Distortion Minimisation by Coding TheoryCSM25 Secure Information Hiding
Dr Hans Georg Schaathun
University of Surrey
Spring 2008
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 1 / 40
Learning Objectives
Get an overview of basic concepts in coding theoryUnderstand the principle of matrix (aka. wet paper) coding
Suggested Reading
Raymond Hill: A first course in coding theory Oxford AppliedMathematics and Computing Science Series. 1986
Suggested Reading
Fridrich et al: ‘Writing on Wet Paper’ IEEE Trans. Signal Proc. Oct2005, pp. 3923-,
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 2 / 40
Coding Theory Overview
Outline
1 Coding TheoryOverview
2 Cosets and syndromesCosetsReducing distortionSyndromeDesign issuesThe Code Parameters
3 Matrix and wet paper codingMatrix codingWet paper codes
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 3 / 40
Coding Theory Overview
OverviewMatrix Coding and Error-Control Coding
Duality betweenError-Control CodingMatrix Coding (e.g. F5)
First: a brief study of error-controlthis will enable us to look further at matrix coding
A linear [n, k ]q code C islinear set (vector space) of qk codewordseach codeword is a vector (c1, c2, . . . , cn)over an alphabet (field) F of q element
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 4 / 40
Coding Theory Overview
OverviewMatrix Coding and Error-Control Coding
Duality betweenError-Control CodingMatrix Coding (e.g. F5)
First: a brief study of error-controlthis will enable us to look further at matrix coding
A linear [n, k ]q code C islinear set (vector space) of qk codewordseach codeword is a vector (c1, c2, . . . , cn)over an alphabet (field) F of q element
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 4 / 40
Coding Theory Overview
OverviewMatrix Coding and Error-Control Coding
Duality betweenError-Control CodingMatrix Coding (e.g. F5)
First: a brief study of error-controlthis will enable us to look further at matrix coding
A linear [n, k ]q code C islinear set (vector space) of qk codewordseach codeword is a vector (c1, c2, . . . , cn)over an alphabet (field) F of q element
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 4 / 40
Coding Theory Overview
Basic definitions
A field F means (essentially) thatMultiplication/division and addition/subtraction are defined on F∀a, b ∈ F , we have a + b ∈ F and a · b ∈ F .∃0, 1 ∈ F , s.t. a + 0 = 0, a · 1 = a, a · 0 = 0 etc.
The binary field F2:Addition is logical XORMultiplication is logical AND
A vector space (linear set) means that∀a, b ∈ C, we have a + b ∈ C.∀a ∈ F ,∀b ∈ C, we have ab ∈ C.
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 5 / 40
Coding Theory Overview
Basic definitions
A field F means (essentially) thatMultiplication/division and addition/subtraction are defined on F∀a, b ∈ F , we have a + b ∈ F and a · b ∈ F .∃0, 1 ∈ F , s.t. a + 0 = 0, a · 1 = a, a · 0 = 0 etc.
The binary field F2:Addition is logical XORMultiplication is logical AND
A vector space (linear set) means that∀a, b ∈ C, we have a + b ∈ C.∀a ∈ F ,∀b ∈ C, we have ab ∈ C.
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 5 / 40
Coding Theory Overview
Basic definitions
A field F means (essentially) thatMultiplication/division and addition/subtraction are defined on F∀a, b ∈ F , we have a + b ∈ F and a · b ∈ F .∃0, 1 ∈ F , s.t. a + 0 = 0, a · 1 = a, a · 0 = 0 etc.
The binary field F2:Addition is logical XORMultiplication is logical AND
A vector space (linear set) means that∀a, b ∈ C, we have a + b ∈ C.∀a ∈ F ,∀b ∈ C, we have ab ∈ C.
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 5 / 40
Coding Theory Overview
Example
The [7, 4]2 Hamming code.Generator matrix
G =
1 0 0 0 1 1 10 1 0 0 1 1 00 0 1 0 1 0 10 0 0 1 0 1 1
(1)
Take any message m = (m1, m2, m3, m4) ∈ F 42
Encoding: c = m · G.Any such c is a codeword
The code is C = {c : ∃m, c = m · G}.
