A computer search for good convolutional codes

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A computer search for good convolutional codes

Citation for published version (APA):

Poel, van der, A. P. M. (1974). A computer search for good convolutional codes. (EUT report. E, Fac. of Electrical Engineering; Vol. 74-E-50). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1974

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by

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AFDELING DER ELEKTROTECHNIEK VAKGROEP TELECOMMUNICATIE

DEPARTMENT OF ELECTRICAL ENGINEERING GROUP TELECOMMUNICATIONS

A computer search for good convolutional codes

by

A.P.M. van der Poel

TH-Report 74-E-50 October 1974

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-(TECHNICAL UNIVERSITY EINDHOVEN)

Afdeling Elektrotechniek

GROEP Telecommunicatie ECA

A computer search

for good convolutional codes

by

A.P.M. van der Poel

Report of the research executed 1n the period 1-3-1973 until 15 11-1973 under guidance of:

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Contents

~:

o

Contents.

Summary.

2 I. Introduction.

3 II. Criteria for selecting good codes.

4

III. Algorithm.

12 IV. Results.

14 V. Simulation.

17 Conclusion.

18 Appendix A, computer programme for selecting best codes. (with comments)

23 Appendix B, computerprogramme for simulation (with comments).

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....

' - '

Summary

A new criterion for the selection of good convolutional codes is gIven. A search algorithm is described that uses this new criterion. The results of the computer search are gIven in a table of selected codes. Subsequent simulation shows the newly found codes to be better' than earlier codes found by Odenwalder and Larsen.

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I Introduction

This report describes the optimization of the distance properties of short rate-~ convolutional codes (ref. 1).

These codes result by shifting a binary data sequence into a shift register of length N, connected with two mod-2 adders, see Fig. 1.

coded sequence mod-2 data sequence Fig. 1: Convolutional Encoder.

The tap connections between the register and the mod-2 adders are characterized by the subgenerator functions PA and PB, represented by polynomials in a formal parameter D, or only the binary coefficients of these polynomials; these binary coefficients can in turn be con-sidered as the binary representation of a number; in this report these numbers will be presented in either decimal or binary form.

The contents of the register are called the state 8; this state can be considered as the binary representation of a number too.

By shifting a digit into the right side of the register, a new state arises; evidently only the right (N-1) digits of the register contents,

. N-1 .

are of lmportance for the new state. I.e. reduction modulo-2 lS

allowed before the new state is determined.

As the new state is causedby shifting a "0" or a "1" into the regis-ter. the new state must be 2*8 or 2*8+1.

Hence a particular excursion (e.g. II = 4, 2I1-1 = 8) could contain the following states: 0, 1,3.7.14 (=6),12 (=4). 8 (=0).

As convolutional codes are group codes. the distance properties of the all zero's path with respect to all the other paths, are the same as those of any other path with respect to all other paths. Hence it is no restriction to call the all zero's path the correct one.

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II 'Criteria for selecting good codes

a) Let P k represent the error probability for two binary codewords at Hamming distance k. V.d. Meeberg (ref. 3) showed that P_2k

=

P

2k-1, k

=

1, 2, ••• Using Viterbi's (ref. 1) bounding argument, V.d. Mee-berg could now upperbound the bit error probability of a vi terbi

de-coder by

"'"

P_b

<

I:

_{(A2k_1}+ _{A2k )} P_{2k- 1}

k=1

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Let kO be such that A_2k_₁+ A2k = 0 for k (k

O' The criterion is now that (for a given constraint length N) codes are selected with maxi-mum kO and among these, codes with minimaxi-mum A 2k -1 + A2k •

o

0

b) One has to check, of course, whether a code, selected according to the above criterion, is noncatastrophic (ref. 1).

The next table shows

1. the free distance achieved by Odenwalder's and Larsen's best codes. 2. the upperbound for the free distance (Heller).

3. the Hamming distances (HD) of interest for this search.

N d

free bound HD N d free bound HD

3

5

5 5,

6 8 10 11 9, 10

4 6 6

5,

6 ₉ 12 12 11 , 12

5

7

8

7,

8 10 12 13 11 , 12

6 8 8

7,

8 11 14 14 13, 14

7

10 10

9,

10 iI!)

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III The algorithm

The algorithm counts the number of bit errors (= number of "ones",

= number of odd states) in the paths that contribute to the first term of (1).

