Multipermutation Codes in the Ulam Metricffh8x/d/t/Ulam-Multiperm-ISIT2014.pdfMultipermutation Codes...
Transcript of Multipermutation Codes in the Ulam Metricffh8x/d/t/Ulam-Multiperm-ISIT2014.pdfMultipermutation Codes...
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Multipermutation Codes in the Ulam Metric
Farzad Farnoud† Olgica Milenkovic∗
†California Institute of Technology1
∗University of Illinois at Urbana-Champaign
ISIT 2014, Honolulu
1Farzad Farnoud was with the University of Illinois at Urbana-Champaign.Farnoud, Milenkovic Ulam Multipermutation Codes 1
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Summary
Novel rank modulation multipermutation codes (MPCs) inUlam metric
Codes correcting translocation and deletion errors
Highlight connection between MPCs in the Ulam andHamming metrics
Capacities or bounds for MPCs in both metrics
Constructions using Steiner systems, BIBDs, and interleaving
Efficienct decoding algorithms
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Rank Modulation for Flash Memory
Rank modulation for flash memory was proposed by Jiang etal. [2008] for dealing with over-injection and charge leakage.
In an array of cells, each cell stores a charge level.Information stored in relative values of charge levels.Data encoded as permutations in blocks of cells.
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81
59
24
7
(6 3 8 1 5 9 2 4 7)Permutation:
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Errors in Rank Modulation: Adjacent Transpositions
Retention errors (charge leakage), cycling errors, andwrite-disturb errors often modeled as adjacent transpositions.
Adjacent
Transposition
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24
7
6
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59
24
3
7
( 6 3 8 1 5 9 2 4 7 ) ( 6 8 3 1 5 9 2 4 7 )
Studied by Barg, Bruck, Cassuto, Hagiwara, Jiang, Mazumdar, Schwartz, Wang,
Yaakobi, Zhou,...
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Errors in Rank Modulation: Adjacent Transpositions
Retention errors (charge leakage), cycling errors, andwrite-disturb errors often modeled as adjacent transpositions.
Adjacent
Transposition
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81
59
24
7
6
81
59
24
3
7
( 6 3 8 1 5 9 2 4 7 ) ( 6 8 3 1 5 9 2 4 7 )
Correcting t adjacent transposition ⇐⇒ min Kendall taudistance ≥ 2t + 1.
Studied by Barg, Bruck, Cassuto, Hagiwara, Jiang, Mazumdar, Schwartz, Wang,
Yaakobi, Zhou,...
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Errors in Rank Modulation: Translocations
Increasing number of charge levels to increase capacity leads tolarger charge fluctuations relative to gap between charge levels.
Farnoud, Skachek, Milenkovic, Transactions on Information Theory, May 2013
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Errors in Rank Modulation: Translocations
Increasing number of charge levels to increase capacity leads tolarger charge fluctuations relative to gap between charge levels.
Large magnitude error leading to a translocation:
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24
7
6
81
59
24
3 7
( 6 3 8 1 5 9 2 4 7 ) ( 6 8 1 5 9 2 4 3 7 )
Translocation
Farnoud, Skachek, Milenkovic, Transactions on Information Theory, May 2013
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Errors in Rank Modulation: Translocations
Increasing number of charge levels to increase capacity leads tolarger charge fluctuations relative to gap between charge levels.
Large magnitude error leading to a translocation:
63
81
59
24
7
6
81
59
24
3 7
( 6 3 8 1 5 9 2 4 7 ) ( 6 8 1 5 9 2 4 3 7 )
Translocation
Correcting t translocations ⇐⇒ min Ulam distance ≥ 2t + 1.
Ulam distance = length – length of longest common subsequence (LCS).
Farnoud, Skachek, Milenkovic, Transactions on Information Theory, May 2013
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Errors in Rank Modulation: Deletions
In harsh situations, such as high P/E cycles, transistor failure,a cell may become unreadable: deletion.
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( 6 3 8 1 5 9 2 4 7 ) ( 6 8 1 5 9 2 4 7 )
Deletion
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Errors in Rank Modulation: Deletions
In harsh situations, such as high P/E cycles, transistor failure,a cell may become unreadable: deletion.
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24
7
6
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24
3
7
( 6 3 8 1 5 9 2 4 7 ) ( 6 8 1 5 9 2 4 7 )
Deletion
Correcting t deletions ⇐⇒ min Ulam distance ≥ t + 1.