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 6 / 40
Coding Theory Overview
Encoding a message
message m = (0101)
[0110] ·
1 0 0 0 1 1 10 1 0 0 1 1 00 0 1 0 1 0 10 0 0 1 0 1 1
= [0100110] + [0010101]
= [0110011]
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 7 / 40
Coding Theory Overview
The Hamming CodeA diagrammatic view
m1m2
m3
m4
����
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
The Hamming CodeA diagrammatic view
p1
m1
p2
m2
p3
m3
m4
����
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
The Hamming CodeA diagrammatic view
p1
m1
p2
m2
p3
m3
m4
s1 s2
����
s3
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
The Hamming CodeA diagrammatic view
p1
m1
p2
m2
p3
m3
m4
s1 = 0 s2 = 0
����
s3 = 0
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
The Hamming CodeEncoding
p1
0
p2
1
p3
1
0
s1 = 0 s2 = 0
����
s3 = 0
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
The Hamming CodeEncoding
0
0
1
1
1
1
0
s1 = 0 s2 = 0
����
s3 = 0
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
The Hamming CodeOne error in channel
0
0
1
0
1
1
0
s1 s2
����
s3
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
The Hamming CodeDecoding
0
0
1
0
1
1
0
1 0
����
1
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
The Hamming CodeDecoding
0
0
1
0
1
1
0
1 0
����
1
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
The Hamming CodeDecoding
0
0
1
1
1
1
0
0 0
����
0
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
Hamming Matrix CodingMessage and host
p1
m1
p2
m2
p3
m3
m4
s1 s2
����
s3
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
Hamming Matrix CodingThe message
p1
m1
p2
m2
p3
m3
m4
0 0
����
1
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
Hamming Matrix CodingThe host signal
0
1
0
1
1
0
1
0 0
����
1
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
Hamming Matrix CodingEmbedded message
0
1
0
1
1
0
1
0 0
����
1
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Coding Theory Overview
Hamming Matrix Coding
0
1
0
1
0
0
1
0 0
����
1
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 8 / 40
Cosets and syndromes Cosets
Outline
1 Coding TheoryOverview
2 Cosets and syndromesCosetsReducing distortionSyndromeDesign issuesThe Code Parameters
3 Matrix and wet paper codingMatrix codingWet paper codes
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 9 / 40
Cosets and syndromes Cosets
A Sample Code
G =
[1 1 1 0 00 0 1 1 1
]
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 10 / 40
Cosets and syndromes Cosets
The Cosets
00000 00001 00010 00100 01000 10000 01001 0101011100 11101 11110 11000 10100 01100 10101 1011000111 00110 00101 00011 01111 10111 01110 0110111011 11010 11001 11111 10011 01011 10010 10001
4 codewords (blue)7 correctable, non-zero error patterns (red)8 co-sets (columns) r + C4 × 8 = 32 words of length 5Look up received word in the table
Go left to find sent wordGo up to find the error
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 11 / 40
Cosets and syndromes Cosets
The Cosets
00000 00001 00010 00100 01000 10000 01001 0101011100 11101 11110 11000 10100 01100 10101 1011000111 00110 00101 00011 01111 10111 01110 0110111011 11010 11001 11111 10011 01011 10010 10001
4 codewords (blue)7 correctable, non-zero error patterns (red)8 co-sets (columns) r + C4 × 8 = 32 words of length 5Look up received word in the table
Go left to find sent wordGo up to find the error
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 11 / 40
Cosets and syndromes Cosets
The Cosets
00000 00001 00010 00100 01000 10000 01001 0101011100 11101 11110 11000 10100 01100 10101 1011000111 00110 00101 00011 01111 10111 01110 0110111011 11010 11001 11111 10011 01011 10010 10001
4 codewords (blue)7 correctable, non-zero error patterns (red)8 co-sets (columns) r + C4 × 8 = 32 words of length 5Look up received word in the table
