Abdel-Ghaffar, Khaled

McEliece, Robert

Solomon, Gustave

Caltech Authors - Main

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SOME PARTIAL UNIT MEMORY CONVOLUTIONAL CODES 
Robert McE!iece Khaled Abdel-Ghaffar 
University of California 
Davis, CA 95616 
California Institute of Technology 
Pasadena, CA 91125 
Gustave Solomon 
10747 Wilshire Blvd. 
Los Angeles, CA 90024 
Summary 
In general, an [n, k, d; m] convolutional code over a field F has gen-
erator matrix G(D) = G0 + G1 D + · · • + GKDK, where each G; is a 
k X n matrix with entries from F. Here n is the branch length, k is 
the dimension per branch, mis the memory (i.e., the total number of 
nonzero rows in the matrices G1, ... , GK), and d is the free distance. 
Thus in this notation an [n,k,d] block code is a [n,k,d;0] convolu-
tional code. A partial unit memory (PUM) convolutional code is one 
for which 1( = l (hence the term "unit memory") and at least one 
of the rows of G1 is zero (hence the term "partial unit memory.") 
Indeed, if the first k - m rows of G1 are all zero, then the resulting 
code is a [n,k,d;m] PUM code. 
In this paper we will give a general construction for partial unit 
memory convolutional codes. This construction may be used to de-
sign efficient finite state codes [2], [3]. Informally, the construction 
goes like this: Suppose C* and Co are two linear block codes of length 
n, with C* <;;: Co. SupposeC* is a [n,k*,d*] code, and C0 is a [n,k,do] 
code. Then almost always we can combine these two codes to make 
a noncatastrophic partial unit memory convolutional code with pa-
rameters [n,k,d;k - k*], where d 2'. min(d*,2d 0 ). Formally, the 
construction is described in the following theorem. 
Theorem 1. Suppose that C0 is an [n, k, do] linear block code, and 
that C1 is an [n, k, d1] linear block code, and C0 =J C1. Suppose 
further that Co and C1 contain a common subcode C* which is a 
[n, k*, d*] code. Then there exists a noncatastrophic [n, k, d; m] PUM 
convolutional code, with m = k - k* and d 2'. min( d*, d0 + d1 ). 
In applications, almost always (but not always) we only need 
two codes, C* and C0 • This is because as a rule the automorphism 
group of C* will contain a permutation 1r that does not fix C0 , and 
we can take C1 = C0 in Theorem 1. The following Corollary spells 
this out. 
Corollary 1. Suppose that Co is an [n, k, do] linear block code, and 
that C* is a [n, k*, d*] code wl1icl1 is a subcode o[C0 • If the automor-
phism group ofC* contains a permutation that does not fix C0 , then 
there exists a [n, k, d; m] PUM convolutional code, with m = k - k* 
and d 2'. min(d*,2do). 
Theorem 1 and Corollary 1 permit us to construct a large num-
ber of PUM codes, many of which are optimal, in the sense of having 
the largest possible drree for the given n, k, and m. Here are two 
Examples. 
Example 1. Let C* be the [8, 1, 8] binary repetition code, and let Co 
be the [8, 4, 4] extended Hamming code. The automorphism group 
of C* is the symmetric group S8 , which plainly does not fix C0 . Thus 
Corollary 1 implies the existence of a [8,4,8;3] PUM code, which 
is optimal. This code was previously known (see e.g. [1]), but it is 
interesting to see how easily our construction finds it. It is also the 
inner code in the well-known Soviet concatenated "Regatta" system. 
Example 2. Let Co be the binary Golay [24, 12, 8] code. It is possi-
ble to show that there is an isomorphic copy of C0 , which we call C1, 
such that the dimension of the intersection Co n C1 is 9. This inter-
section contains both a [24, 5, 12] code, and a [24, 2, 16] code. Thus 
by Theorem 1 we can construct both a [24, 12, 12; 7] PUM code, and 
a [24, 12, 16; 10] PUM code, which are bo~h optimal. 
In the special case that C* is the [n, 1, n] binary repetition code 
(as in Example 1), the automorphism group ofC* contains all permu-
tations on {1, 2, ... , n}. Then unless k = l, n - l, or n, C0 can't be 
fixed by all such permutations. This leads to the following Corollary 
to Theorem 1. 
Corollary 2. IfC 0 is a [n,k,d 0 ] binary block code containing the 
all-ones vector, and if k =J l, n - l, n, then there exists a [ n, k, d; k- 1] 
PUM code with d 2'. 2do. 
Corollary 2 naturally leads one to ask how large can d0 be, given 
that C0 contains the all-ones vector. We do not have a full answer to 
this question, but the following modification of the classic Griesmer 
bound is useful. 
Thus let N ( k, d) denote the minimum length of a binary code 
with Hamming distance 2'. d and dimension k which contains the 
all-ones vector. 
Theorem 2. If k 2'. 2, then 
N(k,d) 2'. d + N(k - l, f d/21)-
Corollary 3. N(l,d) = d, and N(2,d) = 2d, and fork 2'. 3, 
Theorem 2 proves, for example, that there is no [7, 3, 4] binary 
code containing the all-ones vector, although there is a [7, 3, 4] code. 
Similarly, there is no [20, 5, 9] linear code with the all-ones vector, 
although there is an [21, 5, 9] such code. This is of interest, since 
Lauer [l] constructed a [20, 5, 18; 4] PUM code, which therefore can-
not be constructed by our methods. However, all of Lauer's other 
codes, and many others scattered throughout the literature, can be 
constructed by our methods. Theorem 2 also raises the following 
question: Give a bound on the minimum distance of a linear block 
code that contains a known subcode. Except for the special case 
where the subcode is the repetition code, we know practically noth-
ing about this question. 
196 
Acknowledgements 
Abdel-Ghaffar's contribution to this paper was supported by NSF 
Grant NCR 89-08105 and by an IBM Faculty Development Award. 
McEliece's contribution was partially supported by a consulting con-
tract with Caltech's Jet Propulsion Laboratory under contract to the 
National Aeronautics and Space Administration, and also partially 
supported by AFOSR grant 88-0247 and a grant from Pacific Bell. 
Solomon's contribution was supported by a consuling contract with 
Caltech's Jet Propulsion Laboratory under contract to the National 
Aeronautics and Space Administration. 
References 
[1] Lauer, G. S., "Some Optimal Partial-Unit-Memory Codes," 
IEEE Trans. Inform. Theory vol. IT-25 (March 1979), pp. 240-
243. 
[2] Pollara, F., McEliece, R., and Abdel-Ghafar, K., "Finite-State 
Codes," IEEE Trans. Inform. Theory vol. IT-34 (September 
1988), pp. 1083-1088. 
[3] Pollara, F., Cheung, K.-M., and McEliece, R. J., "Further Re-
sults on Finite-State Codes," TDA Progress Report vol. 42-92 
(October-December 1987), pp. 56-62. 


Some Partial Unit Memory Convolutional Codes

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Some Partial Unit Memory Convolutional Codes

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