Binary multilevel convolutional codes (CCs) with unequal error protection (UEP) capabilities are studied. These codes belong to the class of generalized concatenated (GC) codes. Binary CCs are used as outer codes. Binary linear block codes of short length, and selected subcodes in their two-way subcode partition chain, are used as inner codes. Multistage decodings are presented that use Viterbi decoders operating on trellises with similar structure to that of the constituent binary CCs. Simulation results of example binary two-level CC\u27s are also reported

Imai, Hideki

Morelos-Zaragoza, Robert H

English

San Jose State University

San Jose State University
SJSU ScholarWorks
Faculty Publications Electrical Engineering
7-1998
Binary Multilevel Convolutional Codes with
Unequal Error Protection Capabilities
Robert H. Morelos-Zaragoza
LSI LOGIC Corporation, robert.morelos-zaragoza@sjsu.edu
Hideki Imai
University of Tokyo
Follow this and additional works at: https://scholarworks.sjsu.edu/ee_pub
Part of the Electrical and Computer Engineering Commons
This Article is brought to you for free and open access by the Electrical Engineering at SJSU ScholarWorks. It has been accepted for inclusion in Faculty
Publications by an authorized administrator of SJSU ScholarWorks. For more information, please contact scholarworks@sjsu.edu.
Recommended Citation
Robert H. Morelos-Zaragoza and Hideki Imai. "Binary Multilevel Convolutional Codes with Unequal Error Protection Capabilities"
Faculty Publications (1998): 850-853. doi:10.1109/26.701299
850 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 7, JULY 1998
Binary Multilevel Convolutional Codes with
Unequal Error Protection Capabilities
Robert H. Morelos-Zaragoza, Senior Member, IEEE, and Hideki Imai, Fellow, IEEE
Abstract—Binary multilevel convolutional codes (CC’s) with
unequal error protection (UEP) capabilities are studied. These
codes belong to the class of generalized concatenated (GC) codes
[1]. Binary CC’s are used as outer codes. Binary linear block
codes of short length, and selected subcodes in their two-way
subcode partition chain, are used as inner codes. Multistage
decodings are presented that use Viterbi decoders operating on
trellises with similar structure to that of the constituent binary
CC’s. Simulation results of example binary two-level CC’s are
also reported.
Index Terms—Multilevel codes, multistage decoding, punctured
convolutional codes, unequal error protection.
I. INTRODUCTION
GENERALIZED concatenated (GC) codes [1] are a pow-erful family of error correcting codes based on multiple
outer codes, and an inner code and its partition into subcodes.
Due to their multilevel structure [2], GC codes can be designed
as unequal error protection (UEP) codes. The need for UEP
arises in communications systems where part of the source
messages are more important, or error sensitive, than others.
Specific examples include practically all digital speech and
image transmission systems.
Convolutional UEP codes based on the GC code construc-
tion are generalizations of the two-level convolutional codes
(CC’s) introduced independently by Pellizzoni and Spalvieri
[3] and by Hattori and Saitoh [4], and then studied in [5] and
[6]. Moreover, multistage decoding procedures can be devised
for convolutional UEP codes as a natural generalization of the
two-stage decoding of CC’s [3]–[6].
In [3] and [4] multilevel coset (or “superimposed”) codes
with outer CC’s are presented. In [3] two-level CC’s were
constructed and shown to offer improved error performance,
with reduced decoding complexity, compared to previously
known codes with the same number of states. A construction
using punctured convolutional (PC) codes as component codes
in the construction was proposed in [4].
In this letter it is shown that two-level CC’s based on the
construction are special cases of GC codes. This
gives binary multilevel CC’s a rich algebraic structure that
Paper approved by M. Fossorier, the Editor for Coding Theory and
Communication of the IEEE Communications Society. Manuscript received
December 21, 1996; revised February 20, 1998. This paper was presented
in part at the 1996 International Symposium on Information Theory and Its
Applications (ISITA’96), Victoria, B.C., Canada, September 17–20, 1996.
R. H. Morelos-Zaragoza is with LSI LOGIC Corporation, Milpitas, CA
95035 USA.
H. Imai is with the Institute of Industrial Science, University of Tokyo,
Minatoku, Tokyo 106, Japan.
Publisher Item Identifier S 0090-6778(98)05160-5.
is useful in explaining their UEP capabilities and in devising
low-complexity multistage decoding algorithms.
