Given a particular convolutional code C, we wish to find all minimal generator matrices G(D) which represent that code. A standard form S(D) for a minimal matrix is defined, and then all standard forms for the code C are counted (this is equivalent to counting special pre-multiplication matrices P(D)). It is shown that all the minimal generator matrices G(D) are contained within the 'ordered row permutations' of these standard forms, and that all these permutations are distinct. Finally, the result is used to place a simple upper bound on the possible number of convolutional codes

Lumbard, Kim E.

McEliece, Robert J.

English

Caltech Authors - Main

Counting Minimal Generator Matrices 
Kim E. Lumbard h b e r t  J.  McEliece 
California Institute of Technology, Pasadena, CA 91125 
Abstract - Given a particular convolutional code C, 
we wish to And all minimal generator matrices G(D)  
which represent that code. A standard form S ( D )  for a 
minimal matrix will be defined, and then all standard 
forms for the code C will be counted (this is equivalent 
to counting special pre-multiplication matrices P ( D ) ) .  
It will then be shown that all the minimal generator 
matrices G ( D )  are contained within the ‘ordered row 
permutations’ of these standard forms, and that all 
these permutations are distinct. Finally, the result 
will be used to place a simple upper bound on the 
possible number of convolutional codes. 
I .  DEFINITIONS 
Given any k x n polynomial matrix G ( D )  = [g,,(D)], define 
the constmint lengths U, of G ( D )  as the maximum degree of 
any polynomial in row i ,  i.e. U, = maxyzl{deg g , ] ( D ) ) .  
Define a minimal generator matrix G ( D )  for C as does 
Forney [2, p. 4941. A minimal matrix S ( D )  is in standard 
form if the row degrees U, are ordered V I  5 uz 5 . . . 5 uk. 
Note that the set {U,)! of any minimal matrix G ( D )  for a 
fixed (n ,  k, m) convolutional code C is invariant. Call this 
set the Forney Indices {e,)! of C ,  el 5 ez 5 . . . 5 e k .  
Some of the Forney Indices may be equal; let s be the 
number of distinct indices. Label the distinct index values by 
8, with multiplicities a, for J 1; (E j = 1, 2, . , . 3). Note the 
8, are strictly ordered fll < 82 < . . . < 8.. 
11. MAIN THEOREM 
Given a particular (n, k, m) convolutional code C over GF(q) 
with Forney Indices {e,):. The number of minimal generator 
matrices in standard form S ( D )  for C is given by 
, = I  L h=i  J 1=1 
Furthermore, the total number of minimal generator matrices 
G ( D )  for C is ICs, where IC is the s-nomial coefficient 
111. PROOF OF MAIN THEOREM 
Given two minimal matrices in standard form SI (D) and S ( D )  
representing the convolutional code C. We know 
3 P ( D )  3 Si(D) = P ( D ) S ( D )  (3) 
where P ( D )  is a unique k x k unimodular matrix of polynomi- 
als and det P ( D )  # 0. Partition P ( D )  naturally into smaller 
submatrices M , , ,  where each M,, is an a, x crl sized matrix. 
Within each M,, the constraint lengths are all equal; the pre- 
dictable degree property then yields 
degm(”) (D)  5 f l ,  - 8, for i 1; and j 1; (4) 
‘McEliece’s contribution was supported in part by AFOSR 
Grant F49620-94-1-005 and in part by a grant from Pacific Bell. 
where m(’J) (D)  is any polynomial entry in the sub-matrix M a l .  
P ( D )  is seen to be a lower block diagonal matrix 
P ( D )  = ( 5 )  
where 
Q{cr,) 
R{a*’ 
= a. x a, matrix of scalars of full rank cr, 
a, x a, matrix of polynomials with = 
degree 5 p. - 8, 
Since S ( D )  is basic, distinct P ( D )  yield distinct S l ( D ) ;  count- 
ing the standard form matrices S1(D)  is thus equivalent to 
counting the pre-multiplication matrices P ( D ) .  Let N[A] E 
“number of distinct matrices of type A”. Then 
, = I  ,=1 
The Q { a , )  and R{a, ,a , )  are easy to  count, giving Eq (1). 
Recall that our goal is to count minimal matrices. Given 
a particular minimal matrix G ( D ) ,  define an ordered row per- 
mutation(0RP) as one which preserves the sequential order of 
rows with equal constraint length. These ORPs form an equiv- 
alence relation on the set G of all minimal matrices G ( D )  of 
a fixed convolutional code C. Thus G is the disjoint union of 
its equivalence classes. 
The number of equivalence classes is equal to S,  the number 
of standard form matrices S ( D )  (as there is exactly one S ( D )  
to every class). Each equivalence class also has the same size 
IC {from Eq (2)); thus IGI = ICs. 
IV. SIMPLE UPPER BOUND 
Consider the k x n  matrices W ( D )  with ordered {U,)!. Clearly, 
N[W(D)]  = qnm(qn - l )k ,  where m = U,. Any minimal 
matrix in standard form S ( D )  for an (n, k, m) convolutional 
code C with Forney Indices {e,}! = {U,): will be among these 
W ( D ) .  Thus, 
0 - 7803-2015-8/94/$4.00 01994 IEEE - 18 -  
This bound is fairly good for low rates. 
REFERENCES 
[I] G. David Fomey, Jr. “Convolutional Codes I: Algebraic Struc- 
ture” IEEE Trans. on IT, Vol IT-16, pp. 72&738 (Nov 1970) 
[2] G. David Fomey, Jr. “Minimal Bases of Rational Vector Spaces, 
With Applications To Multivariable Linear Systems”, Sinm J. 
Control, Vol 13, No. 3, pp. 495520 (May 1975) 
(31 Thomas J.  Shusta, “Enumerationof Minimal ConvolutionalEn- 
coders” IEEE Trans. o n  IT, Vol IT-23, pp. 127-132 (Jan 1977) 
[4] Bradley W. Didrinson, “A New Characterization of Canonical 
Convolutional Encoders” IEEE Trans. on IT,  Vol IT-22, pp. 
352-354 (May 1976) 


Counting minimal generator matrices

A New Characterization of Canonical Convolutional Encoders”

Convolutional Codes I: Algebraic Structure”

Minimal Bases of Rational Vector Spaces, With Applications To Multivariable Linear Systems”,

http://authors.library.caltech.edu/29418/1/LUMisit94.pdf

Counting minimal generator matrices

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