XII Reunión de Trabajo en Procesamiento de la Información y Control, 16 al 18 de octubre de 2007

Non-punctured non-systematic ½-rate turbo codes

Castiñeira Moreira, J. , Petruzzi, D. M., Liberatori M. C. and Bonadero, J. C.

Laboratorio de Comunicaciones, Facultad de Ingenieria UNMDP

casti@fi.mdp.edu.ar

Abstract— In this paper we present non-punctured three Finite State Sequential Machines (FSSMs)) and of
non-systematic ½-rate turbo codes designed using applying puncturing over these three outputs to convert
Finite State Sequential Machines (FSSMs) that the rate of the code from 1/3 to ½ (Banerjee et al.
operate over the Finite Field GF(4). It is shown in (2005). We show in this paper that the use of FSSMs
this paper that these schemes do not show defined over the Finite Field GF ( 4 ) allows us the
convergence problems in their iterative decoding, as design of NS ½-rate turbo codes that do not need to
it happens to similar schemes defined over the apply puncturing. The structure is shown in Fig. 1, and
binary Field. Non-systematic turbo codes can be it consists on transmitting simply and alternately the
useful for the design of communication systems for corresponding redundancy outputs of two FSSMs.
which error correction and encryption are two In this paper we will show that in general, NS ½-rate
important goals. turbo codes present a slight loss in the Bit Error Rate
A relationship is found between the state (BER) performance with respect to their equivalent
transitions structure of the FSSM and the BER systematic schemes, in the waterfall region of the BER
performance (or equivalently the form of the EXIT performance curve of the turbo code. However,
chart) of the turbo code constructed using that Banerjee et al. (2005) shows that NS 1/3-rate schemes
FSSM. FSSMs defined over GF(4) with all-zero present lower floor effects in the BER performance
input responses of the form of closed cycles are the curve of turbo codes with respect to their equivalent
best options as constituent encoders of non- systematic schemes. We will concentrate our analysis
punctured non-systematic ½-rate turbo codes. A on the waterfall region of the BER performance curve
close relationship is also found between the BER of NP-NS ½-rate turbo codes defined over GF ( 4 ) .
performance and the coefficients that define the In spite of their slight loss in BER performance in the
structure of the FSSM that constitutes the turbo waterfall region, NS ½-rate turbo codes can be used in
code. schemes where not only error-control coding, but also
privacy is an important additional aim in the design of
Key words— Turbo codes. EXIT charts. Non- communications systems. The use of systematic turbo
systematic coding. Non-linear coding. codes is related to the transmission of the message
I. INTRODUCTION symbols that appear in the transmitted output, allowing
a given eavesdropper to directly read this desired
Binary non-systematic turbo codes were introduced information. This is however not strictly speaking a
by Banerjee et al. (2005). In a non-systematic turbo problem in terms of security, because systematic
code, the message symbols are not transmitted, thus information can be externally protected by applying an
only redundancy symbols appear at the output. Non- encryption algorithm over this message information. In
systematic turbo codes can be defined in any Finite a NS mode, message symbols are not present at the
Field of the form GF ( q ) , where usually q = 2 n , and n transmitted output, and this increases the levels of
is an integer, and the most common definition is done privacy with respect to the systematic mode. An
over the Finite Field GF ( 2 ) , also known as the binary application of NS ½-rate turbo codes in secure
Field. In the classic form of a turbo code (Berrou, et. al., communication is given in a previous work of the
1993) the rate of the code is controlled by applying authors (Castiñeira Moreira et al., 2006). A
puncturing, which is essentially performed by means of cryptoanalysis (brute force and differential
an output selector that transmit only determined output cryptoanalysis) is performed in that paper, for schemes
symbols. In this paper we present non-punctured non- where coefficients of the FSSMs that are constituent
systematic (NP-NS) ½-rate turbo codes defined over encoders of a turbo code are randomly varied during
GF ( 4 ) . NP-NS ½-rate turbo codes defined over the transmission. Under certain conditions, the scheme is
shown to be robust against the above described attacks.
binary Field present convergence problems in their
iterative decoding, such that for a suitable NS turbo
code defined over this Field there is a need of
generating three redundancy outputs (by means of the
use of three convolutional encoders, or equivalently
 XII Reunión de Trabajo en Procesamiento de la Información y Control, 16 al 18 de octubre de 2007
II. NON-SYSEMATIC ½-RATE TURBO B. Structure of the state transitions of FSSMs over
CODES OVER GF ( 4 ) GF(4)

