Cryptography Reference
In-Depth Information
equal to 1, which means that if the RTZ sequence is decided instead of the
transmitted "all zero" sequence, only one bit is erroneous. In the case of the
classical code, one sequence at the input has a weight of 1 and another a weight
of 3: one or three bits are therefore wrong if such an RTZ sequence is decoded.
In the case of the recursive systematic code, the RTZ sequences with minimum
weight have an input weight of 3.
Knowledge of the minimum Hamming distance and of the input weight as-
sociated with it is not sucient to closely evaluate the error probability at the
output of the decoder of a simple convolutional code. It is necessary to com-
pute the distances, beyond the minimum Hamming distance, and their weight in
order to make this evaluation. This computation is called the
distance spectrum
.
5.3.3 Transfer function and distance spectrum
The error correction capability of a code depends on all the RTZ sequences,
which we will consider in the increasing order of their weight. Rather than
computing them by reading the graphs, it is possible to establish the transfer
function of the code. The latter is obtained from the state transition diagram
in which the initial state (000) is cut into two states
a
e
and
a
s
,whichareno
other than the initial state and the arrival state of any RTZ sequence.
Let us illustrate the computation of the transfer function with the example
of the systematic code of Figure 5.1, whose state transition diagram is again
represented in Figure 5.15.
(000)
a
e
O²I
OI
(100)
(110)
OI
O
O²I
g
e
O²I
OI
(010)
(101)
O
(111)
O²I
1
c
f
h
1
OI
1
(001)
(011)
O
b
d
O
(000)
a
s
[1
,
1+
D
+
D
3
]
Figure 5.15 - Machine state of the code
, modified for the computation
of the associated transfer function.
Each transition has a label
O
i
I
j
,where
i
is the weight of the sequence coded
and
j
that of the sequence at the input of the encoder. In our example,
j
can take the value 0 or 1 according to the level of the bit at the input of the