II. BINARY MULTILEVEL CONVOLUTIONAL CODES
As usual, let denote a binary linear block code
of length , dimension , and minimum distance The
construction of a binary multilevel CC starts with a binary
code and its two-way partition as a chain of
subcodes such that
, where, for convenience, we define
and Let denote
an binary subcode of , which is a set of coset
representatives of in , of dimension
and minimum Hamming distance Then
is the direct sum For
, let denote a rate- memory- binary
CC with minimum distance Then a binary multilevel CC
is defined as the direct sum
(1)
where denotes a product (or concatenated) code
with as outer code and as inner code, .
is a binary -level CC of rate , memory
, and minimum distance , where is
given by and
A. UEP Capabilities
Because of their structure, multilevel CC’s can be designed
as UEP CC’s. The UEP capability of a linear CC follows
from a necessary and sufficient condition for a linear code
to be a UEP code [7]: the set of minimum-weight codewords
does not span a linear UEP code. From this condition and the
definitions above, it follows that if
(2)
then those coded sequences in correspondence with in-
formation bits have a Hamming distance between them of at
least In the language of UEP codes we
say that a binary multilevel CC is an -level UEP code with
separation vector
and message space
For simplicity, in this letter only two-level UEP CC’s are
dealt with. It is assumed that the -component concatenated
0090–6778/98$10.00  1998 IEEE
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 7, JULY 1998 851
codes are chosen in such a way that the separation vector
is , where
,
with and
III. TWO CONSTRUCTIONS OF BINARY
MULTILEVEL CONVOLUTIONAL CODES
In this section two specific constructions of binary UEP
CC’s, based on generalized concatenation, are presented. For
simplicity, we only consider the cases where the inner block
code is of length or , and two or three binary
CC’s are used as outer codes, although other choices of
and number of component codes are possible.
A. Construction I
For , let be a rate- , memory- , and
minimum distance CC. Let be a binary block (2, 2,
1) code with generator matrix
(3)
so that and Then it follows
from the results of Section II that the code
is a two-level CC of rate , memory
, and minimum distance
Suppose that Then it follows from (2) that
is a two-level UEP CC with separation vector
and message space
Note that two-level CC’s constructed in this fashion are, in
fact, equivalent to -constructed codes.
Example 1: Let be a CC of rate 1/4, memory 2, and
minimum distance 10 (from [9, Table A.5]). Let be a PC
code of rate 3/4, memory 3, and minimum distance 4 (from [9,
Table A.8]). Construction I results in a two-level CC of
rate 4/8, memory 5, and minimum distance 8. From condition
(2), it follows that is a two-level UEP CC with separation
vector and message space Also,
note that the minimum distance of the best rate-1/2 memory-3
CC is equal to 8.
A Two-Stage Decoding Procedure: For practical applica-
tions, in order to reduce the complexity of Viterbi decoding
(at the expense of an increased error coefficient or degradation
in coding gain [3], [6]), suboptimal two-stage decoding
(TSD) may be adopted. Coded bits are Binary phase-shift
keying (BPSK)-modulated and transmitted over an additive
white Gaussian noise (AWGN) channel. The following TSD
procedure is considered.
Stage 1: A decoder for a CC is used in this
stage, where
The trellis structure of code is isomorphic (up to
connections between states) to that of the trellis of the first-
level code The only difference between
and is that contains two parallel subbranches per original
two-bit symbol in a branch of That is, {00, 11} (resp. {01,
10}) for 00 (resp. 01). Once an information sequence is
decoded in this first stage, the most important bits (MIB’s)
are recovered. Sequence is then reencoded by to obtain
a coded sequence From this sequence, a modified received
sequence is obtained, where , and passed on
to the second stage.
Stage 2: Using a trellis for the second component code
, the modified received sequence is decoded and the
less important bits (LIB’s) extracted.
Note that iterative decoding may be used, as proposed in [8].
Also, it is possible to use two-stage decoding with interleaving
[6].
B. Construction II
In this section a new construction method that combines
three CC’s is presented. For , let be a rate-
, memory- , and minimum distance CC. Let
be a binary block (3, 3, 1) code with generator matrix
(4)
and subcodes , , and
Then it follows from the results of Section
II that the code
is a CC of rate , memory