A. An encoder of a Non-systematic ½-rate turbo In this paper we analyse the state transitions structure
code of the 63 cases of FSSMs defined over GF ( 4 ) with
The structure of a NP-NS ½-rate turbo code defined
over GF ( 4 ) is seen in Fig. 1. The structure has a s = 2 , and g ( x ) = a0 + a1 x + a 2 x 2 , and its relationship
random interleaver of size L = 10000 . with the BER performance of the corresponding NP-NS
The encoder is constructed using FSSMs that operate ½-rate turbo code. A similar analysis is done in
(Petruzzi et al., 2007) for systematic turbo codes. The
over GF ( 4 ) , and include a polynomial g( x ) between
use of these FSSMs provides a wider range of
the two memory units of the FSSM. This polynomial is possibilities of turbo coding schemes in comparison
in general of the form: with those defined over the binary Field. This number of
possibilities increases as m and n increase.
g (x ) = a0 + a1 x + ... + a m x m (1) FSSMs under analysis show different responses to the
all-zero input, which are essentially determined by the
where m is an integer, and ai ∈ GF ( q ) , so called current state – next state representation.
FSSMs used in this paper are non-linear, and they have
i = 0 ,1,...,m . Coefficients of this polynomial define the
a non-zero output for the all-zero input.
characteristics of the FSSM and also of the turbo code For each FSSM we have analysed the response for the
constructed using this FSSM (Castiñeira Moreira et al., 16 possible initial state conditions. Responses of FSSMs
2006). An analysis of the randomness of the output under analysis can be classified into different groups.
generated by these FSSMs for the all-zero input case is FSSMs can show responses that are cycles, which can
done in (Petruzzi et al., 2006). Output randomness and be run in a closed form on not. Since this difference in
the state transition structure of the all-zero input the behaviour of the FSSM is relevant to its
response of the FSSM define the behaviour of the NS performance as a constituent encoder of a turbo code,
½-rate turbo code constructed with that FSSM. we want to differentiate these two types of cycles. A
We will consider all the possible turbo codes defined cycle that includes always a given set of states and is
over GF ( 4 ) , with s = 2 memory units, and with run starting from any of these states, will be referred to
polynomials of degree m = 2 or less, of the form as a closed cycle. The number of states that the cycle
g( x ) = a0 + a1 x + a 2 x 2 . The turbo code is always runs is the length of the cycle, and it will be denoted as
constructed using the same FSSM in both constituent LC . When the cycle is reached after a sequence of
encoders. Thus, a given NP-NS ½-rate turbo code states that do not form the cycle, showing a transient
defined over GF ( 4 ) , and with m = 2 , will be denoted behaviour, the response will be referred to as a cycle
with transients, and the number of states of the cycle,
as an (a0 a1 a 2 ) NP-NS ½-rate turbo code. (excluding the number of states of the transient), is the
Therefore, for each turbo code (Fig. 1), length of the cycle, denoted as LT . FSSMs can also
show responses without cycles, and also stationary
g 1 ( x ) = g 2 ( x ) = g ( x ) = a0 + a1 x + a 2 x 2 (2). responses, which can be given at the all-zero state, or at

