A method to define the labels of the state diagram of a linear finite-state code is presented and investigated. This method is particularly suitable for simple hardware implementation since it simplifies the encoder structure. The method can also be applied to the labeling of a state diagram that is not completely connected to obtain a linear finite state code with larger free distance

Cheung, K.-M.

English

NASA Technical Reports Server

TDA Progress Report 42-92
N88-18781
October-December 1987
A Labeling Procedure for Linear Finite-State Codes
K.-M. Cheung
Communications Systems Research Section
In this article a method to define the labels of the state diagram of a linear finite-state
code [1] is presented and investigated. This method is particularly suitable for simple
hardware implementation since it simplifies the encoder structure. The method can also
be applied to the labeling of a state diagram that is not completely connected to obtain
a linear finite state code with larger free distance.
I. Introduction
It was shown in [1] that a finite-state code (FS code)on a
completely connected state diagram with 2m states requires at
least 2m÷l labels. Also, a simple method to define such labels
has been suggested in [1]. However, the codes constructed
using the method in [1] are not linear. In this article, another
method using shift registers to define the labels of the state
diagram of the FS codes is presented. This method is particu-
larly suitable for simple hardware implementations since it
simplifies the encoder structure. The method can also be
applied to the labeling of a state diagram that is not com-
pletely connected to obtain an FS code with larger free
distance. Lastly, a mapping scheme to assign the cosets to the
labels generated by the shift registers is described. It can be
shown that by using the above method, a linear FS code can
be constructed.
In order to facilitate the discussion on FS codes with non-
completely connected state diagrams as well as those with
completely connected state diagrams, the following definition
of FS codes is adopted:
Definition 1: An (n, k, m) finite state code (FS code) on
a c-connected state diagram is a code with the following
properties:
(1) The code has rate k/n.
(2) Its operation can be represented by a state diagram
with 2 rn states.
(3) There are 2c (c _< m) branches going into each state
and 2 c branches going out of each state.
(4) Each branch of the state diagram is associated with a
code (code word length = n and code size = 2k-c), and
any two different codes associated with different
branches are disjoint.
II. Preliminaries
Some important results in the theory of convohitional
codes will now be reviewed. These results will be referred to in
the proofs in later sections.
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https://ntrs.nasa.gov/search.jsp?R=19880009397 2020-03-20T08:21:55+00:00Z
A typicalencoderof an(nl, c, m)convolutional code con-
sists of a linear sequential circuit (with c shift registers) that
accepts c input bits and outputs n I bits. It is well known that
the operation of the encoder can be represented by (1) a state
diagram with 2 m states, 2c branches going into each state, and
2 c branches going out of each state; or (2) a c × n 1 transfer
function matrix (denoted by G [D]) such that the entries of
the matrix are polynomials in D, representing the generator
sequences of the code.
In order to avoid catastrophic error propagation, the trans-
fer function matrix must satisfy Massey and Sain's condition
[2] (a necessary and sufficient condition) on non-catastrophic
codes:
GCD IAi(O ), i = 1,2 .... , (_1)1 = D/
for some l/> 0, where Ai(D), i = 1,2 ..... (rid) are the deter-
minants of the (nd) distinct c × c submatrices of the transfer
function matrix G(D).
III. Generation of Labels by Shift Register
FS code encoders have structural properties very similar
to those of convolutional encoders, and their operation can
be described by a state diagram. In the case of a convohitional
code, each branch of the state diagram is labeled by an n l-bit
output sequence, whereas in the case of a finite-state code
according to Definition 1, each branch is labeled by a code
that is not necessarily linear. Because of the similarities be-
tween convohitional codes and finite state codes, it should be
expected that much of the theory on structural properties of
convolutional codes will be applicable to finite state codes.
In order to guarantee a noncatastrophic finite state code
with good distance properties, the labeling of the branches of
the state diagram must satisfy the following conditions [1] :
(1) different labels out of each state; (2) different labels into
each state; and (3) no disjoint paths with identical labels that
remain unmerged indefinitely.
A method to assign the labels of the state diagram of a
finite state code by using the linear sequential circuit (with
shift registers) of a noncatastrophic (nl, c, m) convolutional
code is now described. Let the c shift registers have lengths
l I, l2 ..... lc where l 1 + l 2 ... + lc = m. The pth row of the
corresponding c × n1 transfer function matrix thus consists
of polynomials in D of degree no greater than lu for 1 _<p _< c.
The state diagram of the convohitional code consists of 2 m
states (each state is defined by the shift register content);
also, there are 2 c branches going into each state and 2 c
