It is shown that (n sub 0, k sub 0) convolutional codes with unit memory always achieve the largest free distance among all codes of the same rate k sub 0/n sub 0 and same number 2MK sub 0 of encoder states, where M is the encoder memory. A unit-memory code with maximal free distance is given at each place where this free distance exceeds that of the best code with k sub 0 and n sub 0 relatively prime, for all Mk sub 0 less than or equal to 6 and for R = 1/2, 1/3, 1/4, 2/3. It is shown that the unit-memory codes are byte-oriented in such a way as to be attractive for use in concatenated coding systems

Lee, L. N.

English

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https://ntrs.nasa.gov/search.jsp?R=19750019703 2020-03-22T20:07:46+00:00Z
I I
(NASA -CO-143100)	 St10RT, UNIT-MEMORY,
HYTE-ORI?NTED, BINARY CONVOLUTIONAL CODES
HAVING MAXIMAL FREE DLS'rANcF ( Notre DAme
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ELECTRICAL ENGINEERING
UNIVERSITY OF NOTRE DAME, NOTRE DAME, INDIANA I
Technical Report No. E1:-754
Lin-nan Lee
Department of Electrical Engincering
University of Notre Dame
Notre Dame, Indiana 46556
Short, Unit-Memory, Byte-Oriented, Binary
Convolutional Codes llaviug Maximal Free Dir;tance*
July 11, 1975
Abstract: It is shown that (n o) ko ) convolutional codes with unit memory always
achieve the largest free distance among all codes of the same rate k o n  and
Mk
^ime number 2 0 o encoder states, where M is the encoder memory. A i;nit-
memory code with maximal free distance is given at each place wher, this free
distance exceeds that of the best code with k  and n o relatively prime, for all
Mk < 6 and for R - 1/2, 1/3, 1/4, 2/3. It is shown that the unit-memory codes
o --
are byte-oriented in such B. way as to be attractive for use in concatenated
coding systems.
* This research was supported by the National Aeronautics and Space
Administration ender NASA Cram NSG 5025 at the University of Notre
Dame in liaison with the Goddard Space Flight Center..
W1 .	 >
2 ---i -
k0
binary unit delay
0 0 0
I -.
— _ LI	 multiply by
 matrix G 
CM
.L
_> 1
2
•	 n
1
I. ;NTRODUC.ION
Let the binary k0-tuplet denote the subblock of information digits
at time t (t - 0, 1,2,...), and let the binary n0-tuple b  denote the encoded
subblock at time t in an (n o , k 0 ) convolutional code. Then, the encoding
equations may be written
b t
 - At Go + t-1Gl + ... + at-D1GTI (t a 0, 1,2,...)
	 (1)
where cacti 
C  
is a k0 x no binary matrix, where M is the code memur , where
the operations are in GF(2), and where, by way of convention, A - 0 for
t < 0. An encodin,; circuit is shown in Figure 1. Note that the encoder has
Mk
2 0 distinct states where the state is taken as the contents of the delay cells
in the encoder. Ve_ shall refer to the number Mk 0 
of binary state variables in
the encode r ?s the state-complexity_ of the convolutional code.
Fig. 1:
	