c1 ( l )
other non-zero states.
u1 ( l ) QC FSSM e ( l )
1
Polar A coefficient is defined in (Petruzzi et al., 2007) that
d1 ( l ) and measures the randomness and properties of the state
binary transitions structure of FSSM and their relationship with
u2 ( l ) e 2 ( l ) format c 2 ( l )
D
the BER performance of the corresponding turbo code.
ed 1 ( l ) d2( l ) It is called the random behaviour coefficient ∆R , and its
D definition is repeated here for clarity:
g1 ( x )
ed 2 ( l )
edgf 1 ( l ) qs
1
D g2( x )
∆R =
q s Lc _ max
∑L C ( i )ηδ ( i ) (3)
edgfd 1 ( l ) i =1

edgf 2 ( l )
Interleaver D where LC ( i ) is the length of the closed cycle present at
edgfd 2 ( l )
the response for the initial condition i , ηδ ( i ) is the
Figure 1. A ½-rate NP-NS turbo code defined over delta function coefficient measured for the all-zero input
GF ( 4 ) . case for the initial condition i , which was defined in
(Petruzzi et al., 2006), and evaluates the response of the
auto correlation of the output of the FSSM for the all-
zero input, and Lc _ max is the maximum length of a
 XII Reunión de Trabajo en Procesamiento de la Información y Control, 16 al 18 de octubre de 2007
closed cycle verified for the corresponding FSSM, This analysis is applied to NP-NS ½-rate turbo codes
evaluated over all the responses. If the response has designed in this paper using the proposed FSSMs under
cycles with transients, or responses without cycles for study. Some relationship is found between the EXIT
the initial condition i , then we set LC ( i ) = 1 . When chart appearance and the structure of the state
the response is a closed cycle, LC ( i ) can adopt any transitions and randomness of the output for the all-zero
input case, quantitatively indicated by the value of the
value in the set {2 ,3,4 ,6 ,15}. The addition in (3) defined random behaviour coefficient ∆R .
excludes stationary responses and all-zero outputs. On the other hand the classic BER performance analysis
This coefficient is evaluated over the 16 responses of can be also used to characterise these turbo codes. It is
the FSSM by calculating a weighted average of the found that NP-NS ½-rate turbo codes over GF ( 4 )
delta function coefficient ηδ for each response. The
have the best BER performance when they are
weight is the length of the closed cycle (if any) present constructed using FSSMs that have the highest values of
at that response. It is found that the higher is the value the random behaviour coefficient, which means that the
of the random behaviour coefficient of the FSSM, the all-zero input responses of these FSSMs are of the form
better is in general the BER performance of the NP-NS of long (in some cases the longest) closed cycles. Turbo
½ rate turbo code constructed with that FSSM. codes constructed using FSSMs with cycles with
transients perform worse than those with closed cycles.
III. EXIT CHART AND BER This can be verified in Fig. 2, where BER performances
PERFORMANCE OF NON-PUNCTURED of several NP-NS ½-rate turbo codes are shown.
NON-SYSTEMATIC TURBO CODES Simulations are done for the transmission of 250 blocks
OVER GF ( 4 ) of 10000 elements of the Finite Field GF ( 4 ) , encoded
by using in all the cases the same random interleaver of
A. BER performance analysis of non-punctured size L = 10000 , decoded using the LOG-MAP BCJR
non-systematic turbo codes over GF ( 4 ) decoder (Bahl et al., 1974), with 8 iterations.
The best schemes among those shown in Fig. 2 are the
The performance of a turbo code can be analysed by NP-NS ½-rate (α 0 α ) turbo code ( ∆R = 0.5058 )
using two well known procedures. One of them is the (
and the NP-NS ½-rate α 2 α 2 0 turbo code )
simulation of the Bit Error Rate, by depicting a curve of ( ∆R = 0.7260 ), that show high values of the random
this parameter as a function of the average bit energy-
behaviour parameter.
to-noise power spectral density ratio, E b / N 0 . Another
These schemes use FSSMs that have closed cycles of
tool for analysing the behaviour of iteratively decoded length 6 and 15 respectively. BER performances of
codes like turbo and LDPC codes has been proposed in these schemes are quite close to that corresponding to
(Ten Brink, 2001). He introduced a very useful tool that one of the best binary systematic schemes. Floor effect
is known as the Extrinsic Information Transfer (EXIT) is not present in this particular simulation for these two
chart. The EXIT chart is an especially good tool for the schemes. The simulation involves the transmission of
analysis of the waterfall region, and also illustrates the
behaviour of the code in the other two regions of the 5 x10 6 bits. Best schemes seen in Fig. 2 have presented
BER performance curve of a turbo code (Hanzo et al., no errors in this simulation, for E b / N 0 = 2 , 2.5 and
2001; Castiñeira Moreira and Farrell, 2006). It 3dB . This could indicate that the measured BER in
describes, for each value of Eb / N 0 , the relationship these cases is less than Pbe < 2 x10 −6 . We should note
between the mutual information of the a priori however that reliable values of the BER obtained by
information and the message bit information, I A , and simulation require that the number of errors present in
the mutual information of the extrinsic information and the simulated case has to be at least one hundred o
the message bit information, I E . This Extrinsic larger. This means that BER performances seen in Fig.
Information Transfer function is defined as (Ten Brink 2 are reliable only for Pbe > 10 −5 . This is however
et al., 2001): enough for our proposes, because we analyse the
waterfall region of the BER performance, typically
I E = Tr ( I A , Eb / N 0 ) (4) present in the range 10 −1 > Pbe > 10 −5 .

