minimum distance bounds for expander codes
TRANSCRIPT
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Minimum Distance Bounds
for Expander Codes
Vitaly SkachekClaude Shannon Institute
University College Dublin
Open Problems Session
Information Theory and Applications Workshop
UCSD
January 28, 2008
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Basic Definitions
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Basic Definitions
Definition
Code C is a set of words of length n over an alphabet Σ.
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Basic Definitions
Definition
Code C is a set of words of length n over an alphabet Σ.
Definition
The Hamming distance between x = (x1, . . . , xn) andy = (y1, . . . , yn) in Σn, d(x, y), is the number of pairs ofsymbols (xi, yi), 1 ≤ i ≤ n, such that xi 6= yi.
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Basic Definitions
Definition
Code C is a set of words of length n over an alphabet Σ.
Definition
The Hamming distance between x = (x1, . . . , xn) andy = (y1, . . . , yn) in Σn, d(x, y), is the number of pairs ofsymbols (xi, yi), 1 ≤ i ≤ n, such that xi 6= yi.
The minimum distance of a code C is
d = minx,y∈C,x 6=y
d(x, y).
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Basic Definitions
Definition
Code C is a set of words of length n over an alphabet Σ.
Definition
The Hamming distance between x = (x1, . . . , xn) andy = (y1, . . . , yn) in Σn, d(x, y), is the number of pairs ofsymbols (xi, yi), 1 ≤ i ≤ n, such that xi 6= yi.
The minimum distance of a code C is
d = minx,y∈C,x 6=y
d(x, y).
The relative minimum distance of C is defined as δ = d/n.
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Linear Code
Definition
A code C over field F = GF(q) is said to be a linear [n, k, d]code if there exists a matrix H with n columns and rankn − k such that
Hxt = 0 ⇔ x ∈ C.
The matrix H is a parity-check matrix.
The value k is the dimension of the code C.
The ratio r = k/n is the rate of the code C.
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Linear Code
Definition
A code C over field F = GF(q) is said to be a linear [n, k, d]code if there exists a matrix H with n columns and rankn − k such that
Hxt = 0 ⇔ x ∈ C.
The matrix H is a parity-check matrix.
The value k is the dimension of the code C.
The ratio r = k/n is the rate of the code C.
Definition
Let C be a code of minimum distance d over Σ.
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Linear Code
Definition
A code C over field F = GF(q) is said to be a linear [n, k, d]code if there exists a matrix H with n columns and rankn − k such that
Hxt = 0 ⇔ x ∈ C.
The matrix H is a parity-check matrix.
The value k is the dimension of the code C.
The ratio r = k/n is the rate of the code C.
Definition
Let C be a code of minimum distance d over Σ.
The unique decoding problem:
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Linear Code
Definition
A code C over field F = GF(q) is said to be a linear [n, k, d]code if there exists a matrix H with n columns and rankn − k such that
Hxt = 0 ⇔ x ∈ C.
The matrix H is a parity-check matrix.
The value k is the dimension of the code C.
The ratio r = k/n is the rate of the code C.
Definition
Let C be a code of minimum distance d over Σ.
The unique decoding problem:
Input: y ∈ Σn.
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Linear Code
Definition
A code C over field F = GF(q) is said to be a linear [n, k, d]code if there exists a matrix H with n columns and rankn − k such that
Hxt = 0 ⇔ x ∈ C.
The matrix H is a parity-check matrix.
The value k is the dimension of the code C.
The ratio r = k/n is the rate of the code C.
Definition
Let C be a code of minimum distance d over Σ.
The unique decoding problem:
Input: y ∈ Σn.Find: c ∈ C, such that d(c, y) < d/2.
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Gilbert-Varshamov Bound
Let Hq : [0, 1] → [0, 1] be the q-ary entropy function:
Hq(x) = x logq(q − 1) − x logq x − (1 − x) logq(1 − x) .
