Improved bounds on the word error probability of RA(2) codes with - - PowerPoint PPT Presentation

Improved bounds on the word error probability of RA(2) codes with linear programming based decoding

Nissim Halabi Tel-Aviv University Joint work with Guy Even

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Outline

• “Turbo-like” codes.

– classic Turbo codes & “turbo-like” codes. – Repeat Accumulate codes.

• Auxiliary graphs and promenades.
• RALP decoding.
• Characterization of RALP failure.

– Non-positive cost minimal promenades. – Skeleton graphs and skeleton promenades.

• Algorithms for error bounds.
• Experimental results.

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• “Turbo-like” codes [Divsalar, Jin, McEliece, 1998]:

parallel, serial concatenated convolutional codes with interleavers.

Turbo Codes [Berrou, Glavieux, Thitimajashima, 1993]:

Information word Interleaver Convolution code 2 Convolution code 1 codeword

y x

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Repeat Accumulate Codes RA(q)

[Divsalar, Jin, McEliece, 1998] Information word: Repeat: Interleave: Accumulate (codeword): RA(2) code: 1 0 1 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0

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• Model for RA(2) codes [Bazzi et al. 2001].
• Undirected graph: path + matching.

– Vertices: codeword bits – Matching edges: interleaver; – Path edges: also called Hamiltonian edges.

• Theorem [BMMS01]: code distance = graph’s girth

(shortest cycle).

• Construct maximal distance RA(2) codes: cubic

graphs with girth Θ(logn) [Erdös & Sachs, 1963][BMMS01].

Auxiliary Graphs of RA(2) codes

e1 e2 e3 e4 e5 e6 e7 x1 x4 x2 x3 x4 x1 x3 x2

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Auxiliary Graphs of RA(2) codes (cont.)

• Error word ↦ costs to Hamiltonian edges [Feldman &

Karger, 2002].

– BSC: – AWGN channel:

• Matching edge: cost ≡ 0.

[ ]

   + − =

• therwise

1 channel by the flipped is bit if 1 i e c

i

[ ]

( )

2

where 1

N i i i

e c , Ν ~ ϕ ϕ + =

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Promenade [FK]

• A Promenade is a closed walk that does not traverse

an edge twice in a row.

• The cost of a promenade is the sum of the costs of the

edges traversed by the promenade.

• Infinitely many promenades.
• At least every second edge is Hamiltonian.

1 + 1 − 1 − 1 − 1 + 1 + 1 +

1 2 3 4

[ ]

1 − = M c

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Promenade [FK]

• A Promenade is a closed walk that does not traverse

an edge twice in a row.

• The cost of a promenade is the sum of the costs of the

edges traversed by the promenade.

• Infinitely many promenades.
• At least every second edge is Hamiltonian.

1 + 1 − 1 − 1 − 1 + 1 + 1 +

1 4 3 2

[ ]

3 + = M c

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Promenade [FK]

• A Promenade is a closed walk that does not traverse

an edge twice in a row.

• The cost of a promenade is the sum of the costs of the

edges traversed by the promenade.

• Infinitely many promenades.
• At least every second edge is Hamiltonian.

1 + 1 − 1 − 1 − 1 + 1 + 1 +

1 2 8 3 4 5 6 7 10 9

[ ]

= M c

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RALP decoding [FK]

• A provably polynomial time algorithm.
• Agrees with ML decoding; or outputs “error”.
• Theorem [FK02]: The RALP decoder succeeds if all

promenades have positive cost. The RALP decoder fails if there is a promenade with negative cost.

– Success: output the original information word.

• Theorem [FK02]:

– Specific, deterministically constructible codes. – Every code length.

( ) ( )

n poly fail 1 Pr ≤

{ }

[ ]

{ }

: promenade Pr Pr ≤ ∃ ≤ M c M fail

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Our Results

• New structural theorem that characterizes the event

that RALP fails.

• Present polynomial time algorithms that, given an

RA(2) code, compute upper and lower bounds on Pw.

• Experiments demonstrate an improvement for bounds
• n Pw.

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NPCM-Promenades

• Non-Positive Cost Minimal Promenade:

A promenade with:

– Non-positive cost – Minimal with respect to inclusion.

• Observation: ∃ NPCM-promenade ⇔ ∃ non-positive

cost promenade.

• The number of NPCM-promenades is finite.

1 + 1 − 1 − 1 − 1 + 1 + 1 −

1 2 3 4

[ ]

1 − = M c

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Skeleton Graphs and Promenades

• A skeleton graph has

the structure of a “tree

• f cycles”.

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Skeleton Graphs and Promenades

• A skeleton graph has the

structure of a “tree of cycles”.

• A skeleton promenade is

a closed Eulerian tour induced by a tree of cycles

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Skeleton Graphs and Promenades

• A skeleton graph has the

structure of a “tree of cycles”.

• A skeleton promenade is

a closed Eulerian tour induced by a tree of cycles

• A skeleton walk is a

sub-walk of a skeleton promenade.

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Characterization of RALP-failure

• characterization → bound
• g ≜ girth (=log n)
• Distinction between two types of promenades:

– Short promenades: length < 2g + 2 → Pshort – Long promenades: length ≥ 2g + 2 → Plong

Theorem: Every NPCM-promenade is a skeleton promenade.