Similarly, an EXIT chart can also describe the mutual The comparison is done with respect to a systematic
information I A as a function of the mutual information binary turbo code considered as that with the best BER
performance possible. As mentioned earlier, NP-NS ½-
I E , which can be useful for understanding the process rate binary turbo codes present convergence problems
in which the extrinsic information of the current and show very poor BER performances, and it is not
iteration becomes the a priori information of the useful to consider them in this comparison..
following iteration (Ten Brink et al., 2001).
 XII Reunión de Trabajo en Procesamiento de la Información y Control, 16 al 18 de octubre de 2007
B. EXIT chart analysis of non-punctured non- with transients, and sometimes there are no stationary
systematic turbo codes over GF ( 4 ) responses, or there is more than one stationary response.
The random behaviour coefficient of schemes of this
A strong relationship is found between the state group is always lees than ∆R < 0.3 .
transitions structure and randomness of the output of the
A strong relationship is found between the state
FSSM for the all-zero input, and the appearance of the
transitions structure and the form of the EXIT chart for
EXIT chart of the corresponding turbo code, in the case
schemes of the so called group A, for NP-NS ½-rate
of ½-rate systematic turbo codes over GF ( 4 ) in
turbo codes. However, this correspondence is not
(Petruzzi et al., 2007). clearly given for schemes of group B. Especially in the
10
0 case of group B, the form of the EXIT chart of a NP-NS
turbo code is rather defined by the values of coefficients
-1
10 a 1 and a 2 .

10
-2 NP-NS ½-rate turbo codes that belong to group A, show
slight differences in the corresponding EXIT charts.
-3 These schemes are the NP-NS ½-rate turbo codes with
Pb

10
the best BER performances, that is, those that show the
10
-4 best EXIT charts. Sub group A.1, characterised by
having FSSMs with the longest closed cycles (Length
10
-5
LC = Lmax = 15 ) have EXIT charts of the form of Fig.
3.
-6
10
0.5 1 1.5 2 2.5 3 1
Eb/No [dB]
0.9
(α α α ) NP-NS ½-rate turbo code 0.8

(0 1 α ) NP-NS ½-rate turbo code 0.7

+ + (α α 2
)
1 NP-NS ½-rate turbo code 0.6
IE, IA
x x (α 2
0 1) NP-NS ½-rate turbo code
0.5