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Gilbert-Varshamov Bound
Let Hq : [0, 1] → [0, 1] be the q-ary entropy function:
Hq(x) = x logq(q − 1) − x logq x − (1 − x) logq(1 − x) .
Theorem
Let F = GF(q), and let δ ∈ (0, 1− 1/q] and R ∈ (0, 1), such that
R ≤ 1 − Hq(δ) .
Then, for large enough values of n, there exists a linear
[n,Rn,≥ δn] code over F.
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Gilbert-Varshamov Bound
Let Hq : [0, 1] → [0, 1] be the q-ary entropy function:
Hq(x) = x logq(q − 1) − x logq x − (1 − x) logq(1 − x) .
Theorem
Let F = GF(q), and let δ ∈ (0, 1− 1/q] and R ∈ (0, 1), such that
R ≤ 1 − Hq(δ) .
Then, for large enough values of n, there exists a linear
[n,Rn,≥ δn] code over F.
The above expression is called the Gilbert-Varshamov
bound.
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Gilbert-Varshamov Bound
Let Hq : [0, 1] → [0, 1] be the q-ary entropy function:
Hq(x) = x logq(q − 1) − x logq x − (1 − x) logq(1 − x) .
Theorem
Let F = GF(q), and let δ ∈ (0, 1− 1/q] and R ∈ (0, 1), such that
R ≤ 1 − Hq(δ) .
Then, for large enough values of n, there exists a linear
[n,Rn,≥ δn] code over F.
The above expression is called the Gilbert-Varshamov
bound.
Denote δGV (R) = H−12 (1 −R).
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Concatenated Codes
[Forney ’66] Ingredients:
A linear [∆, k=r∆, θ∆] code C over F = GF(q) (inner code).
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Concatenated Codes
[Forney ’66] Ingredients:
A linear [∆, k=r∆, θ∆] code C over F = GF(q) (inner code).
A linear [N, RΦN, δΦN ] code CΦ over Φ = Fk (outer code).
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Concatenated Codes
[Forney ’66] Ingredients:
A linear [∆, k=r∆, θ∆] code C over F = GF(q) (inner code).
A linear [N, RΦN, δΦN ] code CΦ over Φ = Fk (outer code).
A linear one-to-one mapping E : Φ → C.
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Concatenated Codes
[Forney ’66] Ingredients:
A linear [∆, k=r∆, θ∆] code C over F = GF(q) (inner code).
A linear [N, RΦN, δΦN ] code CΦ over Φ = Fk (outer code).
A linear one-to-one mapping E : Φ → C.
Concatenated code C of length N = ∆n over F is defined as
C ={
(c1|c2| · · · |cn) ∈ F∆n : ci = E(ai) ,
for i ∈ 1, 2, · · · , n, and (a1a2 · · · an) ∈ CΦ
}
.
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Concatenated Codes
[Forney ’66] Ingredients:
A linear [∆, k=r∆, θ∆] code C over F = GF(q) (inner code).
A linear [N, RΦN, δΦN ] code CΦ over Φ = Fk (outer code).
A linear one-to-one mapping E : Φ → C.
Concatenated code C of length N = ∆n over F is defined as
C ={
(c1|c2| · · · |cn) ∈ F∆n : ci = E(ai) ,
for i ∈ 1, 2, · · · , n, and (a1a2 · · · an) ∈ CΦ
}
.
The rate of C: R = rRΦ.
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Concatenated Codes
[Forney ’66] Ingredients:
A linear [∆, k=r∆, θ∆] code C over F = GF(q) (inner code).
A linear [N, RΦN, δΦN ] code CΦ over Φ = Fk (outer code).
A linear one-to-one mapping E : Φ → C.
Concatenated code C of length N = ∆n over F is defined as
C ={
(c1|c2| · · · |cn) ∈ F∆n : ci = E(ai) ,
for i ∈ 1, 2, · · · , n, and (a1a2 · · · an) ∈ CΦ
}
.
The rate of C: R = rRΦ.