Go left to find sent wordGo up to find the error
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 11 / 40
Cosets and syndromes Cosets
The Cosets
00000 00001 00010 00100 01000 10000 01001 0101011100 11101 11110 11000 10100 01100 10101 1011000111 00110 00101 00011 01111 10111 01110 0110111011 11010 11001 11111 10011 01011 10010 10001
4 codewords (blue)7 correctable, non-zero error patterns (red)8 co-sets (columns) r + C4 × 8 = 32 words of length 5Look up received word in the table
Go left to find sent wordGo up to find the error
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 11 / 40
Cosets and syndromes Cosets
The Cosets
00000 00001 00010 00100 01000 10000 01001 0101011100 11101 11110 11000 10100 01100 10101 1011000111 00110 00101 00011 01111 10111 01110 0110111011 11010 11001 11111 10011 01011 10010 10001
4 codewords (blue)7 correctable, non-zero error patterns (red)8 co-sets (columns) r + C4 × 8 = 32 words of length 5Look up received word in the table
Go left to find sent wordGo up to find the error
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 11 / 40
Cosets and syndromes Cosets
Correctable error patterns
00000 00001 00010 00100 01000 10000 01001 0101011100 11101 11110 11000 10100 01100 10101 1011000111 00110 00101 00011 01111 10111 01110 0110111011 11010 11001 11111 10011 01011 10010 10001
Correctable error-patterns are chosen arbitrarilyWe may want to correct 10010, but then we cannot correct 01001Low-weight error-patterns are more likely,
hence chosen for correction
but how choose between 10010 and 01001?
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 12 / 40
Cosets and syndromes Cosets
Correctable error patterns
00000 00001 00010 00100 01000 10000 01001 0101011100 11101 11110 11000 10100 01100 10101 1011000111 00110 00101 00011 01111 10111 01110 0110111011 11010 11001 11111 10011 01011 10010 10001
Correctable error-patterns are chosen arbitrarilyWe may want to correct 10010, but then we cannot correct 01001Low-weight error-patterns are more likely,
hence chosen for correction
but how choose between 10010 and 01001?
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 12 / 40
Cosets and syndromes Cosets
Correctable error patterns
00000 00001 00010 00100 01000 10000 01001 0101011100 11101 11110 11000 10100 01100 10101 1011000111 00110 00101 00011 01111 10111 01110 0110111011 11010 11001 11111 10011 01011 10010 10001
Correctable error-patterns are chosen arbitrarilyWe may want to correct 10010, but then we cannot correct 01001Low-weight error-patterns are more likely,
hence chosen for correction
but how choose between 10010 and 01001?
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 12 / 40
Cosets and syndromes Reducing distortion
Outline
1 Coding TheoryOverview
2 Cosets and syndromesCosetsReducing distortionSyndromeDesign issuesThe Code Parameters
3 Matrix and wet paper codingMatrix codingWet paper codes
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 13 / 40
Cosets and syndromes Reducing distortion
A matrix code
00000 00001 00010 00100 01000 10000 01001 0101011100 11101 11110 11000 10100 01100 10101 1011000111 00110 00101 00011 01111 10111 01110 0110111011 11010 11001 11111 10011 01011 10010 10001
Assign a message to each column8 messages : 3 bits
4 codewords per message
F3/F4 would flip 1 12 bit on average
Use five coefficients, and embed any word in the columnflip as few bits as possible
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 14 / 40
Cosets and syndromes Reducing distortion
A simpler example
message 00 01 10 11codeword 1 000 001 010 100codeword 2 111 110 101 011
Embed 01.Cover-image has 100.
Hiding 001 requires change of two bitsHiding 110 requires change of one bits
We change the cover to 110 to represent 01.