branches going out of each state. Each branch in the state
diagram is assigned an n1-bit sequence bo, b I ..... bnl-1,
which consists of the n I output bits of the shift registers. Let
us assign to the branches of the state diagram, which are asso-
ciated with the n I -bit sequence bo. b I ..... bn I the label i
such that i = bo + 2b I +... + 2n-I bn -1" Each of these labels
represents one of the disjoint codes. ]'here are 2 nt of them.
This modified state diagram of the convolutional code is used
as the state diagram of an (n, k, m) finite state code on a
c-connected state diagram.
The construction of a shift register circuit that generates
the state diagram of a finite state code that satisfies conditions
l, 2, and 3 is given as follows. It is not hard to see that condi-
tion 1 is satisfied if, for a fixed shift register content, different
inputs to the shift registers produce different outputs. This can
be achieved if there exists at least one c × c submatrix _i(D)
of the transfer function matrix G(D), i= 1,2 ..... (nc_), such
that the term "1" appears exactly once in each row and in
each column of _i(D). Similarly, condition 2 is satisfied if,
for a fixed input, different shift register contents produce
different outputs. This can be achieved if there exists at least
one c × c submatrix f2/(D), j = 1, 2 ..... (nc), such that the
term Dry representing the last shift register stage of the pth
shift register appears exactly once in row p for 1 _<p _<c, and
each of these D l, , D z2 .... , D tc terms appears in different
columns of f2/(D).
It was shown in [3] that if the (nl, c, m) convolutional
code that generates the state diagram of the finite state code is
noncatastrophic, then the labeling also satisfies condition 3.
Thus, the c X n I transfer function matrix G(D)of the convo-
lutional code must satisfy Massey and Sain's condition [2]. It
will be shown in later sections that the minimum value n_
could have is c + 1. Two algorithms to construct a c × c + 1
transfer function matrix G(D) of the convolutional code are
given as follows:
Algorithm 1: Completely connected state diagram, dy =
2 branches.
(1) Construct a c × c matrix G'(D) such that
a_. •
,I(D) = 1. i= 1 ..... c. j = i
= D, i = 1 ..... c, /' = (i+l)modc
= 0 otherwise
(2) Append the column [1 ..... 0] T to G'(D) to obtain a
c X c + 1 matrix G(D).
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An example of a 3 × 4 transfer function matrix G(D) con-
structed using the above algorithm is given in Table 1. It is
obvious that G(D) satisfies conditions 1 and 2. Also, it is not
hard to see that one of the determinants, £xj(D), j = 1 .....
(c÷ 1h equals 1 and the rest are nonzero. Thus
C ]_
C+I)]GCD j(D), j = 1 ..... = 1C
Massey and Sain's condition is satisfied and the state diagram
generated by this transfer function matrix satisfies conditions
1,2, and 3.
Algorithm 2: Non-completely connected state diagram, dy =
3 branches.
(1) Construct a c × c matrix G'(D) such that
Gi/(D = + j = i)' 1 D, i=1 ..... c,
= D 2, i = 1 ..... c, j = (i+l)modc
= 0 otherwise
(2) Append the column [1 ..... 0] T to G'(D) to obtain a
c × c + 1 matrix G(D).
An example of a 3 × 4 transfer function matrix G(D)
constructed using the above algorithm is given in Table 2. Again
it is obvious that G(D) satisfies conditions 1 and 2. Also, it
can be shown that one of the determinants, AI(D), equals
D2(c-1) and the rest are nonzero. Thus
GCD A i (D), j = 1 ..... =
C
. where l is some integer. Massey and Sain's condition is satisfied
and the state diagram generated by G(D) satisfies conditions
1,2, and 3.
IV. Properties
On the basis of the labeling procedure by shift register
above, which is based on a linear sequential circuit, the finite
state code possesses a mathematical structure that facilitates
encoding/decoding and simplifies hardware implementation.
Also, this labeling procedure is applicable to the construction
of finite state codes with incompletely connected state dia-
grams to obtain larger free distance.
Definition 2: Let N be the number of states of a finite state
code. A labeling matrix L of the state diagram is defined to be
an N × N matrix, where L(i, j) denotes the label from state i
to state j.
Let u = (u 1, u2,..., u c) represent the c input bits to the
convolutional encoder. Let D = (D 1, D 2 ..... De) represent
the last c shift register stages of the convolutional encoder.
That is, Dp represents the term Dip in row p for 1 _<p _< c.
In the following theorems, some properties of FS codes which
use the new labeling procedure are revealed.
Theorem 1: For a state diagram with 2m states generated by
G(D) which satisfies conditions 1, 2, and 3, if the graph has
2c branches going into each state and 2c branches going out of
each state, c _< m, at least 2c+1 labels are required.
Proof: Suppose that 2c labels suffice. The transfer function
matrix G(D) of the convolution code that generates the state
diagram of the FS code is then a c × c matrix. By condition 1,
since different labels are coming out of each state, the c out-
put bits can be written as
uA+d
where A is a c X c nonsingular matrix and _d is a constant
binary c-tuple which depends upon the shift register contents
of the encoder. Thus, IAI = a, where a is a nonzero integer.
Thus, the term a is contained in the expression of IMI. Simi-
larly, by condition 2, since different labels are going into each
state, the c output bits can be written as
