ii. linear block codes

II. Linear Block Codes

© Tallal Elshabrawy 2

Last Lecture H Matrix and Calculation of dmin

Error Detection Capability


Error Correction Capability

The “error correction capability” defines the number of errors that are GUARANTEED to be corrected.

For a code with minimum Hamming distance dmin:

The error correction capability is (dmin-1)/2x means floor(x)


Assume w is also a valid codeword in CTriangle Inequality:

Error Correction Capability: Proof

i.e., if an error pattern of t or fewer errors occur, the received vector r MUST BE closer to v than to any other codeword in C

Channelve

r

v: transmitted codeworde: error pattern caused by channelr: received pattern

Assume a Block Code with dminDefine a Positive Integer t such that:

min2t 1 d 2t 2

d d d v,r w,r v,wAssume an error pattern with t’ errors d v,r t'

Given that v and w are codewords d mindv,w

d dd

mind 2t 12t 1 t'

v,r w,rw,r

If t’≤t d mind 1t, t 2w,r


Error Correction & Error Detection Capability

Some codes are designed such that they can GUARANTEE correction of λ or fewer errors AND detection of up to l>λ

For a code with minimum Hamming distance dmin:

If the code could correct λ and detect up to l then dmin≥ λ+l+1Example: If it is required for a code to correct 3 errors AND detect up to 6 errors then dmin must satisfy: dmin ≥10Notes: In the example above,

If the number of errors are 3 or less: you can provide a GUARNTEE to correct all of them.If the number of errors are from 4 to 6: you can provide a GUARNTEE to detect that the number of errors are between 4 to 6 without being able to correct ANY of them.If the number of errors are greater than 6: there is NO GUARANTEE that you would be able to detect or correct the errors


v0=0 v1=1 v2=2 v3=3 v*0=0 v*

1=1 v*2=2 v*

3=3Transmitter Side Receiver Side

Quantization and Analogy to Error Correction and Detection Operation

Correct Reception:The value received is identical to what has been transmitted

Channel


Error Detection:The value received IS NOT EQUAL to any of the valid representation levelsARQ: Request retransmission

v0=0 v1=1 v2=2 v3=3 v*0=0 v*

1=1 v*2=2 v*


Channel



Un-Detected Errors:The value received IS a valid representation level. However, it is NOT what has been transmitted

v0=0 v1=1 v2=2 v3=3 v*0=0 v*

1=1 v*2=2 v*


Channel



Error Correction:Approximate the received value to the closest valid representation level.FEC: The receiver defines decision zones and maps ANY received value to a valid representation level. ARQ cannot be applied

v0=0 v1=1 v2=2 v3=3 v*0=0 v*

1=1 v*2=2 v*


Channel

D0 D1 D2 D3



False Correction:The error pushes the received value outside the decision zone of the representation level that has been transmitted.

v0=0 v1=1 v2=2 v3=3 v*0=0 v*

1=1 v*2=2 v*


Channel

D0 D1 D2 D3



Error Correction and Error Detection:Defines two types of decision zone such that both FEC and ARQ could be used dependent on the received value

v0=0 v1=1 v2=2 v3=3Transmitter Side Receiver Side

Channel

D0 D1 D2 D3

v*0=0 v*

1=1 v*2=2 v*

3=3




Channel

D0 D1 D2 D3

v*0=0 v*

1=1 v*2=2 v*

3=3



Error Correction



Channel

D0 D1 D2 D3

v*0=0 v*

1=1 v*2=2 v*

3=3





Channel

D0 D1 D2 D3

v*0=0 v*

1=1 v*2=2 v*

3=3



False Error Correction



Channel

D0 D1 D2 D3

v*0=0 v*

1=1 v*2=2 v*

3=3



Error Detection and ARQ could be used


Decoding: A rule to partition 2n possible received vectors into 2k

disjoint subsets D1, D2,…,D2k such that the codeword vi is contained in the subset Di, 1≤i ≤2k

Let C be an (n,k) linear block code with codewords v1, v2,…,v2k

Channelve

r Decoder

Which vi was sent??2k codewords 2n n-tuples

Decoding Rule: ivr s Decision i⇒Decoding D∈If i

Correct Decoding: Given vi is transmitted, error pattern e does not push r outside Di

Decoding and Error Correction


Design of a Decoder

How to divide 2n vectors into 2k disjoint subsets such that the error correcting capability is optimal?In other words, which n-tuples should be designed such that they belong to a given subset Di?