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 15 / 40
Cosets and syndromes Reducing distortion
A simpler example
message 00 01 10 11codeword 1 000 001 010 100codeword 2 111 110 101 011
Embed two bitsF3/F4 would flip one bit on averageSometimes two, sometimes one
You never need to flip more than one bit25% of the time no flip is necessary
Flips 0.75 bits on averageMatrix coding reduces detectability by 25%
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 15 / 40
Cosets and syndromes Syndrome
Outline
1 Coding TheoryOverview
2 Cosets and syndromesCosetsReducing distortionSyndromeDesign issuesThe Code Parameters
3 Matrix and wet paper codingMatrix codingWet paper codes
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 16 / 40
Cosets and syndromes Syndrome
Parity check matrix
[n, k ] code CDescribed by k × n generator matrix GAlso, described by (n − k) × n parity check matrix H
where G · HT = 0Note, both G and H have full rank
that is, no linearly dependent rows
C = {c : HcT = 0}
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 17 / 40
Cosets and syndromes Syndrome
Hamming code
G =
1 0 0 0 1 1 10 1 0 0 1 1 00 0 1 0 1 0 10 0 0 1 0 1 1
, (2)
H =
1 1 1 0 1 0 01 1 0 1 0 1 01 0 1 1 0 0 1
(3)
Systematic form:G = [I|P] H = [PT|I]
where I is an identity matrix
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 18 / 40
Cosets and syndromes Syndrome
The Syndrome
Given a received word r,the syndrome is s = rHT
Each syndrome can be mapped to an error vector e ∈ F n
Syndrome decodingError patterns uniquely assigned to a syndrome can be corrected
s = 0 corresponds to no error
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 19 / 40
Cosets and syndromes Syndrome
Syndromes and Cosets
The syndrome is a property of the coset
H =
110000001110101
000 011 010 001 100 101 111 110
00000 00001 00010 00100 01000 10000 01001 0101011100 11101 11110 11000 10100 01100 10101 1011000111 00110 00101 00011 01111 10111 01110 0110111011 11010 11001 11111 10011 01011 10010 10001
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 20 / 40
Cosets and syndromes Design issues
Outline
1 Coding TheoryOverview
2 Cosets and syndromesCosetsReducing distortionSyndromeDesign issuesThe Code Parameters
3 Matrix and wet paper codingMatrix codingWet paper codes
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 21 / 40
Cosets and syndromes Design issues
The decoding problem
Encoding is simple matrix-vector multiplicationDecoding of arbitrary codes is NP-complete
Small codes can easily be decodedExhaustive searchSyndrome lookup table
Intractible for larger codes.
Some classes of codes are known with fast decoding algorithms
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 22 / 40
Cosets and syndromes Design issues
Block length
Compare two codes[400, 200] code C1 able to correct 60 errors[100, 50] code C2 able to correct 15 errors
We can use one codeword of C1 or four of C2C1 can correct any 60 errorsC2 can correct 60 errors if they are evenly distributed over the fourblocks
Since errors are random, they are rarely evenly distributedIn general, larger block lengths give better robustness.
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 23 / 40
Cosets and syndromes Design issues
Block lengths in F5Andreas Westfeld
k n changes information rate bits per change1 1 50.00% 100.00% 2.002 3 25.00% 66.67% 2.673 7 12.50% 42.86% 3.434 15 6.25% 26.67% 4.273 31 3.12% 16.13% 5.163 63 1.56% 9.52% 6.093 127 0.78% 5.51% 7.068 255 0.39% 3.14% 8.039 511 0.20% 1.76% 9.02
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 24 / 40
Cosets and syndromes The Code Parameters
Outline
1 Coding TheoryOverview
2 Cosets and syndromesCosetsReducing distortionSyndromeDesign issuesThe Code Parameters
3 Matrix and wet paper codingMatrix codingWet paper codes
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 25 / 40
Cosets and syndromes The Code Parameters
Hamming Distance
Two vectors
a = (a1, a2, . . . , an),
b = (b1, b2, . . . , bn).