The algorithm 1S applied to all combinations of polynomials, but every time, so in the given description of the algorithm too, PA and PB are fixed.

The explanation and proof of the correctness of the algorithm will be given for the specific case that the constraint length II

=

7 (so the free distance must be 9 or 10, see table page 3) and both subgene-rator functions are of the form 1XXXY~1.

These restrictions are not essential, but they simplify both expla-nation and, I hope, the readability of the proof.

PA and PB load 1n these cases both the steps 0, 1 and 2N-2, 211-1 (=0) with weight 2, so the starting value of the Hamming distance (HD)

=

4.

0 PA

_"

5

4)

PB

_"

0 PA

"

6

5) PA

_"

PB

_"

"

3 PB

_"

3

The first two possibilities are investigated by the "D-path of PA"; the second two by the "D-path of PB"; the third two by the "1-paths of PA" etc.

Note that one has to compensate for the fact that the cases 11) and 13) are identical.

"O-path of PA" is a path that starts at initial state SO and consists of subsequent steps (called "O-steps") that are charged by polynomial PA with weight

o.

Note, that the two steps: S, 2~ and S, 2*S+1 have complementory weight, i.e. one step is charged by PA with weight 0 (O-step) and the other step with weight 1 (1-step). This means there is, starting from any state, always the choice between a 1-step and a O-step.

In a O-path, this choice always is a O-step. 0000 •••

PA

a.

PB

Fig. 2:

PA determines the required symbols a

i to make the out-put of the (top) mod-2 adder O. So PA determines the new state, so PA determines the O-path too.

PA is called the active polynomial, because it determines the path; PE is called the passive polynomial, because it only can add to the HD of this path.

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After every O-step the algorithm investigates whether PB adds to the HD and whether a bit error should be counted; the algorithm reduces

( N-l).

>

",N-l

the state S mod-2 1f S " •

"O-path" stops when one of the stopcondi tions is satisfied:

- the path reaches the terminal state; the algorithm investigates whether the path was interesting or not and produces a new subto-tal of bit errors.

- the current HD exceeds 10; the subtotal of bit errors remains unchanged. - the path is catastrophic, so the code is called bad and all

calcu-lations are stopped.

In the cases 3) and 4), investigated by the "O-path of PB", the two polynomials have changed roles, so PB is the active polynomial and PA the passive one.

"l-paths" chooses k times a O-step (with respect to the active poly-nomial) and pursues from the thus reached state a "l-path".

A "l-path" .starts with a l-step (so HD:=HD+1), determines whether a bit error should be counted and whether FB charges this step, and pursues from this state aD-path.

"l-paths" will terminate, if, while doing the k O-steps,

- the terminal state will be reached (this path was counted before as a O-path)

- the current HD exceeds 10

- one of the paths turns out to be bad (catastrophic or HD <9).

All the steps (so, tqe k O-steps, the l-step and the O-path) will be determined by the active polynomial.

In the cases 5) and 6) PA is active; in the cases 7) and 8) PB 1S

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"2-paths" etc. have the same structure, so consist of k O-steps, followed by a "2-path", i.e. a path consisting of a l-step and from the thus reached state" l-paths" etc.

The proof of the correctness of the algorithm should consist of three parts.

a) the algorithm must terminate, so every path must terminate. b) any path may not be investigated more than once.

c) no path of interest may be forgotten.

Proof

a) The proof of the first part uses the following identity: Fig. A) s r 'l •

.

n = N-l B) a. J p 0000 ••• s r ' l P ~ N a_i+1 ••••

In situation A) a.

=

the modulo-2 sum over the positions p,'l,r,s,

J

In situation B) a. must be a "1" if the modulo-2 sum over the

posi-1

tions p,'l,r,s = 1 and a. must be "0" if this sum were O. So,

1

identical to this sum. Conclusion: the symbols a. and a. are

1 J

a. 18

1

always the same, so the situations A) (Feedback shift register, FSR) and B) are identical.

As is well known from FSR theory (ref. 9), any state S will return

n ~1 .

wi th period k ~ 2 -1 = 2 -1, where the e'luali ty holds lf the so called characteristic polynomial is primitive. So any state will re~

turn into the original state, by doing k O-steps.