Standard Array

Partitioning 2n possible received vectors into 2k disjoint subsets with each subset containing only one codeword

kn-kn-kn-kn-k

k

k

k

22i2222

23i3233

22i2222

2i2

+...+...+..

.

...

.

...

.

.+...+...++...+...+

......=

vevevee

veveveevevevee

vvv0v1

D1 D2 Di D2k


Standard Array


kn-kn-kn-kn-k

k

k

k

22i2222

23i3233

22i2222

2i2

+...+...+..

.

...

.

...

.

.+...+...++...+...+

......=

vevevee

veveveevevevee

vvv0v1e2≠v1,v2,…,v2k

D1 D2 Di D2k


Standard Array


kn-kn-kn-kn-k

k

k

k

22i2222

23i3233

22i2222

2i2

+...+...+..

.

...

.

...

.

.+...+...++...+...+

......=

vevevee

veveveevevevee

vvv0v1

e3≠v1,v2,…,v2k,e2,…,e2+v2k

D1 D2 Di D2k


Standard Array


kn-kn-kn-kn-k

k

k

k

22i2222

23i3233

22i2222

2i2

+...+...+..

.

...

.

...

.

.+...+...++...+...+

......=

vevevee

veveveevevevee

vvv0v1

e2n-k ≠Any previous element in the array

D1 D2 Di D2k


Theorem

1. No two n-tuples in the same row of a standard array are identical.

2. Every n-tuple appears in one and only one row.

PROOF:1. Given a linear block code is considered. By definition of

the standard array construction, ej is not a codeword. Therefore, ej+ vj generate distinct patterns

2. Select row l<m, assume el+vi= em+vj. Therefore,em= el+vi+vjem= el+vs (because vi and vj are codewords)This means that em could be found in row lThis is a contradiction to the construction rule of the standard array


Standard Array


kn-kn-kn-kn-k

k

k

k

22i2222

23i3233

22i2222

2i2

+...+...+..

.

...

.

...

.

.+...+...++...+...+

......=

vevevee

veveveevevevee

vvv0v1

e2n-k ≠Any previous element in the array

D1 D2 Di D2k


Coset of a Code, Coset Leader

Given the standard array. The 2n-k rows are called the cosets of the code. The first n-tuple ej of each coset is called the coset leader


Error Correction

DECODINGFALSE ⇒≠Det Dn the subsis found i+=

++=+=⇒+=

cosetlis in the or patternAssume errDECODING⇒FALSE a is notor patternIf the errDECODING⇒CORRECT is a or patternIf the err

s Decision i⇒Decoding DIf

js

sl

ilj

jil

th

i

rve

vevxvrvex

xleader coset

leader cosetvr i∈

A Linear block code can correct 2n-k-1 error patterns corresponding to 2n-k-1 coset leadersNote that one of the coset leaders is the zero vector


Error Correction MethodologyThe error patterns that are most likely to occur should be used as the coset leaders

Example: A (7,4) code can correct 2(7-4)-1=7 error patternsTo correct a single error. The coset leaders should be selected as:

e1= v1= (0 0 0 0 0 0 0)e2 =(0 0 0 0 0 0 1)e3 =(0 0 0 0 0 1 0)e4 =(0 0 0 0 1 0 0)e5 =(0 0 0 1 0 0 0)e6 =(0 0 1 0 0 0 0)e7 =(0 1 0 0 0 0 0)e8 =(1 0 0 0 0 0 0)


Theorem

1. All 2k n-tuples of a coset have the same syndrome. 2. The syndrome for different cosets are different

PROOF:1. (el+vi)HT=elHT+viHT=elHT (independent of vi)2. Let ej and ei be the coset leaders. If they have the same

syndromeejHT+eiHT=0. Therefore,(ej+ei) is a codewordThis contradicts the definition of a linear block code because ej and ei are not codewords

Advantages: We can use a syndrome decoding table which is much simpler to use than a standard array


Syndrome Decoding

1. Compute syndrome of r (i.e., rHT)2. Locate the coset leader el whose syndrome is

equal to rHT

3. v*= r+el where v* is the codeword output by the decoder

ii. linear block codes

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