General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright
owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
You may not further distribute the material or use it for any profit-making activity or commercial gain
You may freely distribute the URL identifying the publication in the public portal
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from orbit.dtu.dk on: Aug 10, 2022
Short convolutional codes with maximal free distance for rates 1/2, 1/3, and 1/4
(Corresp.)
Larsen, Knud J.
Published in:
I E E E Transactions on Information Theory
Publication date:
1973
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Larsen, K. J. (1973). Short convolutional codes with maximal free distance for rates 1/2, 1/3, and 1/4 (Corresp.).
I E E E Transactions on Information Theory, 19(3), 371-372.
CORRESPONDENCE
371
computed with Q(Xi,Xj) becomes small. Finally, since
1 W(P’) - W(Q’)I becomes small as
max P(x) -
QWI
x
becomes small. As a consequence of (5) it then follows that
max 1 W(P,‘) - W(P’)I L 0 with probability I.
[ET
(6)
The implication of (6) is simply that for all s sufficiently large
we will, with probability I, always pick a tree in T’ if we choose
t(s) such that W(P,‘) is maximum. Using the theorem of Chow
and Liu quoted earlier with (6), (3) now follows.
The same ideas as outlined above also yield the following
statement. If
P
is an arbitrary distribution and
Psf@)
is picked
as before, then
W(P,““‘) 5 max W(P’) with probability 1
tET
TABLE I
RATE 4
CODES
WITH MAXIMUM FREE DISTANCE
A. Noncatastrophic Codes
” N
generators(octa1) dfree
bound
.l
5 6 5
7'
5 5
4 8 15
17'
6 6
5
10
25
35'
7
8
6 12
53 75'
8 8
7
14
155
171'
10
10
8 16
247
571'
10
11
9
,a
561 753'
12
12
10 20 1167
1545
12
15
11 22
2535
3661
14
14
12 24
4335 5723 15
16
13 26
3053
17661
16
16
14 28
21675
27123
16
37
B. Catastrophic Codes
9 N
generators(octa1) dfree bound
5 10 27 35 8
8
12 24 5237 6731 16
16
14 28 21645 371~3 17
‘7 b
even though P,‘@) itself may not converge.
1
This code was
found by Odenwalder [4] and is listed here for
completeness.
ACKNOWLEDGMENT
C. K. Chow wishes to thank Prof. H. Chernoff of Stanford
code is the appropriate criterion of goodness for the convolu-
University for his interest and enlightening suggestions.
tional code used with Viterbi decoding.
REFERENCES
[l] C. K. Chow and C. N. Liu,
“Approximating discrete probability
distributions with dependence trees,” IEEE Trans. Inform. Theory,
vol. IT-14, pp. 462467, May 1968.
The rates of most practical interest for Viterbi decoding on
memoryless channels are
R
= 3, +, and 3.
R
= + codes with
maximal d,,,,
are already known for v 5 9 [4] and
R = 3
codes
with maximal d,,,,
are known for v I 24 [S].
R = +
codes
with maximal d,,,,
are known for v I 8 [4] and with nearly
maximal d,,,,
for v 5 28 [6]. The best
R
= + codes reported
are repetitions of theBahl-Jelinek
R = +
codes [5], i.e., Gc3)(0) =
G(l)(D) and Gc4)(0) = G@)(O). In this correspondence we
report rate +, 3, and + codes with maximal dr,,, for v 5 14.
Short Convolutional Codes With Maximal Free Distance
for Rates +, 3, and $
KNUD J. LARSEN
Abstract-This paper gives a tabulation of binary convolutional codes
with maximum free distance for rates s, f, and t for all constraint
lengths (measured in information digits) v up to and including v = 14.
These codes should be of practical interest in connection with Viterbi
decoders.
A binary convolutional code of rate
R
= l/n and constraint
length v, measured in information digits, is specified by its code
generating polynomials
G’1’(o) = 1 + g,“‘D + g;“)D’ + . . . + g$Dv-’
for 1 5
i
5 n where each gjo) is a binary digit. It is now well
known that the Viterbi decoding algorithm is the maximum-
likelihood decoding rule for the
trellis
defined by such a code
[l] and that surprisingly good performance on memoryless
channels such as the deep-space channel can be obtained for
codes with small enough v, say v 5 10, so that the Viterbi
decoder could actually be implemented [2]. It is also well
known [2]-[4] that the free distance dr,,, of the convolutional
The newly found codes, together with some previously known
codes with maximal dr,,, for rates
R
= +, 3, and t are listed in
Tables I, II, and III, respectively, where we follow the usual
practice of listing the generating polynomials by the octal form
of the binary sequence l,g,(i),g,o),+ . + ,gFll, for 1 I
i 5 n.
The number N =
vR-l
is the total constraint length. The
optimality of d,,,,
for these codes can be established from a
simple upper bound, due to Heller [7],
p
2gq(v
+ k - l)]
(1)
where [ ] denotes integer part of the enclosed expression. This
bound can be improved [8] for some (n,v) using the Griesmer
bound for block codes [9]. The latter bound says that if
do
is
the minimum distance of an
(N,k)
binary linear code, and if
di
=
[(di-1
+ 1)/2], then
do + d, + *a* + dkel I N [=
(v +
k
- 1)n in this case]. Thus by checking for every (n,v)
the bound (1) can in some cases be improved by one. The
resulting upper bound is listed in the Tables I, II, and III.