, and minimum distance
This construction is a permuted version of a “
construction.”
Suppose that and Then, from
condition (2), is a two-level UEP CC with separation
vector and message space
A Three-Stage Decoding Procedure: A decoding procedure
for codes obtained from Contruction II is similar to the two-
stage decoder in the previous section—decoding proceeds in
three stages, passing decoded information from one stage
to the next, with Viterbi decoders for codes , , and
, where
with , and
with
IV. TABLES OF CODES
By means of a computer search, good binary two-level
CC’s were found based on the construction methods presented
in the previous section. The goal of the search was to find
codes with minimum multistage decoding complexity (number
852 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 7, JULY 1998
TABLE I
PARAMETERS OF THE COMPONENT CODES OF BEST TWO-LEVEL CC’S OBTAINED
IN THE COMPUTER SEARCH. EACH ROW LISTS THE PARAMETERS OF A
MEMORY-m, RATE-k=n, AND FREE DISTANCE df CC. LABELS Gi:j REFER TO
PC CODES. Ci ANDDi REFER TO RATE-1/3 AND RATE-1/4 CODES, RESPECTIVELY
TABLE II
TWO-LEVEL CC’S OBTAINED FROM CONSTRUCTION I
of additions and comparisons in a Viterbi decoder). The
search was performed using the parameters (rate, memory, and
minimum distance) of codes from tables of best convolutional
and PC codes [9].
Table I shows the parameters of memory- rate- CC’s
found in the search to be components of the best codes.
The notation used in Table I is as follows: denotes the
free distance, is the path multiplicity, or number of code
sequences at distance , is the polynomial generator matrix
of the code, with polynomials represented as octal numbers,
and is the puncturing matrix, in the case of a PC code.
Tables II and III list the parameters of codes found in
the search. For simplicity, in the three-level constructions the
component first- and second-level CC’s were chosen such
that in order to obtain two-level UEP codes.
In addition, all of the binary component codes were selected
to have equal number of output bits, i.e., or
TABLE III
TWO-LEVEL CC’S OBTAINED FROM CONSTRUCTION II
Fig. 1. Error performance of code S1: MLD and TSD of the MIB’s and
LIB’s.
Fig. 2. Error performance of code S3: MLD and TSD of the MIB’s and
LIB’s.
V. COMPUTER SIMULATION RESULTS
To illustrate the error performance of binary multilevel
CC’s, two codes from Construction I were simulated. In Figs.
1 and 2, MIB-MLD and LIB-MLD are used to denote the bit-
error rates of the MIB’s and LIB’s, with maximum-likelihood
decoding (MLD), respectively. Similarly, MIB-TSD and LIB-
TSD refer to the bit-error rates with TSD.
The label S1 denotes the example code presented in Section
III-A. The code labeled S3 is obtained from code S1 by
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 7, JULY 1998 853
replacing the rate-1/4 component code D1 of Table I by a
more powerful rate-1/4 code D3. It can be seen from Figs. 1
and 2 that the performance of the MIB with MLD improves
drastically when increasing the number of states from code S1
to code S3 (component code D3 needs 16 states as opposed
to four states of D1). The improvement in performance for
the MIB is gained by, in addition to increasing the number of
states, reducing the coding gain for the LIB. A discussion of
the effect of error coefficients in two-stage decoding can be
found in [6] and is not addressed here.
REFERENCES
[1] V. A. Zinoviev, “Generalized cascated codes,” Probl. Peredachi Inf.,
vol. 12, no. 1, pp. 5–15, 1976.
[2] H. Imai and S. Hirakawa, “A new multilevel coding method using error-
correcting codes,” IEEE Trans. Inform. Theory, vol. IT-23, pp. 371–377,
May 1977.
[3] R. Pellizzoni and A. Spalvieri, “Binary multilevel coset codes based on
Reed-Muller codes,” IEEE Trans. Commun., vol. 42, pp. 2357–2360,
July 1994.
[4] M. Hattori and Y. Saitoh, “A coding technique based on punctured
convolutional codes,” Electron. Lett., vol. 30, no. 13, pp. 1042–1043,
June 1994.
[5] M. P. C. Fossorier, S. Lin, and D. Stojanovic, “Research report:
Decomposable convolutional codes and multi-stage decoding,” Dept.
Elec. Eng., Univ. Hawaii, Honolulu, Mar. 1995.
[6] J.-F. Cheng, “Hyperimposed convolutional codes,” in Proc. ICC’96,
Dallas, TX, June 1996, pp. 979–983.
[7] I. M. Boyarinov and G. L. Katsman, “Linear unequal error protection
codes,” IEEE Trans. Inform. Theory, vol. IT-27, pp. 168–175, Mar. 1981.
[8] K. Fazel, “Iterative decoding of generalized concatenated Blokh-
Zyablov codes,” in Proc. ICC’96, Dallas, TX, June 1996, pp. 96–101.
[9] A. Dholakia, Introduction to Convolutional Codes With Applications.
Norwell, MA: Kluwer, 1994.