An encoding circuit for an (no) k0) convolutional
code with memory M.
C!
2
The constraint length K (measured in information digits) of the convo-
lutional code is defined by
K - (M + 1)kc
The rate R of the code is def. , by
R - k /n0 0
In virtually all past applications of convolutional codes, k 0	 0and n have been
takes i. • relatively prime. i.e., gcd 0,09 n0) - 1 where "gcd" denotes "greatest
conmion divisor'.' In fact, the condition gcd (ko , n0 ) - 1 is generally tacitly
assumed so that speaking,for instance, of a convolutional code as being of
rate R = 1.12 woulc' imply k0 - 1 and 11 0 = 2 unless the contrary were explicitly
stated. As will be seen, however, there can be advantages in taking
gcd (k o , n0 ) > 1.
For convenience, met b[t,t,] denote the encoded sequence lbt :h t+1' .. "]t'^
nver time units t through t' and let h	 ) denote the entire semi - infinite
encoded sequence. Let a[t.t,] and a [O,m) be similarly defines. The free dist ance,
dfree' of the convolutional code is the minimum }lamming distance between all
pairs of encoded sequences b [0,-) resulting from pairs of information sequences
a [0,-) chat differ in their time 0 subblock. By the linearity of (1), it
follows that
dfree - min W(b[0,„))
a +0
where W( • ) denotes the Hamming weight of the enclosed binary sequence and where
the minimization is over all a [0,.) such that %f0. The code is non-catastrophic
[1,2] when W(n [Om) )	 m implies that W(b [O ^u) 1 = m , in which case the minimization
in (<) reduces to the minimum over all a [01-) such that a!oO and 140 [0, ^ ) ) < W,
11ie restriction to non-cata.:.rophic codes entails no loss in the achievable value
(2)
II
3
of 
dfree 
for given n o , k  and M, a fortunate situation because the non-
catastrophic property is essential in applications (cf. (11).
Because 
dfree 
is the primary determiner of decoding error probability
	 •
when Viterbi (i.e., maximum likelihood] decoding is used with a non-catastrophic
code, 
dfree 
is the usual criterion of goodness for codes to be used with Viterbi
decoders. Because the number of states of the Viterbi decoder [5] coincides
Mk
with the number of encoder states, viz. 2 0 , pr;.cticality dictates a small
state-complexity. The region Mk  < 6 appear: Lo be about the range where
Viterbi decoding is attractive in applications. Thus, for Viterbi decoding
applications, we are motivated to find, for a given code rate and a given state-
complexity in the above range, a convolutional code with maximun' dfree' In
the next sec t ion, we report the results of our search for such codes and we
also derive a useful upper bound on the attainable dfree'
II. UNIT-M);MORY CODES WITH MAXIMA]. dfree
Any (n o , ko) convolutional code with memory M can be considered as an
(no - Mno , k'0 	Mk 0)code with M' = 1 simply by taking
GO	 G1 ••• GM-1
G4 =
	
0	 Go ... GM--2
0	 0	 ... GO
and
G 	 ....	 0-'
G1 c	 GM-1	 C^m ... 
U
i	 G1	 G2 ... GM
I These two codes a-e entirely equivalent in the sense that the same semi-infinite
binary information sequence produces the n.me semi-infinite binary encoded
sequence, although the diviaion into time subblocks would be different. Since
— 0i	 _ _
	
-
___
F .
4the state-complexity is Mko for both codes, it follows that for a given ntate-
complexity and given rate, the maximum value of dfree is achieved within the
subset of conv olutional codes with M - 1. Hence, we can restrict our search
for optimal codes, for a given rate and state-complexity, to codes with unit-
memory .
For a unit-memory code, equation (1) reduces to
b  - a 
t 
G 0 + at
-1GI	
(t - 0, 1,2,...).	 (3)
When 4#0 is the only non-zero information subblock, then
b (O,lj 
_ a0 (GO :G l ]	 (4)
is the only possibly-non-zero protion of b [0.-) . From (2), it then follows
that the attainable dfree 
of a unit-memory (n o , ko ) convolutional code is upper-
bounded by the .largest minimum distance of an (n = 2no , k - ko) block cede.
We shall call this upper bound the block code upper bo und on dfree' and we note
that McFliece and Rumsey (6] have used similar arguments to derive more
elaborate upper bounds on dfree foL codes where M # 1.
We see that the argume -r used above to establis`i ae block code upper
bound on dfree suggests the following search procec	 or finding a non-
catastrophic unit-memory (n0 , ko ) cede with maximal 
a free :
(i) Set d equal to the largest dmin achievable by any (n = 2no , k - ko)
block code.
(ii) Choose (G 0 :G 1 I as the generator matrix of a (n - 2nv , k = ko ) block
code with dmin - d. If dfree - d and the code is ion-catastrophic, stop.
Otherwise continue with step (ii) until all block erodes with d
min	
d have been
exhausted.
(iii) Reduce d by 1 and return to step (ii).
The above search procedure was carried out to obtain, for rates 1/4, 1/3,
1/2 and 2/3 (which are the usual rates of interest in applications), a
snon-catastrophic urit-memory convolutional code with maximal 
dfree for all
state-complexities of 6 or less. The values of d free obtained are given in
Table I where we also list, for comparison, the largest 
dfree 
attainable by
a code of the same state-complexity having gcd (n o , ko ) - 1. The block code
upper bound on 
dfree 
for each case is also listed.
The codes with gcd (n o , ko ) = 1 that achieve the values of dfree given in
Table I may be foun(. 1 in Larsen [3) and Paaske [4). The values of the block
code upper bound on 
dfree' 
given in Table I, were taken from Calabi and
Myrvaagnes [7) and frum Helgert and Stinoff [8]. In Table II, we give the
matrices G0 and G  of a non-catastrophic unit-memory code with maximal. dfree
at each place where Table I shows that value to exceed the dfree 
available from
the best gcd (no , ko ) - 1 code having the same -ate-complexity.
6Rate	 State Complexity	 Block Code Upper Maximal d fre of	 Maximal df€r 
a 
of
(ko /no ) (No. of State Variables) Bound on
	 dfree	 Unit-Memory Eudes gcd (not K0 - 1
Codes
1/4	 1	 8	 7	 7
2	 10	 10	 10
3	 13	 13	 13
4	 16	 16	 16
5	 ?.0	 20	 18
6	 24	 24	 20
1/3
	 1	 6	 5	 5
2	 8	 b	 8
3	 10	 10	 10
4	 12	 12	 12
5	 15	 15	 13
6	 16	 16	 15
1/2	 1	 4	 3	 3
2	 5	 5	 5
3	 6	 6	 6
4	 8	 8	 7
5	 9	 9	 8
6	 10	 10	 10
2 2/3
	 2	 4	 3	 3
4	 6	 6	 5
	
