A general construction for finite-state (FS) codes is applied to some well-known block codes. Subcodes of the (24,12) Golay code are used to generate two optimal FS codes with d sub free = 12 and 16. A partition of the (16,8) Nordstrom-Robinson code yields a d sub free = 10 FS code. Simulation results are shown and decoding algorithms are briefly discussed

Cheung, K.-M.

Mceliece, R. J.

Pollara, F.

English

NASA Technical Reports Server

TDA Progress Report 42-92
v
N88- 18782
October-December 1987
Further Results on Finite-State Codes
F. Pollara and K-M. Cheung
Communications Systems Research Section
R. J. McEliece
California Institute of Technology
A general construction for finite-state (FS) codes is applied to some well-known block
codes. New subcodes of the (24,12) Golay code are used to generate two optimal FS
codes with dfree = 12 and 16. A partition of the (16,8)Nordstrom-Robinson code yieMs
a d¢_ee = 10 FS code. Simulation results are shown and decoding algorithms are briefly
discussed.
I. Introduction
Future deep-space communication systems will take advan-
tage of powerful error-correcting coding schemes to keep power
• and antenna size requirements within acceptable bounds. Such
codes can be found by computer search or, as considered in
this article, by constructions based on known codes.
In a previous article [1] it was shown how some optimal
Finite-State (FS) codes can be constructed from known block
codes. This article considers new FS codes based on other
block codes and describes performance results obtained by
simulation.
II. Codes Derived From the (24,12)
Golay Code
The basic idea developed in [1] consists in choosing an
(n, kl) block code C 1 with minimum distance all, and then
decomposing C 1 into the disjoint union of cosets generated by
an (n, k2) subcode C 2 of C1, with minimum distance d 2 . By
properly assigning these cosets to the edges of a 2 m-state com-
pletely connected graph, it is possible to construct an (n, k, m)
FS code, with k = m + k 2 and dfree _ min (d2,2dl).
The (24,12) Golay code could be an interesting candidate
for this construction provided that it contains a subcode with
minimum distance d 2 larger than d 1 = 8. The following theo-
rem shows that such a subcode does indeed exist.
%..
Theorem 1. The (24,12) Golay code has a (24,5) subcode with
minimum distance 12.
Proof: The proof is based on the Turyn construction of the
Golay code (p. 587 of [2]). LetA be the (7,3) code with code
words consisting of (0,0,0,0,0,0,0) and the seven cyclic shifts
of (1,1,0,1,0,0,0). Then the (7,4) code H = A L)A, where the
bar denotes the complemented code words, is the (7,4) Ham-
ming code. Similarly, consider the code A* obtained by revers-
ing the order of symbols inA, and the codeH* =A* L) A*. Let
56
https://ntrs.nasa.gov/search.jsp?R=19880009398 2020-03-20T08:21:57+00:00Z
Table 1. Transfer function matrix to generate a completely
connected state diagram with 8 states and 16 labels
1 0 D 1
G(D) : D 1 0 0 df = 2 branches
0 D 1 0
Table 2. Transfer function matrix to generate a non-completely
connected state diagram with 64 states and 16 labels
1 +D 0 D 2 1
G(D) = D 2 1 + D 0 0 dr- = 3 branches
0 D 2 1 +D 0
55
C and C* be (8,4) codes obtained by adding a parity check bit
to H and H*. Then C and C* have Clmin = 4, and the code G
consisting of all vectors
la+xlb+xla+b+xl, a, bEC, xEC* (1)
is the (24,12) binary Golay code with dmi n = 8.
Let B be the subcode of C consisting of the two code words
(0,0,0,0,0,0,0,0) and (1,1,1,1,1,1,1,1). Then the construction
in (1) with a, b E B and x E A* generates code words of the
form
Ixlxlxl, Ixl_l_'l, Ix-lxl_l, Ix'lx'lxl (2)
Code words taken from two distinct subcodes of the four sub-
codes above are at minimum distance 8 + 8 = 16 for fixed x,
and at distance 4 X 3 = 12 forx :_y EA*. Code words in the
same subcode are at minimum distance 4 X 3 = 12. Therefore,
by using all 8 possible choices for x, we have constructed a
(24,5) subcode of the Golay code with dmi n = 12. •
Previously known (24,5) subcodes of the Golay code have
dmi n = 8 [3], [4]. The (24,5) subcode just described can be
represented on a trellis as shown in Fig. 1, where each edge x
or _ corresponds to eight bits. Figure 1 consists of the union
of 8 cosets Di, i = 0, 1 ..... 7, given by (2) withx EA*. Each
coset has 4 code words and is represented by a trellis as shown