0.4
(1 1 0 ) NP-NS ½-rate turbo code
0.3 From:
(α α 2 0 ) NP-NS ½-rate turbo code
2
0.2 0.9 dB to
1.9 dB,
(α 0 α ) NP-NS ½-rate turbo code 0.1 step 0.2 dB
5-7 classic binary systematic ½-rate turbo code 0
0 0.2 0.4 0.6 0.8 1
IA, IE

Figure 2. BER performance curves of several NP-NS ½

rate turbo codes over GF ( 4 )
Figure 3. EXIT chart of the NP-NS (α 2
α 2 0 ) ½-
rate turbo code.
We can classify FSSMs in terms of their state Very similar EXIT charts are found for all the other
transitions structures. There are essentially two groups schemes of the sub group A.1, that are the NP-NS ½-
of FSSMs, which will be called groups A and B. In each rate turbo codes for which a0 = 0 ,1,α ,α 2 , a1 = α ,α 2
group, we can also identify slight differences that are
and a 2 = 0 . A second sub group, called A.2, is
described in the form of subgroups. Groups A and B
differ not only in the length of the cycle but also in the constituted of turbo codes with FSSMs that have 15
type of cycle of the response for each initial condition. responses to the all-zero input of the form of closed
Group A is constituted of FSSMs for which the 15 cycles, 12 of length LC = 6 , 3 of length LC = 3 , and
responses for different initial conditions are always one stationary response. This sub group is constituted of
closed cycles, in some cases of the maximum length, in NP-NS ½-rate turbo codes for which a0 = 0 ,1,α ,α 2 ,
some other cases of considerable long length, and there
is always only one stationary response. FSSMs of this a1 = 0 and a 2 = 1,α ,α 2 . In this sub group, NP-NS ½-
group are those that provide the corresponding non- rate turbo codes for which a0 = 0 ,1,α ,α 2 , a 1 = 0 and
systematic turbo code with the best BER performance,
a 2 = α perform better than those for which
among all the 63 cases under study. They are also
FSSMs with the maximum values of the random a0 = 0 ,1,α ,α 2 , a 1 = 0 and a 2 = 1,α 2 . The set of
behaviour coefficient ∆R .
schemes for which a0 = 0 ,1,α ,α 2 , a 1 = 0 and a 2 = α
A second group is called group B, and in this group
shows a higher minimum distance between the shortest
not all the all-zero input responses are of the form of
closed cycles. Some responses are of the form of cycles all-zero input responses (those with LC = 3 ) with
 XII Reunión de Trabajo en Procesamiento de la Información y Control, 16 al 18 de octubre de 2007

respect to the set of schemes for which a0 = 0 ,1,α ,α 2 , group, NP-NS ½-rate turbo codes of relatively quite
good BER performances are those for which
a1 = 0 and a 2 = 1,α 2 . This indicates differences
a 0 = 0 ,1,α ,α 2 , a1 = α and a2 = α 2 and
between the corresponding trellis structures, which are
closely related to the values of coefficients a1 and a 2 . a 0 = 0 ,1,α ,α 2 , a 1 = α 2 and a 2 = 1 . One of these

An example of the sub group A.2 is the (α 0 α ) NP-

( )
schemes is the α α 2 1 NP-NS ½-rate turbo code,
whose BER performance is seen in Fig. 2. Its Exit chart
NS ½-rate turbo code ( ∆R = 0.5058 ) whose BER is shown in Fig. 6.
performance is seen in Fig. 2. Fig. 4 shows the EXIT
chart for this turbo code. 1

0.9
The “bottleneck” region of the EXIT chart is identified
0.8
as a region in which iterations play an important roll in
the efficiency of the iterative decoding of these codes 0.7

(Ten Brink, 2001). Figs. 3 and 4 show quite similar 0.6

IE, IA
EXIT charts, which correspond to the best NP-NS ½- 0.5
rate turbo codes. 0.4