The relative minimum distance of C: δ ≥ θδΦ.
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Concatenated Codes (Cont.)
Generalized minimum distance (GMD) decoder correctsany fraction of errors up to 1
2δ.
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Concatenated Codes (Cont.)
Generalized minimum distance (GMD) decoder correctsany fraction of errors up to 1
2δ.
[Justesen ’72] For a wide range of rates, concatenated codesattain the Zyablov bound:
δ ≥ maxR≤r≤1
(
1 − Rr
)
H−1q (1 − r).
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Concatenated Codes (Cont.)
Generalized minimum distance (GMD) decoder correctsany fraction of errors up to 1
2δ.
[Justesen ’72] For a wide range of rates, concatenated codesattain the Zyablov bound:
δ ≥ maxR≤r≤1
(
1 − Rr
)
H−1q (1 − r).
[Blokh-Zyablov ’82] Multilevel concatenations of codes(almost) attain the Blokh-Zyablov bound:
R = 1 − H2(δ) − δ
∫ 1−H2(δ)
0
dx
H−12 (1 − x)
.
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Graphs and Eigenvalues
Consider a ∆-regular graph G = (V, E).
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Graphs and Eigenvalues
Consider a ∆-regular graph G = (V, E).
The largest eigenvalue of the adjacency matrix AG of Gequals ∆.
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Graphs and Eigenvalues
Consider a ∆-regular graph G = (V, E).
The largest eigenvalue of the adjacency matrix AG of Gequals ∆.
Let λ∗G be the second largest absolute value of eigenvalues
of AG .
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Graphs and Eigenvalues
Consider a ∆-regular graph G = (V, E).
The largest eigenvalue of the adjacency matrix AG of Gequals ∆.
Let λ∗G be the second largest absolute value of eigenvalues
of AG .
Lower ratios of λ∗G/∆ imply greater values of expansion
[Alon ’86].
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Graphs and Eigenvalues
Consider a ∆-regular graph G = (V, E).
The largest eigenvalue of the adjacency matrix AG of Gequals ∆.
Let λ∗G be the second largest absolute value of eigenvalues
of AG .
Lower ratios of λ∗G/∆ imply greater values of expansion
[Alon ’86].
Expander graphs with
λ∗G ≤ 2
√∆ − 1
are called a Ramanujan graphs. Constructions are due to[Lubotsky Philips Sarnak ’88], [Margulis ’88].
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Graphs and Eigenvalues
Consider a ∆-regular graph G = (V, E).
The largest eigenvalue of the adjacency matrix AG of Gequals ∆.
Let λ∗G be the second largest absolute value of eigenvalues
of AG .
Lower ratios of λ∗G/∆ imply greater values of expansion
[Alon ’86].
Expander graphs with
λ∗G ≤ 2
√∆ − 1
are called a Ramanujan graphs. Constructions are due to[Lubotsky Philips Sarnak ’88], [Margulis ’88].
Let λG be the second largest eigenvalues of AG andγG = λG/∆.
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Barg-Zemor’s Expander Codes ’02
G is bipartite: V = A ∪ B,A ∩ B = ∅, |A| = |B| = n.
Ordering on the vertices and theedges.
Denote by (z)E(u) the sub-block ofz that is indexed by E(u).
Let CA and CB be two linearcodes of length ∆ over F.
Denote N = |E| = ∆n.
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Barg-Zemor’s Expander Codes ’02
G is bipartite: V = A ∪ B,A ∩ B = ∅, |A| = |B| = n.
Ordering on the vertices and theedges.
Denote by (z)E(u) the sub-block ofz that is indexed by E(u).
Let CA and CB be two linearcodes of length ∆ over F.
Denote N = |E| = ∆n.
The code C = (G, CA : CB):
C ={
c ∈ FN : (c)E(u) ∈ CA for v ∈ A
and (c)E(v) ∈ CB for u ∈ B}
.