8	 7	 7
Table 1.	 Maximal dfree for a given state-complexity Mk  of
unit memory convolutional codes and of convolutional
codes with gcc (n 
0 
$ k 
0 ) = 1.
0
r	 -
7G1
10000111
01001011
00101101
00011110
1000011111
0100001111
0010011110
0001011001
0000110101
100j01111011010
010000111.101101
001001011110110
000101101101011
000011110110101
111000110100110000
011100011010011000
001110001101001100
003111100110000110
100011010011000011
110001101001100001
1.0000011111110011000
01000101110111001100
00100110110011100110
00010111011001100011
00001111101100110001
100000111110111010110000
010000011111011101011000
001000101111101110001100
000100110111010111000110
000010111011101011000011
000001111101110101100001
100001
010010
001011
000111
G2
10001011
11100010
10111000
11010001
1111111111
1111000000
0010110100
1010011010
0110101001
000111110010100
001101100101010
01100100110010?
110000011110010
100010111001001
000011000111001011
000110001110010110
001100011100101100
011000111000011001
11000011000 110010
100001100011100101
00011110100001100111
0O1100110101100u1110
01100101101000111100
11000010110011011001
10001101011100010011
001111000101001011010011
011110001010010110100110
111100010100101100001101
111001101000011001011010
110011010001110010110100
100?11100010100101101001
111100
110110
010011
101001
Rate n	 k0 0
1/2	 8	 4
1/2	 10 5
1/3	 15 5
1/3
	 18 6
1/4	 20 5
1/4
	
24 6
2/3
	
6	 4
df ree
8
10
15
16
20
24
6
Table 2.	 The encoding matrices of some non-catastrophic unit-memory
convolutional codes with free distance greater than the
maximal free distance of gcd (no, k 0 ) - i codes of the same
rate and state complexity.
BIII. BYTE-ORIENTED NATURE OF UNIT-MEMORY CODES
We no» show that short, unit-memory convolutional codes are "byte-oriented"
In such a way as to be attractive for use, with Viterbi decoding, as the inner
coding component of a can. .tenated coding system.
an general, the state at titn-- t of the convolutional encoder is the infor-
mation Sequence a(a . • a	 :a	 ) over the preceding L tithe	 a[t-M,t-1)	 -t-M:,t-M+1	 -t-1
units. Note that the successor of this state, namely, 1.2t-M+1'	 'fit-1'a-t)'k
Is already determined up to the 2 ° choices of a t , i.e., each state will have
k
2 o successor: in the "trellis" defined by the convolutional code (5). In
the corresponding Viterbi decoder, the "metric" for the best path to each of
Mk	 k
the 2 o possible states at time t must be relayed to each of its 2 o successors.
Hence, the value of k0 influences the overall complexity of the Viterbi decoder,
although much less strongly then does the state-complexity. Nonetheless, to
determine, for instance, whether the Viterbi decoder for an R - 1/3, M - 1,
k a0	
6 code (state-complexity 6) is simpler than the Viterbi decoder for an
R	 1/3, M - 7, k0 
- 1 code (state-complexity 7) would require a detailed
analysis of the specific decoder design.
Vith Viterbi decoding of convolutional codes, there is a natural segmentation
of the decoded information digits into bytes of k  bits, because the information
byte at acts as a unit in determining the correct state. The typical decoding
"error events" j2) give rise to the incorrect decoding of a small number of
bytes, v17., the average number of non-zero information bytes in an information
sequence a [0,-) that generates an encoded sequence: 1) [0,-) of 11amnting weight
dfree' In an M = 1 code, this average is near 1, although there will generally
be about ko/2 bit errors per byte. In a code with k  - 1, there will generally
be a small number (say, about 3) of byte errors hut, since a byte is a bit
in this case, the same number of bit errors. however, if the information bits
r.
r	 !
9with the latter coete are Fathered into "bytes" of Mk  - M bits, there will
generally be about 1 1	 "byte" errors per error event since the bit decoding
errors are not synchronized to begin at the start of these "bytes." Thus,
even if both codes have the same state-complexity and same dfree' one would
expect the unit-memory code to have a lower byte-error probability for bytes
of Mko hits.
To test the validity of these observations, we simulated Viterbi decoding
oil
	