in Fig. 2. This observation leads to the following result.
Corollaryl. The (24,12) Golay code has a (24,2) subcode
with minimum distance equal to 16.
Proof: This follows directly from Expression (2) with x =
(0,0,0,0,0,0,0,0). Figure 2 shows the trellis representing the
(24,2) subcode with 4 code words. •
Since there are 128 (24,5) cosets in the Golay code, it is
possible to construct a non-catastrophic [1] FS code with up
to 64 states on a completely connected graph. By this con-
struction we obtain a (24,11,6) FS code with dfree = min (12,
2 × 8) = 12. For this code, since d 2 is strictly smaller than
2d t, it is also possible to say that there are exactly 30 error
events at distance 12, the number of code words of the (24,5)
subcode of weight 12. Similarly, by using the 21° (24,2)
cosets, we can construct a (24,11,9) FS code with dfree =
min (16,2 X 8) = 16. These new codes are both optimal in the
sense that they achieve the largest possible free distance, as
predicted by the Plotkin bound for FS codes [1 ].
III. Codes Derived From the Nordstrom-
Robinson Code
In [1] a (16,7,2) FS code was constructed starting from the
nonlinear (16,8) Nordstrom-Robinson code with dmi n = 6,
which is the union of 8 particular cosets of the (16,5) first-
order Reed-Mfiller code with dmi n = 8.
Given that the Nordstrom-Robinson code has many pairs
of code words at distance 10 and that a (16,k) code may have
dmi n = 10 only ifk _<2 (by the Plotkin bound), it is interesting
to see if the Nordstrom-Robinson code can be split into 64 sets
of 4 code words, each with dmi n = 10. The following theorem
proves that this is true.
Theorem2. The (16,8) Nordstrom-Robinson code can be
partitioned into 64 sets, each having dmi n = 10.
Proof: The Nordstrom-Robinson code is the union of
8 cosets of the (16,5) first-order Reed-MiJller code. Let the
8 coset leaders be denoted by a i and bi, i = 0,1,2,3. Then
ao = 0 and the other coset leaders can be taken to be the
following seven bent 1 functions of four Boolean variables
XrX2,X3,X 4 (problem 21, p. 476 of [2]),
a I =xix 2 +XlX 3 +x2x 3 "lx2x 4
a 2 =xix 2 -I-x3x 4
a 3 =x1x 4 +x2x 3 +x2x 4 +x3x 4
b 0 =x1x 2 +x1x 3 +x1x 4 +x2x 4
b 1 =XlX 2 +x1x 4 +x2x 3
b 2 =x1x 3 +x2x 4 +x3x 4
b 3 =X1X 3 +X1X 4 +X2X 3 +X3X 4
If instead the coset leaders are taken to be:
A0=0
A t =a 1 +x 3 + 1
A 2 =a 2 +x 2 +x 4 + 1
A 3 =a 3 +x 1 +x 2 + 1
B o = bo + x 2 + x 3 + x 4
B 1 = b 1 +x] +X 4
B 2 = b2 + x 3 + x 4 + 1
B 3 = b3 +x 1 +X 2 +x 3 + 1
1These Boolean functions are so called because they are in some sense
furthest away from linear functions.
57
then dJst (A i, Aj) = 10, dist (Bi, B/) = 10, and dist (Ai, BI) = 6
for all i and j, i q: ]. Then, for each word w in the (16,5) first-
order Reed-Mfiller code (,40 + w,A 1 + w,A 2 + w,A 3 + w)
and (B o + w, B 1 + w, B 2 + w, B 3 + w) are subsets of the
Nordstrom-Robinson code with distance 10. There are 64
such subsets and they exhaust the Nordstrom-Robinson
code. •
Table 1 shows the code words of the (16,5) first-order
Reed-MiJller code and the 8 coset leaders that generate the
64 subsets used for the FS code construction. By assigning
the 64 subsets to the edges of a 32-state completely con-
nected graph a (16,7,5) FS code can be constructed. This
code has dfree = rain (2 × 6,10) = 10, which meets the Plotkin
bound.
IV. Simulation Results and Decoding
Algorithms
An existing software simulation for FS codes has been
adapted to the newly found codes. Simulation results showing
the probability of bit error versus EJN o are given in Fig. 3.
The (24,11,6) FS code with dfree = 12 and the (16,7,5) FS
code with dfree = 10 are compared for reference to the (2,1,6)
Voyager convolutional code.
These results are obtained by a soft, maximum-likelihood
decoder based on the Viterbi algorithm. The decoder performs
two basic steps:
(1) Each received word (24 or 16 symbols) is compared to
the code words in each coset (128 or 64) and the clos-
est code word in each coset is stored together with its
distance.
(2) At each state, the decoder further selects the closest
code word among those chosen in step 1 for the cosets
assigned to branches reaching that state.
For the (24,11,6) code, the total number of bit operations
per decoded bit involved in the decoding process is (24/11) 212,
where 212 is the total number of branches in one stage of the
decoder trellis. It is interesting to note that the same number