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Barg-Zemor’s Expander Codes ’02
G is bipartite: V = A ∪ B,A ∩ B = ∅, |A| = |B| = n.
Ordering on the vertices and theedges.
Denote by (z)E(u) the sub-block ofz that is indexed by E(u).
Let CA and CB be two linearcodes of length ∆ over F.
Denote N = |E| = ∆n.
The code C = (G, CA : CB):
C ={
c ∈ FN : (c)E(u) ∈ CA for v ∈ A
and (c)E(v) ∈ CB for u ∈ B}
.
0
1
1
10
1
1
0
0CA
∆
CB
∆
vn
v2
v0
v1
un
u2
u0
u1
A B
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Barg-Zemor Expander Codes ’03
‘Dangling edges’ are introduced[Barg Zemor ’03].
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Barg-Zemor Expander Codes ’03
‘Dangling edges’ are introduced[Barg Zemor ’03].
Mimics behavior of concatenatedcodes.
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Barg-Zemor Expander Codes ’03
‘Dangling edges’ are introduced[Barg Zemor ’03].
Mimics behavior of concatenatedcodes.
Can be viewed as a concatenationof two codes [Roth Skachek ’04].
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Barg-Zemor Expander Codes ’03
‘Dangling edges’ are introduced[Barg Zemor ’03].
Mimics behavior of concatenatedcodes.
Can be viewed as a concatenationof two codes [Roth Skachek ’04].
Another construction with similarproperties [Guruswami Indyk ’02].
0
1
1
1
0
1
1
0
0
CA
∆
Cǫ
CB
∆1
vn
v2
v0
v1
un
u2
u0
u1
A B
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Analysis in [Barg Zemor ’04]
Analysis of the codes in [Barg Zemor ’02] and [Barg Zemor ’03].
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Analysis in [Barg Zemor ’04]
Analysis of the codes in [Barg Zemor ’02] and [Barg Zemor ’03].
Lower bounds on the relative minimum distance
(i)
δ(R) ≥ 1
4(1 −R)2 · min
δGV ((1+R)/2)<B<12
g(B)
H2(B),
where the function g(B) is defined in the next slides.
(ii)
δ(R) ≥ maxR≤r≤1
minδGV (r)<B<
12
(
δ0(B, r) · 1 −R/r
H2(B)
)
,
where the function δ0(B, r) is defined in the next slides.
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Definition of the Function g(B)
These two families of codes surpass the Zyablov bound.
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Definition of the Function g(B)
These two families of codes surpass the Zyablov bound.
Let δGV (R) = H−12 (1 −R), and let B1 be the largest root of the
equation
H2(B) = H2(B)
(
B − H2(B) · δGV (R)
1 −R
)
= − (B − δGV (R))·log2(1−B) .
Moreover, let
a1 =B1
H2(B1)− δGV (R)
H2(δGV (R)),
and
b1 =δGV (R)
H2(δGV (R))· B1 −
B1
H2(B1)· δGV (R)) .
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Definition of the Function g(B) (Cont.)
The function g(B) is defined as
g(B) =
δGV (R)
1 −R if B ≤ δGV (R)
B
H2(B)if δGV (R) ≤ B and R ≤ 0.284
a1B + b1
B1 − δGV (R)if δGV (R) ≤ B ≤ B1 and 0.284 < R ≤ 1
B
H2(B)if B1 < B1 ≤ 1 and 0.284 < R ≤ 1
.
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Definition of the Function δ0(B, r)
The function δ0(B, r) is defined to be ω⋆⋆(B) for δGV (r) ≤ B ≤ B1,where
ω⋆⋆(B) = rB + (1 − r)H−12
(
1 − r
1 − rH2(B)
)
,
and B1 is the only root of the equation
δGV (r) = w⋆(B) ,
where
w⋆(B) = (1−r)
(
(2H2(B)/B + 1)−1 +B
H2(B)
(
1 − H2
(
(2H2(B)/B + 1)−1))
)
.