The third sub group in group A is the sub group A.3, 0.3 From:
constituted of NP-NS ½-rate turbo codes for which 0.2 0.9 dB to
1.9 dB,
a0 = 0 ,1,α ,α 2 , a1 = 1 and a 2 = 0 . Schemes of this sub 0.1 step 0.2 dB
0
group perform worse than those of sub groups A.1 and 0 0.2 0.4 0.6 0.8 1
A.2. The BER performance of the (1 1 0 ) NP-NS ½- IA, IE

rate turbo code that belongs to sub group A.3 is seen in

Fig. 2. Its Exit chart is shown in Fig. 5. Figure 5. EXIT chart of the NP-NS (1 1 0 ) ½-rate
turbo code.
1

0.9 1
0.8 0.9
0.7 0.8
0.6 0.7
IE, IA
0.5 0.6
IE, IA
0.4 0.5
0.3 From: 0.4
0.2 0.9 dB to From:
0.3
1.9 dB,
0.1 0.9 dB to
step 0.2 dB 0.2
1.9 dB,
0 0.1
0 0.2 0.4 0.6 0.8 1 step 0.2 dB
IA, IE 0
0 0.2 0.4 0.6 0.8 1
IA, IE

Figure 4. EXIT chart of the NP-NS (α 0 α ) ½-rate

turbo code. Figure 6. EXIT chart of the NP-NS α α 2 ( )
1 ½-rate
An undesirable characteristic of an EXIT chart, depicted turbo code.
for a given value of E b / N 0 , is the presence of what we Exit charts of Figs. 5 and 6 look quite similar. Slight
call anticipated intersecting points. An anticipated differences in the shape of the EXIT chart justify
intersecting point is one for which the curve correspondingly differences in the BER performance.
I E = Tr ( I A , Eb / N 0 ) intersects the curve NP-NS ½-rate turbo codes with the worst BER
I A = Tr ( I E , E b / N 0 ) before the point performance, or equivalently with the worst EXIT
charts, belong also to group B. These are the NP-NS ½-
I E = Tr ( I A , E b / N 0 ) = I A = Tr ( I E , E b / N 0 ) = 1 .
rate turbo codes for which a0 = 0 ,1,α ,α 2 , a1 = α ,α 2
In this sense, group A presents no anticipated
and a 2 = α , and also for which a0 = 0 ,1,α ,α 2 ,
intersecting point in the simulated range for the EXIT
charts of its NP-NS ½-rate turbo codes. a1 = α ,α 2 and a 2 = α . The (α α α ) NP-NS ½-
There is however a close relationship between the form rate turbo code ( ∆R = 0.2977 ) is representative of this
of the EXIT chart and the set of coefficients of g( x ) , sub group, and its BER performance is seen in Fig. 2.
The corresponding EXIT chart is shown in Fig. 7.
{a1 ,a 2 } especially for schemes of group B. In this FSSMs used in this sub group shows all-zero input
 XII Reunión de Trabajo en Procesamiento de la Información y Control, 16 al 18 de octubre de 2007
responses that sometimes are of the form of closed Since message symbols are “hidden” in the non-
cycles of lengths LC = 3 and LC = 6 , and some other systematic case, this on one side improves the
times are of the form of cycles with transients of lengths encryption properties of the transmission, but on the
LT = 3 and LT = 6 . The presence of responses of the other side is in a trade-off with the BER performance.
The loss is however acceptable in practical applications.
form of cycles with transients identifies NP-NS ½-rate
turbo codes of rather poor BER performances. This is Schemes that belong to group A are those with the best
coincident with the existence of anticipated intersecting performance. These schemes use FSSMs with all-zero
points in the corresponding EXIT charts, as seen in Fig. input responses that are always of the form of closed
7. Schemes of this sub group present EXIT charts with cycles, excepting the stationary response, and also have
anticipated intersecting points at random behaviour coefficients that are always higher