additive white Gaussian noise (AWGN) channel with several values of the
energy per information bit to one-sided noise power spectral density ratio,
Kb /No , for several convolutional codes of rate R m 1/3. The codes tested were
(i) the k  - 6 unit-memory code of Table II, (ii) the k o M 1, M - 6 code given
by Larsen [3), and (ii'_) the k0 
a 1, M - 7 code given by Larsen [3]. The byte
size was 6 bits. The results of the simulation are given in Table III. The
unit-memory code (i) had a Jecoding byte-error probability about one-Half that
of the gcd (not k0) - 1 code (ii) with the same state-complexity (Mk o = 6).
This superiority of code (i) over code (ii) is due mainly to its greater free
distance, 16 as opposed to 15. But we also see from Table II? that the decoding
byte-error probability of code (i) is about two-thirds that of code (iii)
which has the same free distance 16 and greater state-complexity, 7 versus 6.
This superiority of code (i) over code (iii) is due entirely to its byte•
oriented nature.
We conclude that the unit-memory codes, because of their byte-oriented
nature, appear ttractive for use as the inner code in concatenated coding
systems [9] where the outer code is a Reed-Solomon (RS) code over the alphabet
k
GF(2 0 ), i.e. the bytes of the convolutional code are single digits for the RS
code. For instance, code (i) above would be used with an RS code over (:F'(2 6).
We expect in the future to report in detail on the effectiveness of unit-
memory codes in such applications.
L21.
10
Fb /No
(dB)
1.00
1.25
1.50
1.7j
M-1, (18,6) code
Byte- 95% Con-
Error	 fidence
Prob.
0.02950 ±0.00533
0.0192 1) +0.00435
0.01100 +0.00329
0.00625 +0.00250
MM6, (3,1) code
Byte-	 95% Con-
Error
	
fidence
Prob.
O.U4375 +0.00681
0.03250 +0.00561
0.02325 +0.00477
0.01275 +0.00350
M-7 (3,1) code
Byte-	 95% Con-
Error	 fidence
Prob.
0.04000 +0.00619
0.02250 40.00469
0.01400 +0.00372
0.01025 +0.00319
Table III.	 Byte-error probability for Viterbi decoding
of three R - 1/3 convolutional codes on a
simulated AWGN channel.
.
F r
ii
ACKNOWJLE:DG UIENT
The author would like to thank Professor James L. MRssey who not only
suggested to the author the possibility of constructing unit-memory convo-
lutional coders superior to gcd (n o , ko) - 1 codes of the same state- complexity,
but also supervised this work during the course of this research.
t
li
REFERENCES
[1] J. L. Ma.:sey &nd M. K. Sain, "Inverses of Linear Sequential Circuits,"
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(2] G. D. Forney, Jr., "Convolutional '.odes I: Algebra'c Structure," IEEE
Transactions on Information Theory, Vol. IT-16, pp. 120-738, November
1970.
[3] Larsen, K. J., "Short Convolutional Codes w' h Maximal Free Distance for
Rates I12, 1/3, and 1/4," IEEE Transactions on Information Theory,
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(4] Paaske, E., "Short Binary Convolutional Codes with Maximal Free Distance
for Rates 2/3, and 3/4," IEEE Transactions on Information Theory,
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[5] A. J. Viterbi, "Convolutional Codes and Their Periormance in Communication
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[6] Mv_' l.iece, R. J. and Rumsey, H. C., "Capabilities of Convolutiona: Codes,"
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[7] Calabi, L. and 0yrva—,nes, E., "On the Minimal Weight of Binary Group
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Short, unit-memory, Byte-oriented, binary convolutional codes having maximal free distance

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