for the Golay code is (24/12) 212, which is very close, but the
FS code has dfree = 12 compared to a dfree = 8 of the Golay
code. Similarly, the decoding of the (16,7,5) code involves
(16/7) 210 bits per decoded bit.
V. Conclusion
In this article we have described FS codes based on parti-
tions of the Golay and Nordstrom-Robinson codes, which did
not appear in the literature.
The comparison of these new codes to known codes, block
and convolutional, is complicated by the fact that both the
performance and the decoding complexity must be taken into
account, and the complexity is intimately related to the partic-
ular hardware architecture used for the decoder. We feel that
the proposed codes may take greater advantage of parallel
VLSI architectures than conventional convolutional codes
with no structure. Also, the trellis representation of cosets as
in Figs. 1 and 2 can be used to reduce the number of compari-
sons to select the closest code word with methods similar to
those described in [3].
Figure 4 summarizes the present knowledge on FS codes by
showing the Plotkin or Hamming bound (whichever is tighter)
on the free distance achievable for a given encoder memory
and for two classes of FS codes, the (24,11,m) and the (16,7,m)
classes. The (16,7,2) code has been reported in [1]. The Voy-
ager code is also shown for comparison as a member of the
(2,1,m) class of convolutional codes. More work needs to be
done in constructing yet more powerful FS codes, especially
those based on graphs that are not completely connected.
58
References
[1] F. Pollara, R. J. McEliece, and K. Abdel-Ghaffar, "Constructions for Finite-State
Codes," TDA Progress Report 42-90, April-June 1987, Jet Propulsion Laboratory,
Pasadena, California, pp. 42--49, August 15, 1987.
[2] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,
Amsterdam: North-Holland, 1978.
[3 ] J. H. Conway and N. J. A. Sloane, "Soft Decoding Techniques for Codes and Lattices,"
1EEE Trans. Information Theory, vol. IT-32, January 1986.
[4J G. D. Forney, "Coset Codes I and II," IEEE Trans. Information Theory, to be pub-
lished in 1988.
59
Table 1. Code words of (16,5) first-order Reed-
Mbller code and coset leaders
O000000000000000w o
1111111111111111w]
0000000011111111w 2
11 1 11 1 1 100000000w 3
000011 1 1000011 llw 4
111 10000111 lO000w 5
0011001100110011w 6
llO0110011001100w 7
OlOlOlOlOlOlOlOlw 8
lOlOlOlOl0101010w 9
0 0 0 01 1 1 1 11 1 10 0 0 0 Wlo
1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 wll
0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 w12
1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 w13
0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 w14
1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 w15
1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 w16
0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 w17
0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 w18
1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 w19
0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 w20
1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 w21
1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 w22
0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 w23
0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 w24
1 0 0 1 1 0 0 11 0 0 1 1 0 0 1 w2s
0 1 1 0 0 1 1 01 0 0 1 1 0 0 1 w26
1 0 0 1 1 0 0 1 0 1 1 0 0:1 1 0 w27
1 0 0 1 01 1 0 1 0 0 1 0 1 1 0 w28
0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 w29
1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 w30
0 1 1 01 0 0 1 1 0 0 1 0 1 1 0 w31
0000000000000000A o
11001010111 I0110A I
i011010010111011A 2
11 I001 ll01001101A 3
0110110000000101B 0
0101011011110011B 1
1000110110111 ll0B 2
1101000101001000B 3
6O
XB.v_........__-- ."."_'X XXX_
,_ XX><><_
D o
,_-\ D1
_- D5
-.J/ \-2'
Fig. 1. The (24,5) subcode, x E A*
X
SUBCODE D O
X
Fig. 2. The (24,2) subcode, x = (0,0,0,0,0,0,0,0)
100
10-1
10-3
10-4
1 I 1
1, 6)
5)
(24, 11, 6)
0 1 2 3
Eb/No, dB
Fig. 3. Bit error probability of two new FS codes
61
28
26
24
22
20
18
_ 16
_- 14
12
10
I I I I , I I I 1 I I l I I 1 ]
t-- BOUNDSD KNOWN FS CODES
_ ,i _
- 7 //
.... 124 1
Z, ,,,,,,,,, ,,,1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Fig. 4. Free distance bounds for two classes of FS codes
compared to (2,1,m) convolutional codes
62


Further results on finite-state codes

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