{ } [ ] { }

& promenade skeleton Pr Pr ≤ ∃ ≤ M c M fail

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• Promenade is simple ⇒
• Chernoff bounds ⇒

– Long promenades are “easy” – Short promenades are “hard” Problem: repetitions (dependency).

Short and Long promenades - Intuition

[ ]

{ }

1 1 Pr << = − = p e c

i

[ ]

{ } ( )

2 1 . .

2 1

>> − ⋅ ≥ p prom prom c E

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Short NPCM-Promenades (length < 2g+2)

• Few Hamiltonian edges; Few errors ⇒ non-positive

cost

• Claim: every short NPCM-promenade is a simple

cycle, namely:

• For a cycle C with h Hamiltonian edges:

[ ]

{ }

: cycle short simple Pr ≤ ∃ = C c C P

short

[ ]

{ } ( )

 

∑

= −

−       = ≤

h i i h i

h

p p i h C c

2

1 Pr

Majority of Hamiltonian edges are negative.

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• One can enumerate all short simple cycles in

polynomial time.

• Lower bound:

– Consider cycles with fewest Hamiltonian edges. – Deal with intersections of cycles: Compute using inclusion-exclusion principle.

Short NPCM-Promenades (Cont.)

depth = log n → 2log n = n paths

[ ] { }

: cycle Pr ≤ ∃ C c C

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Long NPCM-Promenades (length ≥ 2g+2)

• Lemma: If there exists a long NPCM-promenade M,

then there exists a non-positive cost skeleton walk that contains g + 1 Hamiltonian edges (with repetitions).

• Computed similarly to the tree-bound of Feldman et
• al. [FKW02].

[ ]

( ) { }

1 & : alk W skeleton w Pr + = ≤ ∃ ≤ g W ham W c P

long

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Experimental Results

• 5
• 4
• 3
• 2

p

• 12.52
• 9.51
• 6.39
• 1.42

skeleton-bound

• 12.52
• 9.52
• 6.52
• 3.53

Pw Lower Bound

• 17.53
• 12.49
• 7.19
• 1.43

Plong Upper Bound

• 12.52
• 9.51
• 6.47
• 3.13

Pshort Upper Bound

• 8.75
• 5.75
• 2.75

No Bound Tree-bound [FKW02]

n = 1024, g = 10 ; values in log scale (log10)

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• NPCM-promenades characterization → Improve

previous bounds by ~×1000.

Experimental Results

• 5
• 4
• 3
• 2

p

• 12.52
• 9.51
• 6.39
• 1.42

skeleton-bound

• 12.52
• 9.52
• 6.52
• 3.53

Pw Lower Bound

• 17.53
• 12.49
• 7.19
• 1.43

Plong Upper Bound

• 12.52
• 9.51
• 6.47
• 3.13

Pshort Upper Bound

• 8.75
• 5.75
• 2.75

No Bound Tree-bound [FKW02]

n = 1024, g = 10 ; values in log scale (log10)

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• Upper and lower bounds are close ( p → 0) .

Experimental Results

• 5
• 4
• 3
• 2

p

• 12.52
• 9.51
• 6.39
• 1.42

skeleton-bound

• 12.52
• 9.52
• 6.52
• 3.53

Pw Lower Bound

• 17.53
• 12.49
• 7.19
• 1.43

Plong Upper Bound

• 12.52
• 9.51
• 6.47
• 3.13

Pshort Upper Bound

• 8.75
• 5.75
• 2.75

No Bound Tree-bound [FKW02]

n = 1024, g =10 ; values in log scale (log10)

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Experimental Results

• 5
• 4
• 3
• 2

p

• 12.52
• 9.51
• 6.39
• 1.42

skeleton-bound

• 12.52
• 9.52
• 6.52
• 3.53

Pw Lower Bound

• 17.53
• 12.49
• 7.19
• 1.43

Plong Upper Bound

• 12.52
• 9.51
• 6.47
• 3.13

Pshort Upper Bound

• 8.75
• 5.75
• 2.75

No Bound Tree-bound [FKW02]

n = 1024, g =10 ; values in log scale (log10)

• Short promenades determine Pw. (

)

w p short

P P   →  →0

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Experimental Results

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Experimental Results

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Experimental Results

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Experimental Results

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Experimental Results

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Conclusion

• New characterization of RALP decoding failure.
• Efficient algorithms for computing upper- & lower-

bounds on Pw.

• Experimental results:

– Pw smaller by ~×1000. – Lower bound close to upper bound.

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Open problems

• Bound for specific RA(3) codes.
• Coding theorem for RA(3).

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Experimental Results

• Applying universal bounds to specific RA(2) codes →

Minor improvement.

Experimental Results

• 5
• 4
• 3
• 2

p

• 12.52
• 9.51
• 6.39
• 1.42

skeleton-bound

• 12.52
• 9.52
• 6.52
• 3.53

Pw Lower Bound

• 17.53
• 12.49
• 7.19
• 1.43

Plong Upper Bound

• 12.52
• 9.51
• 6.47
• 3.13

Pshort Upper Bound

• 8.75
• 5.75
• 2.75

No Bound Tree-bound [FKW02]

• 8.93
• 5.93
• 2.93

No Bound exp-tree-bound

n = 1024, g = 10 ; values in log scale (log10)

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Experimental Results