An Algorithm for Construction of Error-Correcting Symmetrical

Reversible Variable Length Codes

Chia-Wei Lin, Ja-Ling Wu,

Jun-Cheng Chen

Presented by Jun-Cheng Chen 2004/09/24

Outline

• Introduction

• Notations and preliminaries

• The proposed algorithm

• Experimental results

• Conclusion

Introduction to RVLC (1/3)

• Variable length codes (VLC) are of prime importance in the efficient transmission of digital signals.– High compression efficiency

• There are other criteria that may be important in the application environment .– Channel bit-error resilience , maximum codeword length li

mitation

• Reversibility of variable length codes makes instantaneous decoding possible both in the forward and backward directions .

Introduction to RVLC (2/3)

• A reversible variable length code (RVLC) must satisfy the prefix-free and suffix-free condition for instantaneous forward and backward decoding .

Symbol Probability C1 C2

ABCDE

0.330.300.180.100.09

000111

100101

0011

010101

0110

Average code length 2.19 2.46

Table 0: Huffman code and reversible variable length codes(RVLCs)

C1: Huffman code C2: Symmetrical RVLC

Introduction to RVLC (3/3)• If a bit error occurs, VLC will propagate the bit error and

the data after the bit error becomes useless. However, RVLC can recover data after the bit error.

VLC

RVLC

Bit error

Notations and preliminaries (1/3)

• n source symbols: • probabilities of source symbols:• n codewords: • length of codeword :•

– : codeword sequences

– : hamming distance

naaa ,...,, 21

niap i ,..,1),( },...,,{ 21 nccc

ic nili ,..,2,1, },...,2,1,,,:)(min{ , NjiFffffHd Njijifree

ji ff ,

),( ji ffH

Notations and preliminaries (2/3)

• (minimum block distance)

• In the case of RVLCs

Note: if a VLC does not have equal-length codewords, then its minimum block distance is undefined.

),2min(

2

bfree d

d, if is undefined

, if is defined

bd

bd

bd

codewords

00

11

010

101

0110

1001

Block distance:2

Block distance:3

Block distance:4

(minimum block distance): 2

bd

Find minimum blcok distance

Minimum Block Distancebd

Find minimum hamming distance for each level

Notations and preliminaries (3/3)• R(c, CLd, d) is the replacement of a codeword c in the code

word list CLd

– children(c) are defined by all of the first symmetrical codewords on paths from the codeword c to leaf codewords.

– CLd is a codeword list in which the constraint db d holds for all it

s codewords.

– children(c, d) is a subset of the symmetrical children of a symmetrical codeword c in which the constraint db d holds for all the chil

dren in the subset.

– children(c, CLd, d) is a subset of symmetrical children of c that the

union of the subset and CLd is still a codeword list in which the con

straint db d holds for all its codewords.

00

codeword ’00’

000

children(‘00’) = {000}children(‘00’, 2) = {000}

T-List: (1,00,010, 0110)

children(‘00’,T-List, 2) = {}

0000

R(‘00’,T-List,2) ={0000}

0 1

The Proposed Algorithm (1/4)

• Goal: The free distance of the proposed RVLC is always greater than one, which can result in certain improvement in symbol error rate relative to VLCs with free distance one.

The Proposed Algorithm (2/4)

• The algorithm– Step1: Assign the initial codeword list (“1”, “00”, “010”,

“0110”,…) to the target list, T-List.– Step2: If any codeword c in T-List satisfies the condition

of replacement, replace the codeword c with R(c, T-List, 2) that results in the smallest average codeword length. If the number of codewords in T-List is more than n, the number of source symbols, keep the first n codewords in T-List and discard the others. (Notice that the block distance is still greater than one after the codeword replacement)

– Step3: Repeat Step2 until there is no codeword in T-List satisfying the condition of replacement.

The Proposed Algorithm (3/4)

• Six symbols with probabilities (0.286, 0.214, 0.143, 0.143, 0.071) are given

Table 1: An example of the proposed algorithm, where PU(U) denotes the probability of

the source symbol U.

U PU(U) After step 1 After step 2

a1 0.286 1 00

a2 0.214 00 11

a3 0.143 010 010

a4 0.143 0110 101

a5 0.143 01110 0110

a6 0.071 011110 1001

Average codeword length 2.856 2.714

The Proposed Algorithm (4/4)

• Table 2: Some temporary results of the proposed algorithm.

• T-list: (1, 00, 010, 0110, 01110, 011110)

Iteration #1

c R(c,T-List, 2)

Mergeresult

Average code lengthafter replacement

1 11,101,

1001,10001,100001

00, 11,010, 101,

0110, 1001,01110, 10001,

011110, 100001

2.714

00 0000,00100,001100

1, 0100000, 0110

00100, 01110001100, 011110

3.142

 

Others are 3.285, 2.856, 2.856 and 2.856

Experimental Results (1/3)

• Test corpuses are from Canterbury Corpus File Set. (available in http://corpus.canterbury.ac.nz/)

• Table 3 respectively lists various symmetrical RVLCs constructed by other algorithms, and the proposed algorithm for the English alphabet set.

• Table 4 lists the results of the Canterbury Corpus file set compressed by using other algorithms and the proposed algorithm

Experiment Results (2/3)Table 3: Symmetrical RVLCs for the English alphabet set

Experiment Results (3/3)Table 4: the average codeword lengths of various symmetrical RVLCs for the Canterbury Corpus file set

Conclusion (1/2)• The optimal design of RVLC with good distance p

roperties and compression efficiency is still an open problem.

• The proposed algorithm is suitable for any VLC-involved applications to enhance the error-correcting capability of critical time-constrained applications.

• It should be pointed out that symmetrical RVLCs with cannot be surely obtained by the proposed algorithm.

3freed

Conclusion (2/2)

• The major contribution of the proposed algorithm is that it yields more efficient symmetrical RVLCs with the same free distance than other known algorithms.

Level 4 No. of Available Candidate Codewords at level 5

No. of Available Candidate Codewords atlevel 6

No. of Available Candidate Codewords at level 7

Candidatecodewords

MSSL

0110 1 8 6 15

1001 1 8 6 15

0000 3 7 6 14

1111 3 7 6 14

0

10

110

0110

MSSL:1

0000

0

00

000MSSL:3

MSSL (Maximum Symmetrical Suffix Length)

.

not symmetrical