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Multipermutation Codes I
Multipermutation (MP): a rearrangement of elements of amultiset: [2, 1, 1, 2] for the multiset {1, 1, 2, 2}.
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Multipermutation Codes I
Multipermutation (MP): a rearrangement of elements of amultiset: [2, 1, 1, 2] for the multiset {1, 1, 2, 2}.
Each r cells have the same rank: programmed with roughlysame charge levels.
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Multipermutation Codes I
Multipermutation (MP): a rearrangement of elements of amultiset: [2, 1, 1, 2] for the multiset {1, 1, 2, 2}.
Each r cells have the same rank: programmed with roughlysame charge levels.
Example: r = 2, cells 2 and 3 have rank 1 andcells 1 and 4 have rank 2: MP [2,1,1,2]
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1
2
4
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Multipermutation Codes I
Multipermutation (MP): a rearrangement of elements of amultiset: [2, 1, 1, 2] for the multiset {1, 1, 2, 2}.
Each r cells have the same rank: programmed with roughlysame charge levels.
Example: r = 2, cells 2 and 3 have rank 1 andcells 1 and 4 have rank 2: MP [2,1,1,2]
3
1
2
4
r -regular MPs: each element (rank) appears r times.
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Multipermutation Codes I
Multipermutation (MP): a rearrangement of elements of amultiset: [2, 1, 1, 2] for the multiset {1, 1, 2, 2}.
Each r cells have the same rank: programmed with roughlysame charge levels.
Example: r = 2, cells 2 and 3 have rank 1 andcells 1 and 4 have rank 2: MP [2,1,1,2]
3
1
2
4
r -regular MPs: each element (rank) appears r times.
Beneficial since the number of possible ranks is limited.
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Multipermutation Codes II
In the literature:
Multipermutation re-write codes [En Gad’12]Multipermutation codes in Chebyshev metric [Shieh’10,’11]Multipermutation codes in Kendall tau metric [Buzaglo’13][Sala’13]Multipermutation codes in Hamming metric [Luo’03] [Ding’05][Huczynska’06] [Chu’06]. Aka, constant composition codes,frequency permutation codes.Concatenated permutations [Heymann’13].
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Multipermutation Codes II
In the literature:
Multipermutation re-write codes [En Gad’12]Multipermutation codes in Chebyshev metric [Shieh’10,’11]Multipermutation codes in Kendall tau metric [Buzaglo’13][Sala’13]Multipermutation codes in Hamming metric [Luo’03] [Ding’05][Huczynska’06] [Chu’06]. Aka, constant composition codes,frequency permutation codes.Concatenated permutations [Heymann’13].
This work: codes in the Ulam metric for correctingtranslocation and deletion errors.
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Multipermutation Codes II
In the literature:
Multipermutation re-write codes [En Gad’12]Multipermutation codes in Chebyshev metric [Shieh’10,’11]Multipermutation codes in Kendall tau metric [Buzaglo’13][Sala’13]Multipermutation codes in Hamming metric [Luo’03] [Ding’05][Huczynska’06] [Chu’06]. Aka, constant composition codes,frequency permutation codes.Concatenated permutations [Heymann’13].
This work: codes in the Ulam metric for correctingtranslocation and deletion errors.
We consider permutations and multipermutationssimultanously by considering equivalence classes ofmultipermutations.
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Multipermutation Codes III
3
1
2
44Noise
[2,1,1,2]
3
1
2
4
Writing
to Memory
Multi-
Permutation[2,1,1,2]
Reading
Permutation
from Memory (2,3,4,1)
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Multipermutation Codes III
3
1
2
44Noise
[2,1,1,2]
3
1
2
4
Writing
to Memory
Multi-
Permutation[2,1,1,2]
Reading
Permutation
from Memory (2,3,4,1)
31
2
4Multi-
Permutation[2,1,2,1]
Reading
Permutation
from Memory (2,4,1,3)More Noise
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Permutations and Multipermutations
Same information (same MP): [1, 2, 1, 2]:
31 313131
24 2
4 24
2 4
(3,1,4,2)(1,3,4,2)(3,1,2,4)(1,3,2,4)
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Permutations and Multipermutations
Same information (same MP): [1, 2, 1, 2]:
31 313131
24 2
4 24
2 4
(3,1,4,2)(1,3,4,2)(3,1,2,4)(1,3,2,4)
r -regular MPs divides Sn into equivalence classes.