If the HD stayed unchanged, between the two times the path passes the state S, the code must be catastrophic, so will be called bad.

If the HD is increased, it will do so every k O-steps, so after a finite number of steps, the HD will exceed 10, so the path will be truncated too.

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b) No path will be counted more than once.

Proof: Evidently a O-path is unique (because of the complementary weights (see page 5), there always is only one way to choose

a O-step); hence, a path terminating with a O-path could on-ly be counted twice , i f the preceding traject is investigated twice. This preceding traject however is as unique as the O-path, because it is one of the 1-paths, i.e. it consists of k O-steps ( k is in-creased every time, so k never 1S the same) and one 1-step.

The same arguments hold for the case that 1-paths are the "tails" of 2-paths, so we may conclude that (in general) k-paths are counted

on-ly once.

c) 110 path of interest has been forgotten.

Proof: Say, there is a path with for instance weight 5, that has been forgotten. Polynomial PA took k of this weight and PB took 5-k; if k is smaller than 5-k there has been searched for this path via the k-path of PA (k = 0, 1, 2); if 5-k is smaller, there has been searched for it via the (5-k)-path of PB (5-k = 0, 1, 2), so if the path has been forgotten, it must have been forgotten in the k-path

(resp. (5-k)-path).

In the most complicated case, k=2, the forgotten path consists of X 0-steps, 1 1-step , YO-0-steps, 1 1-step and Z O-steps.

1/-2

I 1I0where the path passes state 2 ,because in that case the path had not been forgotten, but had been truncated.

II The code is not catastrophic (a path of interest had been forgotten!).

Starting from state 1, m O-steps are chosen and, because of reason I) and II), the path has not been truncated. Evidently, the weight has not exceeded 6 yet (so the total HD has not exceeded 10 yet), sO this gives no reason either to cut off the path. By increasing m, the same arguments hold, so at a certain moment m = X.

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After every m a-steps, so after m=X a-steps too, a 2-path is investi-gated.

For the same arguments that held before, the 2-path is not truncated. In the same wB;! it can be proved that after Y a-steps a 1-path is 1n-vestigated etc. So it is impossible to forget a path of interest.

conclusion: the algorithm investigates all the paths that could

pos-sibly be of interest and does so exactly once.

Fig. 4 (page 10) shmTs a flmT chart of "k_paths" and "k_path" and Fig. 5 (page 11) shows a flow chart of "a-path".

One should notice that all procedures are value procedures, so calling a "lower" procedure can not cause a change of the values of the "higher" procedures.

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-1? -17 2) OK? - HAM~ DH? (Hamming distance

no not too much?)

no BITF "f -1?

k-path .. S"fM~S "f M + 1?

(terminal state not reached?)

set new l-step

3)

count Hamming distance (HAM)

BITF and local bit errors (BF)

4)

if code is catastrophic then

O-step count BITF:=-l

I ~ 0 I no 2) reduce mod-M k-paths:

BITF count

3)

(k-1 )-paths

set new Fig.

4:

reduce

N-1 mod-M = 2

BITF _FLOIo/ _{CHART for}

"k-paths" (left)

C.E.P.

4)

k-path:= "k_path" (right)

BITF k-path

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yes

Fig.

5:

FLOW CHARr for

"O-path" BF:= BF+1 O-step count reduce mod-M C.B.P. 3) D-path:= BITF

1) Here the bit error 0, 1 is counted 2) OK? - HAN ~ DH (Hamming distance not

too much?)

_ HAN;<-l (code is not catastrophic?) _ S ;< N

!!:ill!

S " N+ 1 (M=2N- 1)

(terminal state not reached?) 3) If code is catastrophic then HAM:=-l

no BITF:= BITF+BF ? yes I ~ ~ I

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IV Results

With the predescribed algorithm all the codes have been investi-gated through constraint length 8. For length 9. 10 and 11 some preselections had to be made because of the limited computertime.

Table of best codes:

PA PE _{bit errors}

N dec. b1nary dec. b1nary d bit _lo~er

free errors es codes

3 5 101 7 111 _{5. 6} 5 5 4 11 1011 15 1111 _{5. 6} 2 2 5 19 10011 29 11101 7. 8 16 16 19 10011 27 11011 _7. 8 16 6 35 100011 53 110101 _7. 8 1 2 7 77 1001101 103 1010111 9. 10 14 36 69 1000101 123 1111011 _{9. 10} 14 8 141 10001101 179 10110011 9. 10 1 2 155 10011011 197 11000101 9. 10 1

for N

=

_{9. 10} and 11. see remarks belo,,:

----.--- --.