For B1 ≤ B ≤ 12 , the function δ0(B, r) is defined to be a tangent to the
function ω⋆⋆(B) drawn from the point(
12 , ω⋆( 1
2 ))
.
Vitaly Skachek Minimum Distance Bounds
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Minimum Distance Bounds
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Code rate
Rel
ativ
e m
inim
um d
ista
nce
Comparison of Bounds
Zyablov bound Barg−Zemor bound 1 Barg−Zemor bound 2 Blokh−Zyablov bound Gilbert−Varshamov bound
Vitaly Skachek Minimum Distance Bounds
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Generalized Expander Codes
G = (V = A ∪ B, E) be a bipartite∆-regular, as before
Vitaly Skachek Minimum Distance Bounds
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Generalized Expander Codes
G = (V = A ∪ B, E) be a bipartite∆-regular, as before
B = B1 ∪ B2, B1 ∩ B2 = ∅. Let|B2| = ηn, |B1| = (1 − η)n,η ∈ [0, 1].
Vitaly Skachek Minimum Distance Bounds
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Generalized Expander Codes
G = (V = A ∪ B, E) be a bipartite∆-regular, as before
B = B1 ∪ B2, B1 ∩ B2 = ∅. Let|B2| = ηn, |B1| = (1 − η)n,η ∈ [0, 1].
CA, C1 and C2 are linear[∆, rA∆, δA∆], [∆, r1∆, δ1∆] and[∆, r2∆, δ2∆] codes over F,respectively.
Vitaly Skachek Minimum Distance Bounds
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Generalized Expander Codes
G = (V = A ∪ B, E) be a bipartite∆-regular, as before
B = B1 ∪ B2, B1 ∩ B2 = ∅. Let|B2| = ηn, |B1| = (1 − η)n,η ∈ [0, 1].
CA, C1 and C2 are linear[∆, rA∆, δA∆], [∆, r1∆, δ1∆] and[∆, r2∆, δ2∆] codes over F,respectively.
The code code C = (G, CA, C1, C2):
C ={
c ∈ FN : (c)E(u) ∈ CA for u ∈ A,
(c)E(u) ∈ C1 for u ∈ B1
and (c)E(u) ∈ C2 for u ∈ B2}
Vitaly Skachek Minimum Distance Bounds
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Generalized Expander Codes
G = (V = A ∪ B, E) be a bipartite∆-regular, as before
B = B1 ∪ B2, B1 ∩ B2 = ∅. Let|B2| = ηn, |B1| = (1 − η)n,η ∈ [0, 1].
CA, C1 and C2 are linear[∆, rA∆, δA∆], [∆, r1∆, δ1∆] and[∆, r2∆, δ2∆] codes over F,respectively.
The code code C = (G, CA, C1, C2):
C ={
c ∈ FN : (c)E(u) ∈ CA for u ∈ A,
(c)E(u) ∈ C1 for u ∈ B1
and (c)E(u) ∈ C2 for u ∈ B2}
0
0
1
1
1
1
1
0
0
0
1
0
CA
∆
C1
∆
C2
∆vn
v2
v0
v1
un
u2
u0
u1
A B
B1
B2
ηn
Vitaly Skachek Minimum Distance Bounds
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Properties of Generalized Expander Codes
The rate: R ≥ rA + (1 − η)r1 + ηr2 − 1.
Vitaly Skachek Minimum Distance Bounds
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Properties of Generalized Expander Codes
The rate: R ≥ rA + (1 − η)r1 + ηr2 − 1.
Assume
η <δA − γG
√
δA/δ2
1 − γG− γ
2/3G .
Then, the relative minimum distance:
δ > δA(δ1 − 12γ
2/3G ) .
⇒ The code C attains the Zyablov bound.
Vitaly Skachek Minimum Distance Bounds
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Properties of Generalized Expander Codes
The rate: R ≥ rA + (1 − η)r1 + ηr2 − 1.