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Permutations and Multipermutations
Same information (same MP): [1, 2, 1, 2]:
31 313131
24 2
4 24
2 4
(3,1,4,2)(1,3,4,2)(3,1,2,4)(1,3,2,4)
r -regular MPs divides Sn into equivalence classes.
Rr (π): equivalence class of π
R2 (1, 3, 2, 4) = {(1, 3, 2, 4) , (3, 1, 2, 4) , (1, 3, 4, 2) , (3, 1, 4, 2)}.
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Multipermutation Codes
In multipermutation coding:
information stored in terms of MPs,but physical error process described in terms of permutationsof change orderings.
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Multipermutation Codes
In multipermutation coding:
information stored in terms of MPs,but physical error process described in terms of permutationsof change orderings.
An r -regular MP code of length n, MPC(n, r), is a subset C ofSn such that if π ∈ C , then Rr (π) ⊆ C .
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Multipermutation Codes
In multipermutation coding:
information stored in terms of MPs,but physical error process described in terms of permutationsof change orderings.
An r -regular MP code of length n, MPC(n, r), is a subset C ofSn such that if π ∈ C , then Rr (π) ⊆ C .
Size of C is the number of equivalence classes it contains.
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Multipermutation Codes in Ulam and Hamming Metrics
An MPC(n, r) has minimum Ulam distance d if for all π and σnot in the same equivalence class, du(π, σ) ≥ d .
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Multipermutation Codes in Ulam and Hamming Metrics
An MPC(n, r) has minimum Ulam distance d if for all π and σnot in the same equivalence class, du(π, σ) ≥ d .
Corrects t translocation errors iff d ≥ 2t + 1.
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Multipermutation Codes in Ulam and Hamming Metrics
An MPC(n, r) has minimum Ulam distance d if for all π and σnot in the same equivalence class, du(π, σ) ≥ d .
Corrects t translocation errors iff d ≥ 2t + 1.
An MPC(n, r) has min Hamming distance d if for all π, σ theHamming distance between their MP representation is ≥ d .
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Multipermutation Codes in Ulam and Hamming Metrics
An MPC(n, r) has minimum Ulam distance d if for all π and σnot in the same equivalence class, du(π, σ) ≥ d .
Corrects t translocation errors iff d ≥ 2t + 1.
An MPC(n, r) has min Hamming distance d if for all π, σ theHamming distance between their MP representation is ≥ d .
An MPC with minimum Ulam distance d is an MPC withminimum Hamming distance at least d .
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Multipermutation Codes in Ulam and Hamming Metrics
An MPC(n, r) has minimum Ulam distance d if for all π and σnot in the same equivalence class, du(π, σ) ≥ d .
Corrects t translocation errors iff d ≥ 2t + 1.
An MPC(n, r) has min Hamming distance d if for all π, σ theHamming distance between their MP representation is ≥ d .
An MPC with minimum Ulam distance d is an MPC withminimum Hamming distance at least d .
We present construction of Ulam codes using codes inHamming metric.
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Capacity of MP Codes in the Hamming Metric
In a given metric, let A(n, r , d) denote the maximum size of anMP code of length n, regularity r , and minimum distance d .
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Capacity of MP Codes in the Hamming Metric
In a given metric, let A(n, r , d) denote the maximum size of anMP code of length n, regularity r , and minimum distance d .
The capacity is defined as
C (r , d) = limn→∞
lnA (n, r , d)
ln n!.
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Capacity of MP Codes in the Hamming Metric
In a given metric, let A(n, r , d) denote the maximum size of anMP code of length n, regularity r , and minimum distance d .
The capacity is defined as
C (r , d) = limn→∞
lnA (n, r , d)
ln n!.
Theorem [FM2014]
The capacity of multipermutation codes in the Hamming metricwith parameters r and d , with ρ = limn→∞
ln r
ln nand δ = limn→∞
d
n,
is given byCH (r , d) = (1 − ρ) (1 − δ) .
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Capacity of MP Codes in the Ulam Metric
In a given metric, let A(n, r , d) denote the maximum size of anMP code of length n, regularity r , and minimum distance d .
The capacity is defined as
C (r , d) = limn→∞
lnA (n, r , d)
ln n!.