~~f~-~ .-.--~

9 373 101110101 391 110000111 11. 12 14 33

10 563 1000110011 933 1110100101 11. 12 1 14

11 1255 10011100 111 1733 11011000101 13. 14 43_ 92

As no theoretical selecting criterion is available. a practical one has been taken: in all the best codes through length

8.

both PA and PE are of the shape 1XXX1 "hile most times one of them is primitive. This has been used as a criterion for preselection for length 9. 10

and 11.

For II = 9 all such codes have been investigated and a considerable improvement "ith respect to former best codes has been achieved. though the found code is not necessarily the best.

For if = 10 only a part of such codes has been investigated. but the best code (or at least one of the equally best codes) has been found.

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(Less than 1 bit error 1S impossible, otherwise the free distance

would have been 13 and Larsen (ref. 4) would have found it too.) For N=ll only a small part of such codes (only 10% of the primitive polynomials) has been investigated and there has been found already a code with a considerable improvement (by factor

2).

with respect to former results.

Probably the algorithm can be improved, so all best codes can be found for N ~ 9 and at least good codes for N ~ 12.

With the same algorithm systematic codes have been investigated too. (PA

=

0000001

=

1).

Results'

.

PB

N PA dec. binary d bit

free errors 4 1 11 10 11 3, 4 1 5 1 23 10111 5, 6 8 1 27 110 11 5, 6 8 6 1 39 100111 5, 6 3 1 43 101011 5, 6 3 7 1 83 1010011 5, 6 1 8 1 183 10110111 7, 8 17 9 1 423 110100111 7, 8 6 10 1 739 1011100011 7, 8 3 1 791 1100010111 7, 8 3 1 807 1100100111 7, 8 3 1 843 1101001011 7, 8 3 11 1 1319 10100100111

7,

8 1 1 1443 10110100011 7, 8 1 1 1579 11000101011 7, 8 1 1 1675 11010001011 7, 8 1

Note, that for the optimal codes of length 4, 6,8 and 10 and for the best systematic codes of length 4, 7 and 11, there is only 1 "critical excursion", i.e. there is only one path that offers a contribution to the first term of (1), (see page 3) so there can be no overlap of pro-babili ties.

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So in these cases, the first term of (1) is a lowerbound for the bit error probability, while (1) itself is the upperbound.

For very small transition probabilities p. however. these bounds are the same.

V Simulation

In order to test some of the found best codes, the vi terbi decoding algorithm (ref. 1) was simulated on a Burroughs B6700 computer.

The following starting points have been taken: - the all zero's sequence 1S sent

}

i. e. in simulation: a random

- the transmission channel is a BSC of p "ones" at the receiver - the transition probability is p side.

%

- it is not necessary to store the decoded sequences themselves; just counting the number of bit errors (= number of "1"s) in them will do. - the in the B6700 available procedure "RANOOH" is supposed to be good

enough to generate the random "ones" at the receiver side.

- in case two metric values are equal, the procedure RANOOM 1S called

too (with another starting value) for tossing between the two equi-probable possibilities.

The execution time required for one "received bit pair" is:

0,003 sec. for N = 3

0,025 sec. for N = 6 (8 times more states)

0,1 sec. for N = 8

With the computerprogramme any code of length N can be simulated, but the simulation must be restricted (because of the limited computertime) to "bad channels", i.e. p> 1/40 (roughly).

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The simulation has been executed:

for II

=

6

for N = 8

best code with first term in (1): 1.F 7

Odenwalders best codes, with first term ln (1): 2.F 7 best systematic code, with first term in (1): 17.F7

Of course, for very bad channels (p ~ 1/20) the first term is not dominant yet, so the correct value of the error probability is about 10 times the lower bound (= first term, see page 13).

In the case of the systematic code, the first term is more dominant, because the coefficient ie17 instead of 1 (resp. 2).

An important conclusion however is, that for p< 1/50 (roughly), the correct values of the error probability of the newly found code are below the lower bound of Odenwalders best code.