DB+e
where B is a c × c nonsingular matrix and e is a constant
binary c-tuple, depending upon the input bits and the shift
register contents other than D],..., D c. Again, [B[ = 13for
some nonzero integer 13.Therefore the term 13D m = 13Dll+'"+lc
is contained in the expression of IMI. Thus, IMI = 13Dm + ...
+ a and IMI is not of the formD t for some l_> 0. This violates
Massey and Sain's condition and the convolutional code is
catastrophic. This in turn implies that the state diagram
generated by this convolutional encoder is catastrophic and
thus at least c + 1 output bits for the convolutional encoder
are needed. This implies that at least 2c+1 labels are needed in
the state diagram. •
In fact, Algorithm 1 and Algorithm 2 in Section II show
that c + 1 output bits are sufficient to guarantee that condi-
tions 1,2, and 3 are satisfied.
Theorem 2: Let L be the labeling matrix of a state diagram
generated by G(D) which satisfies conditions 1, 2, and 3.
Row i and row j (column i and column j), i 4: j, of L have
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either the same set of labels or a completely different set of
labels.
Proof: The state of the convolutional encoder that generates
the required state diagram of the finite state code is defined as
the shift register contents of the encoder. For an (nl, c. m)
convolutional code, let the binary m-tuple [D l ..... Dm]
denote the state that corresponds to the shift register stages
D1,... , D m of the encoder. Note that the encoder is con-
structed in such a way that for a fixed state [D 1 .... , Dm],
different inputs to the shift registers produce different outputs
(condition 1). If [D 1 ..... Dm] = [0 .... , 0], the set of all
possible binary n 1-tuples (labels) that represent the output
bits of the encoder forms a c-dimension subspace K of an
n l-dimension vector space over GF(2) (because the encoder
is a linear sequential circuit). This set K is isomorphic to the
row of the labeling matrix L that corresponds to the state
[0 ..... 0]. Now, if [D 1 ..... Dm] 4:[0 ..... 0], then
it is not hard to see that the set of all possible output bi-
nary n 1-tuples (output bits of the encoder) is of the form
K + £, where e is a binary n 1-tuple (constant) determined by
[D 1 ..... D m ]. If _e_ K, then K and K +_e are disjoint (since
K is a c-dimensional subspace in an n j-dimensional vector
space). If e E K, then K = K + £. A similar argument holds
for the case ofK+e 1 andK+e 2,wheree 1 and_e 2arebinary
nl-tuples determined by different [D l ..... Dm]'S. That is,
if el q_ K + £2, then K +_e 1 and K +_% are disjoint. If_e 1 E
K + £2 then K + e 1 = K + g2' This proves that any two rows of
a labeling matrix L have either the same set of labels or a com-
pletely different set of labels. The proof for the case of the
columns is similar to the one above. •
V. Assignment of Cosets to Labels
A code C over GF(q) is said to be linear if and only if the
following condition is satisfied:
V_a, _bEC and VT, 6EGF(q), 7g+6bEC
In an FS code. even though we have a linear convolutional
structure (labels are generated by outputs of shift registers),
the overall code may not be linear if the cosets are not prop-
erly assigned to the outputs of shift registers. There may exist
two code word sequences such that their sum is not a legal
code word sequence. In order to generate a linear FS code the
following well-known theorem in linear algebra can be used:
Theorem 3 (without proof): If C is a vector space and S is a
proper subspace of C, then there exists a subspace W of C
such that
S+W=C
sn w = {0}
dimS+dimW = dimC
The following discussion describes a way to generate a
linear FS code. The labeling of an FS code can be divided into
two parts: (1) generation of labels to the branches in the
state diagram; and (2) assignment of cosets to the labels.
Part 1 was taken care of by using a convolutional encoder to
generate labels to the state diagram of the FS code. For
part 2, the method proceeds as follows. Let C be the parent
(n, k) code. Let S be an (n, k I ) subcode of C. By Theorem 3.
there exists a subcode W of C (W is an [n, k - k I ] code) such
that
S+W=C
Sn W = {Q}
dimS+dim IV = dimC
The 2 k-k, cosets are constructed by adding each word in
W to S. That is,
_w+S V_wEW
Note that the set of all binary k - kl-tuples is isomorphic
to W. Let {_wo, _w1..... --Wg-k1-1)be a basis of IV. Let bo,
bl .... , bk-k,-I be the k- k I output bits of the convolutional
encoder. Let the coset assigned to the branches labeled by the
binary (k - kl)-tuples bo, b 1..... bk-k,-I be denoted by
L (b o, b I ..... bk_kt_l ). Let us assign
L(bo, b I ..... bg_k_ 1) = S+ {bo_Wo+b l_w1 +...
+ bk_kl_ 1 W-k-kx-1 )
This assignment of cosets to the branches in the state diagram
guarantees the linearity of the FS code.
53
References
[1] F. Pollara, R. McEliece, and K. Abdel-Ghaffar, "Constructions for Finite State
Codes," TDA Progress Report 42-90, vol. April-June 1987, Jet Propulsion Labora-
tory, Pasadena, California, pp. 42-49, August 15, 1987.
[2] J. Massey and M. Sain, "Inverse of Linear Sequent!al Circuit," IEEE Trans. Comput.,
vol. C-17, 1968.
[3] K.-M. Cheung, "Error-Correction Coding for Data Storage Systems," PhD thesis,
California Institute of Technology, 1987.
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A labeling procedure for linear finite-state codes

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