Theorem [FM2014]
The capacity of multipermutation codes in the Ulam metric withparameters r and d , with ρ = limn→∞
ln r
ln nand δ = limn→∞
d
n, is
given by
(1 − 2ρ) (1 − δ) ≤ CU (r , d) ≤ (1 − ρ) (1 − δ) .
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Code Construction: Almost-disjoint Sets
Lemma
Let C be an MPC(n, r), and 2t < r . If for all π, σ ∈ C , each rankof π and σ are either identical or have less than r − 2t elements incommon, then C can correct t translocation errors.
Simple example for t = 1, r = 6, n = 12 (in MP form)
C ={[1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2] ,
[1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2] ,
...
[2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1]}.
The intersection between each two ranks is of size3 < 4 = r − 2t.
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Steiner Systems and BIBDs
A k-(n, r , λ)-design is a family of r -subsets of a set X of sizen, each called a block, such that every k-subset of X appearsin exactly λ blocks.
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Steiner Systems and BIBDs
A k-(n, r , λ)-design is a family of r -subsets of a set X of sizen, each called a block, such that every k-subset of X appearsin exactly λ blocks.
Such a design is resolvable if its blocks can be grouped into m
classes, such that each class forms a partition of X .
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Steiner Systems and BIBDs
A k-(n, r , λ)-design is a family of r -subsets of a set X of sizen, each called a block, such that every k-subset of X appearsin exactly λ blocks.
Such a design is resolvable if its blocks can be grouped into m
classes, such that each class forms a partition of X .
A Steiner system S(k , r , n) is a k-(n, r , 1)-design.
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Steiner Systems and BIBDs
A k-(n, r , λ)-design is a family of r -subsets of a set X of sizen, each called a block, such that every k-subset of X appearsin exactly λ blocks.
Such a design is resolvable if its blocks can be grouped into m
classes, such that each class forms a partition of X .
A Steiner system S(k , r , n) is a k-(n, r , 1)-design.
A Balanced incomplete block design (BIBD) with parameters(n, r , λ) is a 2-(n, r , λ)-design.
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Code Construction: Resolvable Steiner Systems
Proposition
If a resolvable Steiner system S(k , r , n) exists, then for oddd ≤ r − k + 1, there exists an MPC(n, r) with min Ulam distanced , of size (
n−1k−1
)
(
r−1k−1
)
(
n
r
)
!.
Proof outline:
The blocks in a class of the Steiner system are assigned as theelements of the ranks of the multipermutation.
Each two blocks have less than k elements in common.
Size of code follows from the number of classes and the factthat blocks can be assigned to ranks arbitrarily.
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Code Construction: Resolvable BIBDs
A resolvable BIBD is a resolvable Steiner system where everyk = 2 elements occur in only λ = 1 block.
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Code Construction: Resolvable BIBDs
A resolvable BIBD is a resolvable Steiner system where everyk = 2 elements occur in only λ = 1 block.
For prime r and n = r2, [Khare,Federer’81] give a simpleconstruction.
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Code Construction: Resolvable BIBDs
A resolvable BIBD is a resolvable Steiner system where everyk = 2 elements occur in only λ = 1 block.
For prime r and n = r2, [Khare,Federer’81] give a simpleconstruction.
Example: r = 3, each row is a block, each table is a class.
1 2 34 5 67 8 9
1 4 72 5 83 6 9
1 5 92 6 73 4 8
1 6 82 4 93 5 7
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Code Construction: Resolvable BIBDs
A resolvable BIBD is a resolvable Steiner system where everyk = 2 elements occur in only λ = 1 block.
For prime r and n = r2, [Khare,Federer’81] give a simpleconstruction.
Example: r = 3, each row is a block, each table is a class.
1 2 34 5 67 8 9
1 4 72 5 83 6 9
1 5 92 6 73 4 8
1 6 82 4 93 5 7
Proposition
Suppose that r is an odd prime. Then, there is an MPC(r2, r) withminimum Ulam distance r − 2 and size (r + 1)r !.
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Code Construction: Interleaving I
Let ◦ denote interleaving:
(1, 3, 2) ◦ (6, 4, 5) ◦ (8, 7, 9) = (1, 6, 8, 3, 4, 7, 2, 5, 9).
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Code Construction: Interleaving I
Let ◦ denote interleaving:
(1, 3, 2) ◦ (6, 4, 5) ◦ (8, 7, 9) = (1, 6, 8, 3, 4, 7, 2, 5, 9).