Any other code will have a leading coefficient that will be defi-nitely> 2, so the new code turns out to be the best indeed.

For the codes that have a coefficient ~3 in the first term of (1), we only may conclude, that it is very probable the newly found

codes have a better performance than former best codes.

Figure 6 shows the simulation results and the first term ln (1) for the cases:

A: II =

6,

best code, first term = _{l.P 7}

B: II =

6,

Odenwalders best code, first term = 2.P7

c:

II = 8, systemati c best code, first term = 17 'P 7

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Conclusion

Using V.d. l4eeberg's criterion (ref. 3), new optimal convolutional codes have been found.

The newly found codes for which the first coefficient in (1) is equal to either 1 or 2, yield the lowest bit error probability P B for small p (p ~ 1 /50 for N = 6).

110 such strong statement can be made for codes for which this coefficient is ~ 3.

Note, that in the case where the leading coefficient of (1) is either 1 or 2, the first term in the error bound yields asymptotically

for small p the correct value l10r the bit error probabili ty P B.

Acknowledgement

I would like to thank Prof. Schalkwijk for allowine me to do this research the way I liked it and for his enthusiasm which stimula-ted me very much, though I gratefully acknowledge that lr. Verlijs-donk could protect me sometimes against his enthusiasm too.

I also thank Prof. Schalkwijk and A. Vinck, for the many useful suggestions for improving the written, respectively oral presen-tation of the report.

October 1973

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,

.. --+

_i

\ !

f-~- . t--\: \ _)~ _L . I . ! \ ' Ii , I

\\

_\ \

,

--+.- _. --.-. - -. -,.-'-.. [ -., ..

«~=="T'T'~

...

p ! , ' ___ 1 . -; ---. . . - .. l{)",2~c-r

. 1 .

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- . ( .

\

- t · t- - - .

lit 'simulation results for

best 11

=

6 code

idem for Odenwalders best II

.11 i·dem for best systematic 11

=

6 code = 8 code -3 10

.-i

\ I I \

•

\ \ '

\

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.,

• \

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.

\ \

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I, • . I

₃ -10

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(23)

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=

~ = = =

=

= = = = = =

=

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~ ~

=

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tF"'Jf)tl ;tFNnt

tFLC;Etf\IlILDtn:=-J I :tFl\1nt~JULPA()l

"f.HITFGfP1tPQnCFOUPF:tFFNPAO (~O, 8 ITf ,PA ,P~ .Hf),nL .OH ,N,M) , +'Vl\l.UFtC;O"BTTF,DA,PA,Hn.OL.nH,N,t.A'

tINTFt;FQtSO. 8TH· .DA, OR .. HlhOL ,r)H ,~. M,

tRFGTNttTNT~r;rQtC;,c;nM.T,HAM.RF1C;:=SOlRF:=DLI

;t! T FtA JT F ""IFr)"t-l tTHF:~J:t

·n:u:r. I ~lt:t T F:t C;=I~ l?tOIi tC;=3*M/2'tTHFN-tf~NPAn: =R T TF"tFL <:;F':t

~AF~I~;tC;:~?*c;nIH~M:=Hf)+ll 'I TFtr)I)TP~ IT (c. ,P/\. ~,,,...?);:-tl tTHF"f'I.!t tr~r,TN~C;:=~+l;gF:=RF+ll BLOCK IS DATA I 10 BLOCK I~ 2 ?O 30 0UTPUT I~ 3 40 50 70 4 7! 80 5 90 6 1100 ~ 120 5 lOS 4 140 ISO OUTPUT! 004) 3 1M I DB lAO NULSTAP IS 3 190 200 ?IO NULSTAP(OOS) 3 1500 1510 190 1511}

~UL CYCl 'IS

"

3 1'540 It:;4C:; 1050 1060 1070 1080 4 1090 5 1600 1~20 16:10 1640 1660 1,:;10 16An 4 1~90 1700 NlILCYCLIlS(Ol)f..) 3 230 840 250 2M NUL PAD IS 3 270 2M 4 291 300 310 5 320 330 33? 334 335 340 5 350 360 370 5 1"0 3"5 5 400 410 4 420 ~!ULPj\O (001) 3 450 460 470 fF.NPAr) I~ 1 400 4 500 <; ₅₁₀ 5?n