Binary Multilevel Convolutional Codes with Unequal Error Protection Capabilities

SJSU ScholarWorks

San Jose State UniversitySJSU ScholarWorksFaculty Publications Electrical Engineering7-1998Binary Multilevel Convolutional Codes withUnequal Error Protection CapabilitiesRobert H. Morelos-ZaragozaLSI LOGIC Corporation, robert.morelos-zaragoza@sjsu.eduHideki ImaiUniversity of TokyoFollow this and additional works at: https://scholarworks.sjsu.edu/ee_pubPart of the Electrical and Computer Engineering CommonsThis Article is brought to you for free and open access by the Electrical Engineering at SJSU ScholarWorks. It has been accepted for inclusion in FacultyPublications by an authorized administrator of SJSU ScholarWorks. For more information, please contact scholarworks@sjsu.edu.Recommended CitationRobert H. Morelos-Zaragoza and Hideki Imai. "Binary Multilevel Convolutional Codes with Unequal Error Protection Capabilities"Faculty Publications (1998): 850-853. doi:10.1109/26.701299850 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 7, JULY 1998Binary Multilevel Convolutional Codes withUnequal Error Protection CapabilitiesRobert H. Morelos-Zaragoza, Senior Member, IEEE, and Hideki Imai, Fellow, IEEEAbstract—Binary multilevel convolutional codes (CC’s) withunequal error protection (UEP) capabilities are studied. Thesecodes belong to the class of generalized concatenated (GC) codes[1]. Binary CC’s are used as outer codes. Binary linear blockcodes of short length, and selected subcodes in their two-waysubcode partition chain, are used as inner codes. Multistagedecodings are presented that use Viterbi decoders operating ontrellises with similar structure to that of the constituent binaryCC’s. Simulation results of example binary two-level CC’s arealso reported.Index Terms—Multilevel codes, multistage decoding, puncturedconvolutional codes, unequal error protection.I. INTRODUCTIONGENERALIZED concatenated (GC) codes [1] are a pow-erful family of error correcting codes based on multipleouter codes, and an inner code and its partition into subcodes.Due to their multilevel structure [2], GC codes can be designedas unequal error protection (UEP) codes. The need for UEParises in communications systems where part of the sourcemessages are more important, or error sensitive, than others.Specific examples include practically all digital speech andimage transmission systems.Convolutional UEP codes based on the GC code construc-tion are generalizations of the two-level convolutional codes(CC’s) introduced independently by Pellizzoni and Spalvieri[3] and by Hattori and Saitoh [4], and then studied in [5] and[6]. Moreover, multistage decoding procedures can be devisedfor convolutional UEP codes as a natural generalization of thetwo-stage decoding of CC’s [3]–[6].In [3] and [4] multilevel coset (or “superimposed”) codeswith outer CC’s are presented. In [3] two-level CC’s wereconstructed and shown to offer improved error performance,with reduced decoding complexity, compared to previouslyknown codes with the same number of states. A constructionusing punctured convolutional (PC) codes as component codesin the construction was proposed in [4].In this letter it is shown that two-level CC’s based on theconstruction are special cases of GC codes. Thisgives binary multilevel CC’s a rich algebraic structure thatPaper approved by M. Fossorier, the Editor for Coding Theory andCommunication of the IEEE Communications Society. Manuscript receivedDecember 21, 1996; revised February 20, 1998. This paper was presentedin part at the 1996 International Symposium on Information Theory and ItsApplications (ISITA’96), Victoria, B.C., Canada, September 17–20, 1996.R. H. Morelos-Zaragoza is with LSI LOGIC Corporation, Milpitas, CA95035 USA.H. Imai is with the Institute of Industrial Science, University of Tokyo,Minatoku, Tokyo 106, Japan.Publisher Item Identifier S 0090-6778(98)05160-5.is useful in explaining their UEP capabilities and in devisinglow-complexity multistage decoding algorithms.II. BINARY MULTILEVEL CONVOLUTIONAL CODESAs usual, let denote a binary linear block codeof length , dimension , and minimum distance Theconstruction of a binary multilevel CC starts with a binarycode and its two-way partition as a chain ofsubcodes such that, where, for convenience, we defineand Let denotean binary subcode of , which is a set of cosetrepresentatives of in , of dimensionand minimum Hamming distance Thenis the