2 = [8589, k = 7697]2 R1 = 0.9397 > R2 = 0.8961 Correctable error patterns
❝1 ∈ C1. Corrects 38 errors. ❝2 = (❝1, ❛2
2) ∈ C2. First sub-block (8191 bits) corrects 61 errors.
Second sub-block (398 bits) corrects 11 errors.
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BCH component codes C2
1 = [8191, 7398]2 (t = 61) ⊂ C1 1 = [8191, 7697]2 (t = 38)
A2
2 = [398, 299]2 (t = 11)
2-level rate-compatible codes C1 = [8191, k = 7697]2 ≺ C2 = [8589, k = 7697]2 R1 = 0.9397 > R2 = 0.8961 Correctable error patterns
❝1 ∈ C1. Corrects 38 errors. ❝2 = (❝1, ❛2
2) ∈ C2. First sub-block (8191 bits) corrects 61 errors.
Second sub-block (398 bits) corrects 11 errors.
For comparison, we use a shortened BCH code C3 = [8593, 7697]2 (t = 64), whose code length and rate are similar to those of C2.
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4000 5000 6000 7000 8000 9000 10000 10
−4
10
−3
10
−2
10
−1
10 Program/Erase (P/E) Cycle Count FER
[8191, 7697], r=0.9397 [8589, 7697], r=0.8961 [8593, 7697], r=0.8957
4000 5000 6000 7000 8000 9000 10000 10
−4
10
−3
10
−2
10
−1
10 Program/Erase (P/E) Cycle Count FER
[8191, 7697], r=0.9397 [8589, 7697], r=0.8961 [8593, 7697], r=0.8957
Left: lower page. The code C2 extends lifetime around 3500 cycles. Right: upper page. The code C2 extends lifetime around 2000 cycles. The code C2 is comparable to the shortened BCH code C3.
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LDPC component codes
Use Reed-Solomon based construction1. Nested and 4-cycle free structure. C2
1 ⊂ C1 1 are two nested binary LDPC codes.
C1
1 is a (4, 64)-regular [8192,7697] LDPC code, r = 0.9396.
C2
1 is a (7, 64)-regular [8192,7400] LDPC code, r = 0.9033.
The auxiliary code A2
2 is a (4, 15)-regular [405,300] LDPC code.
2-level rate-compatible codes C1 = [8192, k = 7697]2 ≺ C2 = [8597, k = 7697]2 R1 = 0.9396 > R2 = 0.8953
1Ryan and Lin, Channel Codes: Classical and Modern, Cambridge Univ. Press, 2009. Huang et al. Syndrome-Coupled Rate-Compatible Codes
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10 Program/Erase (P/E) Cycle Count FER
[8192,7697], r=0.9396 [8597,7697], r=0.8953
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10 Program/Erase (P/E) Cycle Count FER
[8192,7697], r=0.9396 [8597,7697], r=0.8953
Left: lower page. The code C2 extends lifetime around 4000 cycles. Right: upper page. The code C2 extends lifetime around 3000 cycles.
Huang et al. Syndrome-Coupled Rate-Compatible Codes
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10 Program/Erase (P/E) Cycle Count
FER
[8192,7697], r=0.9396, LDPC [8597,7697], r=0.8953, LDPC−RC [8191, 7697], r=0.9397, BCH [8589, 7697], r=0.8961, BCH−RC [8593, 7697], r=0.8957, S−BCH
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10 Program/Erase (P/E) Cycle Count
FER
[8192,7697], r=0.9396, LDPC [8597,7697], r=0.8953, LDPC−RC [8191, 7697], r=0.9397, BCH [8589, 7697], r=0.8961, BCH−RC [8593, 7697], r=0.8957, S−BCH
At a higher rate of 0.94, the BCH code outperforms the LDPC code. The opposite conclusion holds for a lower rate of 0.90.
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1
Introduction and Background
2
Syndrome-Coupled Code Construction
3
Correctable Patterns and Decoding
4
Capacity-Achieving Rate-Compatible Codes
5
Applications to MLC Flash Memories
6
Summary
Huang et al. Syndrome-Coupled Rate-Compatible Codes
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Introduced rate-compatible error-correcting codes. Developed a new, general algebraic construction based upon syndrome-coupled encoding, with a sequential decoding algorithm. Studied the minimum distance properties and correctable error patterns of the codes. Proved the capacity-achieving property of our construction and equivalence to recent capacity-achieving polar code schemes. Applied 2-level rate-compatible codes to MLC flash memories.
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Huang et al. Syndrome-Coupled Rate-Compatible Codes
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