,

₅₁₀ , q<; ItOttO' SEGMENT IS _a'It' e.?tftft' ~fGMENT O'3*"~'

'-'3*"'"

C:;F:fiMENT "'4t"~' 0'4tM' ~~4U~' '~4t~~

"..,4 • .--'

-"4l,,'"

"'4.,,,,

o«4tH '~4tM

"1"4tO"

"'4tf'll!' IS O~I nft3U~ ""3"~ ~4J'.0 ~FGM~NT •• 5UO A!'StI't8 '~';t"O I' oa~ fJ .. 't~~ M~3tfll" :'\ft:u~'" te3. " SEGMENT ft'fl,fll'I "n(,e-:lfl "-6t,'r, 'e6t·tIo_,

"''''''tnt'

"P6tM' IUt6t,." ft'-"t"r ~"f,tf'!··

"6,·.,..

~·6't"f 't)61 11 ( ~fII61 !)~ f'ln6tl)r ."'6tt .. Gft"t~r IS OOe ""3t",. '03t~' ""~t"l ~!t3t'H SEGMEN"

""7'."

e~1"1'!( Gnl~' ""7.1":' !t·7!'~ fI~7t"l ""7'"

""71""

007:"

nne

,171" '0711 U7U "~71' &~7.e .'1711 .ntO ftt7t~ '~7t· lS or

"1"

"ft)t" !UI't~

1"",.A

C:;Fr.t.!EN '"stt "flRtr ~""Rt .. .e ... .'~t" u~ .. ~~III'"

(24)

~~Fr,INt#JNTfr,F~~~,~OM,T,HAM,AF'~:=~O'AF:~OL' ~IFtRtTF~N~Qt-l~TH~~~ ':/~J:r. I N1 ~ I F "*C;=~-1/,?t-OQt<;=3.M 12tT~FNtE[NPAn:::R T TF"tF"LSft ~RtGtN~~:=?*~nIH.a4:=Hn+l1 tTFtOUTPUT(~,PA..~,M"2)=f')"THFNt ~R€~TNj5:=S+I;~F:=~F+l; tp'.jn:t;

t t F;tOlJTPUT (S, pq. ~,?*Ml = I tTHF.:NtHU~: =HA ... + I I

t'F'<;>M~THfNtS!=~-~' F:FI\ID.ACI:=NlJlPAD{C;,~tTF,PA,PR,HA"1,qF,nH,N,"')' tFN01; JlFNf)t t~LSf.~~F.NPA~:=-l; tnW)tFFNP AD I

t T"JTFG,:pf-tPRnCF01JPFtFJ::"JP AOF"J (o::;n, H T TF ,PA ,p~, Hn, nL, OH, N, M) • tVALIJFtC;O,8YTF,PA,PR.Hn.nL.nH.N,Mt

~ Tfl.ITFfiFPtSO ,A ITF, DA. PR.Hf),r"J{ ,OM ,N,,,,;

tRFr,T~t~INTFGFR1S,~OM,T,HA~,RF,tl~L6AELtHERH~AL;

tT~tPYT~tNfnt-'tT~~Nt

:tR~GTf\.!tO::;OM:~~~~PAn(o::;~,qTTF.DA.PA.Hn.nL.nH,N,M)' O::;:~O::;OIHAM:=Hn;PF:=nL:T:=O:

HF"PH/I,,6L: S:=~I"'-~TAP(S,P",f\.l,~1 C 1TI=";tC):tNF"O:tt.4

t 6"m:t C;;t~lfQ:t~+ 1 ~ !lNntHA"'~lFo:+.nH-l t ANntSOMtI'llI="Ot-l 't'TI-IF't>,l't'

'I8~r.. T ~-;. t T F tOI.ITPI IT C,:>. DR, ~J. ~*? 1=1 tTH':NtHAtA: =HA"'4+ 1 ,

t t Ftc;:tn T Vi "*;?tI'l~()l'C; t. T~Plt~F: =AF' + 1 , tTl="tS>~tTH~~'1'C):=S-M:T:=t+ll tTFtC;=C;~t~~n;tHA~=HOtTHE~11c;nM:=-1' ~TFtt>(nH+l)·M/?:tTHI="N:tsn~:=-ll c;n":=~FNPAn(~,C;nM,PA,PA.!iA~,RF.OH,N,M)' :t(;nTOtHF.RH,AAL; tfp.Jn~1 FF.NPAnp.J: =C;O~' ;tENn;!! 'EL~F'E~NPAOFN:=-11 tF."In;tFFNPAOFN' tINTEr,EPttPRnr.EOUREtTWFFPAO(50,AITF',PA,P~,HD,DL,DH,N,M "