From Tables II and III it is seen that optimal codes (in the
sense of maximum
d,,,)
achieving the bound for rates
R = 3
and $ were found for all constraint lengths (up to and including
14). These codes, all of which are noncatastrophic [lo], were
found by judicious choosing of the generating polynomials
Manuscript received June 19, 1972; revised September 11, 1972.
The author is with the Laboratory for Communication Theory, Technical
followed by a computer verification of their
d,,,,
using a corrected
University of Denmark, Lyngby, Denmark.
version of the algorithm given by Bahl et al. [Ill, [12].
372
IEEE TRANSACTIONS ON INFORMATION THEORY, MAY 1973
TABLE 11
RATE f NONCATASTROPHIC CODES WITH MAXIMUM FREE DISTANCE
N generators (octal)
3 9
4 12
5 15
6 18
1
21
8 24
9 21
IO 30
11 33
12 36
13 39
14 42
5 7 7’
I3 15 11’
25 33 37’
41 53 75’
133 145 1752
225 331 367’
557 663 711
,117 7365 1633
2353 2671 3175
4161 5723 6265
10533 10675 17661
21645 35661 37133
d
I-Pee
bound
J
8 8
IO 10
12
12
I3 13
15 15
16 16
1s 18
20 20
22 22
24 24
24 24
26 26
1 This code was found by Odenwalder [4] and is listed here for
completeness.
z This code was also found by Odenwalder [4], but was overlooked.
The corresponding code in [4] has free distance only 14.
TABLE III
RATE 2 NONCATASTROPHIC CODES WITH MAXIMUM FREE DISTANCE
3
12
4
16
5
20
6 24
1
28
8
32
9 36
IO 40
‘1 44
12 48
13 52
14 56
13 15
25
27
53 67
135 135
235 275
463 535
1117
1365
we?
2353
4767 5723
11145 124ll
21113
23175
7 7
IO
10
15 17
13
15
33 37
16 16
71 75
18
18
147
163
20
20
313
357
22
22
133
745 24
24
1633 1653 27 27
2671
3175
29
29
6265 7455
32
32
15573 76727
33
33
35521 35537
36
36
The noncatastrophic rate + codes (Table I-A) are all optimal
(i.e., maximum d,,,,) but some of them (V = 5, 8, 10, 12, and
14) do not achieve the bound. The optimality is here established
through a complete search covering all possibly optimal codes.
If we allow the codes to be catastrophic, which might be of
interest in connection with framing of input data, we can find
codes achieving the bound for v = 5, 12, and 14, too, if the
definition of
d,,,,
[3] is slightly modified:
d,,,,
is the weight of the
minimum weight path in the trellis that diverges from the state
0 and later
reconverges
to this state; this reconvergence is not
required in [3]. For a noncatastrophic code the two definitions
are identical. The catastrophic codes for v = 5, 12, and 14 are
listed in Table I-B.
ACKNOWLEDGMENT
The author is grateful to Dr. J. L. Massey (Guest Professor)
and Dr. E. Paaske of the Laboratory for Communication
Theory, Technical University of Denmark, for their supervision
of the work reported here.
HI
PI
[31
[41
[51
[61
[71
181
[91
[lOI
[Ill
[121
REFERENCES
J. K. Omura, “On the Viterbi decoding algorithm,”
IEEE Trans.
Inform.
Theory
(Corresp.), vol. lT-15, pp. 177-179, Jan. 1969.
J. A. Heller and I. M. Jacobs, “Viterbi decoding for satellite and space
communication,”
IEEE
Trans. Commun. Technol., pt. II, vol. COM-19,
pp. 835-848, Oct. 1971.
J. L. Massey and D. J. Costello, Jr.,
“Nonsystematic convolutional
codes for sequential decoding in space applications,” IEEE Trans.
Commun. Technol., pt. II, vol. COM-19, pp. 806-813, Oct. 1971.
J. P. Odenwalder, “Optimal decoding of convolutional codes,” Ph.D.
dissertation, Dep. Syst. Sci., Sch. Eng. Appl. Sci., Univ. California,
Los Angeles, 1970.
L. R. Bahl and F. Jelinek, “Rate 3 convolutional codes with com-
plementary generators,” IEEE Trans. Inform. Theory, vol. IT-17, pp.
718-727, Nov. 1971.
S. J. Curry,
“Selection of convolutional codes having large free
distance,” Ph.D. dissertation, Dep. Syst. Sci., Univ. California, Los
Angeles, 1971.
J. A. Heller “Sequential decoding: Short constraint length convolu-
tional codes,” Jet Propulsion Lab., California Inst. Technol., Pasadena,
Space Program Summary 37-54, vo!. 3, Dec. 1968, pp. 171-174.
J. Layland and R. McEliece, “An upper bound on the free distance
of a tree code,” Jet Propulsion Lab., California Inst. Technol.,
&&ma, Space Program Summary 37-62, vol. 3, Apr. 1970, pp.
J. H. Griesmer. “A bound for error-correcting codes,” IBM J. Res.
Develop., vol. 4, no. 5, 1960.
J. L. Massey and M. K. Sain, “Inverses of linear sequential machines,”
IEEE Trans. Comput., vol. C-17, pp. 330-337, Apr. 1968.
L. R. Bahl. C. D. Cullum. W. D. Frazer, and F. Jelinek, “An efficient
algorithm ‘for computing the free distance,” IEEE Trans. Inform.
Theorv. vol. IT-18. oo. 437-439. Mav 1972.
K. J. Larsen Cor;e&on to “Ah effi;ient algorithm for computing the
free distance:” Rep. Lab. Commun. Theory, Technical Univ. Denmark.
Lyngby, 1972.