E b / N 0 = 0.9 , 1.1, 1.3dB , and sometimes also at than ∆R > 0.40 . In this group there is a relatively strong
E b / N 0 = 1.5 dB . correspondence between the state transitions structure of
the FSSM, and the form of the EXIT chart of the
1 corresponding turbo code. This correspondence is
0.9 however stronger in the systematic case than in the non-
0.8 systematic case. This correspondence is more clearly
0.7 given between the form of the EXIT chart and the set of
0.6 coefficients {a 1 , a 2 } . This trend is even more evident in
IE 0.5 group B.
0.4
REFERENCES
0.3 From:
0.9 dB to Bahl, L., J. Cocke, F. Jelinek y J. Raviv, “Optimal
0.2
1.9 dB, decoding of linear codes for minimising symbol
0.1 step 0.2 dB error rate,” IEEE Transactions on Information
0 Theory, vol. IT–20, 284–287, (1974).
0 0.2 0.4 0.6 0.8 1
IA
Banerjee, A., F. Vatta, B. Scanavino, and D. J. Costello,
“Nonsystematic Turbo Codes,” IEEE Trans. on
Figure 7. EXIT chart of the NP-NS (α α α ) ½-rate Comm., Vol. 53, no. 11, 1841-1849, (2005).
turbo code. Berrou, C., A. Glavieux, and P. Thitimajshima, “Near
In general schemes of group B never reach the optimum Shannon limit error-correcting coding and
performance for NP-NS ½-rate turbo codes, which are decoding: turbo codes,” Proc.1993 IEEE
verified for schemes that belong to group A. These are International Conference on Communications,
those characterised by all-zero input responses of the Geneva, Switzerland, 1064-1070, (1993).
corresponding FSSM of the form of closed cycles, and Castiñeira Moreira, J. , D. M. Petruzzi, M. C. Liberatori,
also by having the highest values of the random and B. Honary, “Trellis hopping turbo coding”,
behaviour parameter ∆R . IEE Proceedings - Communications. 153(6), 966-
975, (2006).
IV. CONCLUSIONS
EXIT chart analysis performed in section III indicates Castiñeira Moreira, J. and Patrick G. Farrell, Essentials
of Error-Control Coding, John Wiley and Sons,
that the use of FSSMs defined over GF ( 4 ) allows us
(2006).
the design of NP-NS ½-rate turbo codes that do not
show convergence problems in their iterative decoding, Hanzo, L., Liew, T. H. and Yeap, B. L., Turbo Coding,
as happens to the same schemes defined over the binary Turbo Equalisation and Space-Time Coding, for
field GF ( 2 ) . The inclusion of the polynomial g ( x ) in Transmission over Fading Channels, IEEE Press,
the structure of the FSSM, both operating over GF ( 4 ) , Wiley, (2001).
provides much more cases of study than in the case of Petruzzi, D. M., M. C. Liberatori, J. C: Bonadero and J.
the binary field, allowing a deeper analysis of this type Castiñeira Moreira, “Non-linear turbo coding over
of turbo codes. GF(4),”, sent to XII RPIC, Rio Gallegos, (2007).
Non-systematic schemes can be useful for applications Petruzzi, D. M., J. Castiñeira Moreira y D. G. Levin,
where also encryption is an important goal of the "Quasi-Chaotic coding over GF(q)", IEEE
design. The best proposed schemes perform with a loss Transactions on Communications, Vol. 54, no. 3,
in BER performance that is less than 0.5 dB with 462-468, (2006).
respect to the best known binary systematic turbo codes. Ten Brink, S., “Convergence behaviour of iteratively
This loss is also present between a given (a0 a1 a 2 ) decoded parallel concatenated codes,” IEEE Trans.
NP-NS ½-rate turbo code and its systematic version, in On Comm., vol. 49, 1727-1737, (2001).
general terms.