tVAL IJftSO, At TF ,PA ,PA, Hn. nL ,fJH,t<.I,1.f1

tINTFGFP:tSO.RTTF'.o~,PP.HO,DL,nH,N,M' tAE~INttJNTFGFQt~,C;OM,T,HAI.f,AF1S:=S01RF:=OLt tTF'tBITl="t~~f)t-ltTH~~t tAEGTNttTFtS=M/?tf)c:t~=3·M/?tTHENtTWt:FPAf):=BITF~ELsft tAfGINt~:=2·50;HAM:~Hn+l; ;tTF:tOUTPIITfS.DII,N,2'1tM'=OtTHF:N'I' t8EGT~t~:=C;+I:RF:=qF+ll tfNntt

;t t F;tnUTPtlT (S. PP ,1\1, ""!*2) = 1 tTH!:NtHAM: =HAM+ 1 •

;r!IF'~C;>~tTH~NtS:=S-I.f' TWfEPAQ:~Ff.NPAnFN(s,qtTF',PA,PA,HA~,~F,OH,N,M); tFN!}t ;t~~m~ ;tF.:L C:EtnIF~OAO: =-1' iFNr) __ nlF.:~PAI) I tt~TFGF.:QttPDnr.EDURftTW~~PAOFN(SO,RtTF,PA,PR,Hn.nL,nH,N .Ml1

tVA! ,UI="~~O.R tTF, DA ,PR ,Hn,nL. nH,N ,to! I

~I~Tf~~P~SO,~ITF,DA.PR.~n,Ol,OH.N,~l

tRF:G T Nt~ INTF"GfQ'I!5, 50M, ~A"'" AF, T, J 'tLAAfL tOPNT FI IW I

~tFtAtTFtNFnt-ltTHF~t

tRfr,INt~OM:=TWF~PAn (o::;n,RTTF,PA,PR,Hn,nL.I)H,N.M),

~:=~OIHA4:=~nl~~:=~L;T:=OI

OPIIJTF:tl\ll: C:::=NUL5TAP(S,?A,,,,,M, ':tJF;tSINFtJt~

t 0\ ~n~ C::tNt:Q<tt.4+ 1 t Mll)tH At.41LFf)tnH-2t.ANOt.S()MtNFQt.-l tTHF:N1 tRFt; I "Itt. TF ;tnUTPtlT (S, PR, ~J, 4.7) = 1 tTHI:'N-tHA."'1: =HA~+ 1 ,

~TF'tC;tnTv~.,.?tNfQt~tTHFN~~~:~BF+l' IT~tS>~~T~~Nt~:=C;-~fT:~T+lf ;tTFt~=~nt~NntHA~::HntTHElIJt5n"":=-1' 1Tftr>(nH~ll0~!?tTHFNt~~M:=-11 5n~:~TWFfoAnf~,SOu,PA,PR,HAM.RF,nH,N,M)1 ;tGOTn:toPNrFU~: ;t~Nnf' TWfF.'p.anFJI.': =o::;nM I lENni: tElSFtTWEFP/\,nF"I: =-11 tC:Nn-tT..,.FFPII.nn l t

,

fENPAO IS ~f 3 490

_?OO" f

(25)

-"

;!El~F;tTWEfPAf)E~:=-l' ;!':;Nf);tTWt:EPA.OtN' .T,lJ.V.WI ~O!=1;~tTF:=O;HD:=4;nL:=n'N:=~I~:= Ih:OH:= A, x:=16, ;!~Oq~PA:=17tSTF.P;!7#'INTII_;!31;tnOt ;tfl.f.(j Pi;! \o/PITfIOIlHc:;PACfP)ll ,

",RITF fnUT, < 1I ... p A. ... PR ... A ANT AL .. C:HTFOllTFN .. NA .. II> I ;

'fiR I TF. f o.n • < "~llJl.P AO, "NULPAf), "FPJPAnEN. "EF:Nc AnEN .. FN1. I"» &

WQITF(OlIT{GPACE/1)})'