direct sum For, let denote a rate- memory- binaryCC with minimum distance Then a binary multilevel CCis defined as the direct sum(1)where denotes a product (or concatenated) codewith as outer code and as inner code, .is a binary -level CC of rate , memory, and minimum distance , where isgiven by andA. UEP CapabilitiesBecause of their structure, multilevel CC’s can be designedas UEP CC’s. The UEP capability of a linear CC followsfrom a necessary and sufficient condition for a linear codeto be a UEP code [7]: the set of minimum-weight codewordsdoes not span a linear UEP code. From this condition and thedefinitions above, it follows that if(2)then those coded sequences in correspondence with in-formation bits have a Hamming distance between them of atleast In the language of UEP codes wesay that a binary multilevel CC is an -level UEP code withseparation vectorand message spaceFor simplicity, in this letter only two-level UEP CC’s aredealt with. It is assumed that the -component concatenated0090–6778/98$10.00  1998 IEEEIEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 7, JULY 1998 851codes are chosen in such a way that the separation vectoris , where,with andIII. TWO CONSTRUCTIONS OF BINARYMULTILEVEL CONVOLUTIONAL CODESIn this section two specific constructions of binary UEPCC’s, based on generalized concatenation, are presented. Forsimplicity, we only consider the cases where the inner blockcode is of length or , and two or three binaryCC’s are used as outer codes, although other choices ofand number of component codes are possible.A. Construction IFor , let be a rate- , memory- , andminimum distance CC. Let be a binary block (2, 2,1) code with generator matrix(3)so that and Then it followsfrom the results of Section II that the codeis a two-level CC of rate , memory, and minimum distanceSuppose that Then it follows from (2) thatis a two-level UEP CC with separation vectorand message spaceNote that two-level CC’s constructed in this fashion are, infact, equivalent to -constructed codes.Example 1: Let be a CC of rate 1/4, memory 2, andminimum distance 10 (from [9, Table A.5]). Let be a PCcode of rate 3/4, memory 3, and minimum distance 4 (from [9,Table A.8]). Construction I results in a two-level CC ofrate 4/8, memory 5, and minimum distance 8. From condition(2), it follows that is a two-level UEP CC with separationvector and message space Also,note that the minimum distance of the best rate-1/2 memory-3CC is equal to 8.A Two-Stage Decoding Procedure: For practical applica-tions, in order to reduce the complexity of Viterbi decoding(at the expense of an increased error coefficient or degradationin coding gain [3], [6]), suboptimal two-stage decoding(TSD) may be adopted. Coded bits are Binary phase-shiftkeying (BPSK)-modulated and transmitted over an additivewhite Gaussian noise (AWGN) channel. The following TSDprocedure is considered.Stage 1: A decoder for a CC is used in thisstage, whereThe trellis structure of code is isomorphic (up toconnections between states) to that of the trellis of the first-level code The only difference betweenand is that contains two parallel subbranches per originaltwo-bit symbol in a branch of That is, {00, 11} (resp. {01,10}) for 00 (resp. 01). Once an information sequence isdecoded in this first stage, the most important bits (MIB’s)are recovered. Sequence is then reencoded by to obtaina coded sequence From this sequence, a modified receivedsequence is obtained, where , and passed onto the second stage.Stage 2: Using a trellis for the second component code, the modified received sequence is decoded and theless important bits (LIB’s) extracted.Note that iterative decoding may be used, as proposed in [8].Also, it is possible to use two-stage decoding with interleaving[6].B. Construction IIIn this section a new construction method that combinesthree CC’s is presented. For , let be a rate-, memory- , and minimum distance CC. Letbe a binary block (3, 3, 1) code with generator matrix(4)and subcodes , , andThen it follows from the results of SectionII that the codeis a CC of rate , memory, and minimum distanceThis construction is a permuted version of a “construction.”Suppose that and Then, fromcondition (2), is a two-level UEP CC with separationvector and message spaceA Three-Stage Decoding Procedure: A decoding procedurefor codes obtained from Contruction II is similar to the two-stage decoder in the previous section—decoding proceeds inthree