Hamming distanced(a, b) = #{i : ai 6= bi}Number of positions where the vectors differ
The Hamming distance is a metric (distance) in mathematicalterms
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 26 / 40
Cosets and syndromes The Code Parameters
The Noisy ChannelClosest Neighbour Decoding
Alice sends cBob receives rAssumption: few errors more likely than many errors
i.e. probably d(c, r) is smallif r 6∈ C, look for c
minimising d(c, r).This is called closest neighbour decoding
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 27 / 40
Cosets and syndromes The Code Parameters
Minimum distance
[n, k , d ]q code CMinimum distance d is
smallest distance between two distinct codewordsd := mina,b∈C d(a, b) where a 6= b
Let e =⌊d−1
2
⌋spheres of radius e around each c ∈ Cnon-overlapping spheresclosest neighbour decoding returns centre of sphere
Up to e errors are uniquely decodable.
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 28 / 40
Cosets and syndromes The Code Parameters
Covering radius
DefinitionThe covering radius ρ is the largest distance between a word a 6∈ Cand the closest codeword c ∈ C.
Draw spheres of radius ρ around each codeword c ∈ CThese spheres covers the entire spaceSmaller spheres would not cover the space
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 29 / 40
Cosets and syndromes The Code Parameters
Perfect codes
DefinitionIf ρ = b(d − 1)/2c (= e), we say that the code is perfect.
The decoding spheres of radius e covers the spaceIt is impossible to add another codeword without
reducing the error-correction capacity.
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 30 / 40
Cosets and syndromes The Code Parameters
ExerciseCoding parameters
Let a code be given by generator matrix
G =
111111110000111
(4)
What are the parameters [n, k , d ]q?Find nFind kFind dFind q
Hint: you may want to list all possible codewords first.What is the covering radius ρ?
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 31 / 40
Matrix and wet paper coding Matrix coding
Outline
1 Coding TheoryOverview
2 Cosets and syndromesCosetsReducing distortionSyndromeDesign issuesThe Code Parameters
3 Matrix and wet paper codingMatrix codingWet paper codes
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 32 / 40
Matrix and wet paper coding Matrix coding
Matrix coding
[n, k ] matrix code C with low covering radius.k -bit message transmitted as the syndrome s.n-bit host signal corresponds to codeword/received wordEncoder: change as few bits as possible, such that
the resulting syndrome match the message
Decoder: multiply host by H to get syndrome s.
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 33 / 40
Matrix and wet paper coding Matrix coding
The main problem
Coding theory has focused onmaximising minimum distancecorrect many errors
Matrix coding requiresminimising covering radiusminimise distortion
Hence, less theory available for matrix coding
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 34 / 40
Matrix and wet paper coding Matrix coding
Implementation issues
The hard job is at the Sender.Embedding requires solving the decoding problem
which is NP-completelarge codes with good decoding algorithms
few (if any) known with low covering radius
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 35 / 40
Matrix and wet paper coding Wet paper codes
Outline
1 Coding TheoryOverview
2 Cosets and syndromesCosetsReducing distortionSyndromeDesign issuesThe Code Parameters
3 Matrix and wet paper codingMatrix codingWet paper codes
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 36 / 40
Matrix and wet paper coding Wet paper codes
Matrix code vs. wet paper code
Matrix codingforbids more than P changes in the hostallows changes in arbitrary positions
Wet paper codingallows changes in specified positions in the hostforbids changes in remaining positions
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 37 / 40
Matrix and wet paper coding Wet paper codes
Random codes
Random codes generally perform wellOnly problem is complex decoding
Fridrich et al proposed random codes for wet paper.
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 38 / 40
Matrix and wet paper coding Wet paper codes
The encoding problem
The hard job is at the encoderSlightly different problem from matrix codingHas to solve the equation
Hc = m
where c is the coverm is the message
Some entries of c are locked,remaining entries are unknowns in the equation
Solvable by Gaussian eliminationPolynomial O(n3)
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 39 / 40
Matrix and wet paper coding Wet paper codes
Simplification
Divide host signal into smaller blocksembed independently in each block
Reduce problem size at decoder (O(n3))Faster decoding
Shorter blocksLess optimal, i.e. slightly increased distortion
Dr Hans Georg Schaathun Distortion Minimisation by Coding Theory Spring 2008 40 / 40