Assume
A partition {P1, . . . ,Pr} of [n] into sets of equal size.Let Ci , i ∈ [r ], be permutation codes of minimum Ulamdistance d ≤ n/r over Pi .Construct C by interleaving codewords of Ci .
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Code Construction: Interleaving I
Let ◦ denote interleaving:
(1, 3, 2) ◦ (6, 4, 5) ◦ (8, 7, 9) = (1, 6, 8, 3, 4, 7, 2, 5, 9).
Assume
A partition {P1, . . . ,Pr} of [n] into sets of equal size.Let Ci , i ∈ [r ], be permutation codes of minimum Ulamdistance d ≤ n/r over Pi .Construct C by interleaving codewords of Ci .
Proposition
The code C is a MPC(n, r) with minimum Ulam distance d .
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Code Construction: Interleaving I
Let ◦ denote interleaving:
(1, 3, 2) ◦ (6, 4, 5) ◦ (8, 7, 9) = (1, 6, 8, 3, 4, 7, 2, 5, 9).
Assume
A partition {P1, . . . ,Pr} of [n] into sets of equal size.Let Ci , i ∈ [r ], be permutation codes of minimum Ulamdistance d ≤ n/r over Pi .Construct C by interleaving codewords of Ci .
Proposition
The code C is a MPC(n, r) with minimum Ulam distance d .
Assuming optimal component codes, if lim rd
n= 0, then C is
capacity achieving.
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Code Construction: Interleaving II
Let ◦r denote interleaving blocks of r elements:
(1, 3, 4, 2) ◦2 (6, 7, 8, 5) ◦2 (12, 10, 9, 11) =(1, 3, 6, 7, 12, 10, 4, 2, 8, 5, 9, 11).
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Code Construction: Interleaving II
Let ◦r denote interleaving blocks of r elements:
(1, 3, 4, 2) ◦2 (6, 7, 8, 5) ◦2 (12, 10, 9, 11) =(1, 3, 6, 7, 12, 10, 4, 2, 8, 5, 9, 11).
Assume
n/r even, d ≤ r , P =[
n
2
]
, and Q = [n]\P .C ′
1is an MPC( n
2, r) with min Ulam distance d over P
C1 is an MPC( n2, r) with min Hamming distance d over Q.
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Code Construction: Interleaving II
Let ◦r denote interleaving blocks of r elements:
(1, 3, 4, 2) ◦2 (6, 7, 8, 5) ◦2 (12, 10, 9, 11) =(1, 3, 6, 7, 12, 10, 4, 2, 8, 5, 9, 11).
Assume
n/r even, d ≤ r , P =[
n
2
]
, and Q = [n]\P .C ′
1is an MPC( n
2, r) with min Ulam distance d over P
C1 is an MPC( n2, r) with min Hamming distance d over Q.
Proposition
The code C = C ′1 ◦r C1 is an MPC(n, r) with minimum Ulam
distance d .
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Code Construction: Interleaving II
Let ◦r denote interleaving blocks of r elements:
(1, 3, 4, 2) ◦2 (6, 7, 8, 5) ◦2 (12, 10, 9, 11) =(1, 3, 6, 7, 12, 10, 4, 2, 8, 5, 9, 11).
Assume
n/r even, d ≤ r , P =[
n
2
]
, and Q = [n]\P .C ′
1is an MPC( n
2, r) with min Ulam distance d over P
C1 is an MPC( n2, r) with min Hamming distance d over Q.
Proposition
The code C = C ′1 ◦r C1 is an MPC(n, r) with minimum Ulam
distance d .
With nested construction, we may only use codes in theHamming metric.
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Code Construction: Interleaving II
Let ◦r denote interleaving blocks of r elements:
(1, 3, 4, 2) ◦2 (6, 7, 8, 5) ◦2 (12, 10, 9, 11) =(1, 3, 6, 7, 12, 10, 4, 2, 8, 5, 9, 11).
Assume
n/r even, d ≤ r , P =[
n
2
]
, and Q = [n]\P .C ′
1is an MPC( n
2, r) with min Ulam distance d over P
C1 is an MPC( n2, r) with min Hamming distance d over Q.
Proposition
The code C = C ′1 ◦r C1 is an MPC(n, r) with minimum Ulam
distance d .
With nested construction, we may only use codes in theHamming metric.
Assuming optimal Hamming codes, C is capacity achieving(for d ≤ r).
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Thank you!
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