stages, passing decoded information from one stageto the next, with Viterbi decoders for codes , , and, wherewith , andwithIV. TABLES OF CODESBy means of a computer search, good binary two-levelCC’s were found based on the construction methods presentedin the previous section. The goal of the search was to findcodes with minimum multistage decoding complexity (number852 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 7, JULY 1998TABLE IPARAMETERS OF THE COMPONENT CODES OF BEST TWO-LEVEL CC’S OBTAINEDIN THE COMPUTER SEARCH. EACH ROW LISTS THE PARAMETERS OF AMEMORY-m, RATE-k=n, AND FREE DISTANCE df CC. LABELS Gi:j REFER TOPC CODES. Ci ANDDi REFER TO RATE-1/3 AND RATE-1/4 CODES, RESPECTIVELYTABLE IITWO-LEVEL CC’S OBTAINED FROM CONSTRUCTION Iof additions and comparisons in a Viterbi decoder). Thesearch was performed using the parameters (rate, memory, andminimum distance) of codes from tables of best convolutionaland PC codes [9].Table I shows the parameters of memory- rate- CC’sfound in the search to be components of the best codes.The notation used in Table I is as follows: denotes thefree distance, is the path multiplicity, or number of codesequences at distance , is the polynomial generator matrixof the code, with polynomials represented as octal numbers,and is the puncturing matrix, in the case of a PC code.Tables II and III list the parameters of codes found inthe search. For simplicity, in the three-level constructions thecomponent first- and second-level CC’s were chosen suchthat in order to obtain two-level UEP codes.In addition, all of the binary component codes were selectedto have equal number of output bits, i.e., orTABLE IIITWO-LEVEL CC’S OBTAINED FROM CONSTRUCTION IIFig. 1. Error performance of code S1: MLD and TSD of the MIB’s andLIB’s.Fig. 2. Error performance of code S3: MLD and TSD of the MIB’s andLIB’s.V. COMPUTER SIMULATION RESULTSTo illustrate the error performance of binary multilevelCC’s, two codes from Construction I were simulated. In Figs.1 and 2, MIB-MLD and LIB-MLD are used to denote the bit-error rates of the MIB’s and LIB’s, with maximum-likelihooddecoding (MLD), respectively. Similarly, MIB-TSD and LIB-TSD refer to the bit-error rates with TSD.The label S1 denotes the example code presented in SectionIII-A. The code labeled S3 is obtained from code S1 byIEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 7, JULY 1998 853replacing the rate-1/4 component code D1 of Table I by amore powerful rate-1/4 code D3. It can be seen from Figs. 1and 2 that the performance of the MIB with MLD improvesdrastically when increasing the number of states from code S1to code S3 (component code D3 needs 16 states as opposedto four states of D1). The improvement in performance forthe MIB is gained by, in addition to increasing the number ofstates, reducing the coding gain for the LIB. A discussion ofthe effect of error coefficients in two-stage decoding can befound in [6] and is not addressed here.REFERENCES[1] V. A. Zinoviev, “Generalized cascated codes,” Probl. Peredachi Inf.,vol. 12, no. 1, pp. 5–15, 1976.[2] H. Imai and S. Hirakawa, “A new multilevel coding method using error-correcting codes,” IEEE Trans. Inform. Theory, vol. IT-23, pp. 371–377,May 1977.[3] R. Pellizzoni and A. Spalvieri, “Binary multilevel coset codes based onReed-Muller codes,” IEEE Trans. Commun., vol. 42, pp. 2357–2360,July 1994.[4] M. Hattori and Y. Saitoh, “A coding technique based on puncturedconvolutional codes,” Electron. Lett., vol. 30, no. 13, pp. 1042–1043,June 1994.[5] M. P. C. Fossorier, S. Lin, and D. Stojanovic, “Research report:Decomposable convolutional codes and multi-stage decoding,” Dept.Elec. Eng., Univ. Hawaii, Honolulu, Mar. 1995.[6] J.-F. Cheng, “Hyperimposed convolutional codes,” in Proc. ICC’96,Dallas, TX, June 1996, pp. 979–983.[7] I. M. Boyarinov and G. L. Katsman, “Linear unequal error protectioncodes,” IEEE Trans. Inform. Theory, vol. IT-27, pp. 168–175, Mar. 1981.[8] K. Fazel, “Iterative decoding of generalized concatenated Blokh-Zyablov codes,” in Proc. ICC’96, Dallas, TX, June 1996, pp. 96–101.[9] A. Dholakia, Introduction to Convolutional Codes With Applications.Norwell, MA: Kluwer, 1994.

https://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=1012&amp;context=ee_pub

Binary Multilevel Convolutional Codes with Unequal Error Protection Capabilities

Abstract

Similar works

Full text

Available Versions

San Jose State University

SJSU ScholarWorks