Assume
η <δA − γG
√
δA/δ2
1 − γG− γ
2/3G .
Then, the relative minimum distance:
δ > δA(δ1 − 12γ
2/3G ) .
⇒ The code C attains the Zyablov bound.
A linear-time decoding algorithm: if δ1 > 2γ2/3G and η as
above, the decoder corrects any error pattern of size JC,
JC
△=
12δ1 − γ
2/3G
(
1 +
√
2(
δ1 − 2γ2/3G
)
)
1 − γG· δA∆n .
The number of correctable errors is (almost) half of theZyablov bound.
Vitaly Skachek Minimum Distance Bounds
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Properties of Generalized Expander Codes (cont.)
Theorem
Let |F| be a power of 2. There exists a polynomial-time
constructible family of binary linear codes C of length N = n∆,
n → ∞, and sufficiently large but constant ∆ = ∆(ε), whose
relative minimum distance satisfies
δ(R) ≥ maxR≤rA≤1
{
minδGV (rA)≤β≤1/2
(
δ0(β, rA)1 −R/rA
H2(β)
)}
− ε .
Vitaly Skachek Minimum Distance Bounds
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Properties of Generalized Expander Codes (cont.)
Theorem
Let |F| be a power of 2. There exists a polynomial-time
constructible family of binary linear codes C of length N = n∆,
n → ∞, and sufficiently large but constant ∆ = ∆(ε), whose
relative minimum distance satisfies
δ(R) ≥ maxR≤rA≤1
{
minδGV (rA)≤β≤1/2
(
δ0(β, rA)1 −R/rA
H2(β)
)}
− ε .
Consider a code C with parameter η = 0. Then, |B2| = 0, andthe code C coincides with the code in [Barg Zemor’02].
Vitaly Skachek Minimum Distance Bounds
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Properties of Generalized Expander Codes (cont.)
Theorem
Let |F| be a power of 2. There exists a polynomial-time
constructible family of binary linear codes C of length N = n∆,
n → ∞, and sufficiently large but constant ∆ = ∆(ε), whose
relative minimum distance satisfies
δ(R) ≥ maxR≤rA≤1
{
minδGV (rA)≤β≤1/2
(
δ0(β, rA)1 −R/rA
H2(β)
)}
− ε .
Consider a code C with parameter η = 0. Then, |B2| = 0, andthe code C coincides with the code in [Barg Zemor’02]. Theminimum distance:
δ(R) ≥ 1
4(1 −R)2 · min
δGV ((1+R)/2)<B<12
g(B)
H2(B).
Vitaly Skachek Minimum Distance Bounds
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Minimum Distance Bounds
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Code rate
Rel
ativ
e m
inim
um d
ista
nce
Comparison of Bounds
Zyablov bound Barg−Zemor bound 1 Barg−Zemor bound 2 Blokh−Zyablov bound Gilbert−Varshamov bound
Vitaly Skachek Minimum Distance Bounds
![Page 57: Minimum Distance Bounds for Expander Codes](https://reader034.vdocuments.net/reader034/viewer/2022052105/6286faebfbb7cf7623320c5c/html5/thumbnails/57.jpg)
Open Problems
Further improvements on the minimum distance
bounds.
Vitaly Skachek Minimum Distance Bounds
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Open Problems
Further improvements on the minimum distance
bounds.
Bounds on the error-correcting capabilities of thedecoders.
Vitaly Skachek Minimum Distance Bounds
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Open Problems
Further improvements on the minimum distance
bounds.
Bounds on the error-correcting capabilities of thedecoders.
Could other types of expander graphs yield betterproperties?
Vitaly Skachek Minimum Distance Bounds
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Open Problems
Further improvements on the minimum distance
bounds.
Bounds on the error-correcting capabilities of thedecoders.
Could other types of expander graphs yield betterproperties?
Do the generalized expander codes have any
advantage over the known expander codes?
Vitaly Skachek Minimum Distance Bounds