~F0PtK:=O~GTrp;tl~U~JTTI_;t1;tnOt

;tAf~tNxP~:=?*K+l+MI

A: =NutPM) (<:;n. R ITF .PA, PR ,HI). Of., I1H.N .~) & tIF_~>ytTHrM;!A:=-ll

1J.:=NIII.PAO(G1. A,PR,PA,.Hf)'Ol,OH,N.M),

tJFtR>YtTHFMtr::l::=_l'

r:!=rp,p"(1nJ f SO ,r::l:, PA, OR ,Hn, nL ,OI-f.N, M) &

tTF;tC>XtTHFNtC:=-ll n:=~F."JPAnE~J(~O,C,P~,PA,HO.f"')l ,m·',N,,"," tTFtn>X;tTI-f~~tf):~-l' F:=TWFFPAnF~(S0,n,PA.p~,Hn,nl.nH.N.M)1 tTFtF;!!'-IFQ;t-J "l!THFNt F:: =1: -H'FF.:Dt,nnHGO. ('I.PA ,P8 ,Hn,nL ,nH-l ,N, 1.4) • trFtF;tNF~t-ltTHFN~ 1\ : =NlJL CYClIl'5 (PR, DA, f.."

H, ;

trFtftNFQt-J~THEN~ tAF~I~~WPIT~(OlJT{SP4CF:(I) )); WRITF(OUT.<15(I5,Xl,>,PA,PB,A,~~~.O,EI' '1FNI');! 'ENO' ~E'NOt: WPTTF(OlJT,<IIIIIF8.2~IITrJrlTNst:CONOF:N">,ARS(Ir.4YSF.LF".PROCE5STI"'E/41blili1))' tF"NF)t ;tf:Nl')f. AfP Or FQDnPC; OETFCTED = 0000. 1200 l?ln 4 1220 TWFEPADE,.,t COOA' 3 2000 BLOCK IS J 2000 2000 ?OOO 2000 2074 4 2000 2074 2000 5 210S 210~ 2105 210A 2110 2115 2120 2125 2DO 2130 2125 2130 2130 2130 21M 2220 22?S 6 22300 S 2?400 RLOr.K!OOC) 3 22500 RLOCK(003) BLOCK(002) OATA _fA "~R JS eO:1 SE(jM AAC

"C

_PAC UC o~e ~~c

"c

_ue Me 8~e 80e e~c .fe ft,e Me ~'C I.C ftle UC «"te O,"C ~AC MC UC 'fC "flr. ~,~ .~C 8.C ~~C ~.C 15 8ft) ~~3 "~3 J5 15 IS .

REP OF 5Fr,~F.NT~

=

013. TOTAL SEGMENT SIZE

=

000536 W0RDS. r.ORF. fSTTMATE ~ 001898 WORDS. STACK FSTTMATF=' r.RAM SI7F = 000174 CARDS. OO?I?6 SYNTACTTC ITFMS. OOOO?7 OIS~ SEG~ENTC;.

mUM FILF t.J4MF: PJEOECI( I047/QlJ"'l/E/!:CA/vnpO~L/OPTCOf)F IVrP5tE3 'LATION TTMF = 000007 SECONns F.LAPSEO, 000004 sECOt.,JDS PQOCFS"C;INf;.

(26)

Comments to the computerprogramme for computing best codes, see appendix A, page 1-3.

Line nr.:

10: Output (si, pi, n, 2~) computes the output of a mod-2 adder connected with the register according to a pOlynomial pi; the state is si, N is the constraint length.

p1 = 1011 = 11 ( dec)

Sl = 1101 = 13 N = 4

N_1

14 = 2 (in the procedure

heading for reduction of the number of calculations)

Hhen p1 and si both have a "one" at a particular position, the counter SOM is increased by 1; if finally SOM = even, OUTPUT: = 0; if SOM = odd, OUTPUT: = 1.

160: NULSTAP (Ostep): if 2Hl gives OUTPUT:=1 then 2*,+1 gives OUTPUT: =0 and v.v. So NULSTAP:= 2 .. S (or 2 .. -8+1).

1500: NULCYCLUS = controll afterwards for CEP (catastrophi c error propagation) •

The procedure has one Shortcoming: it could come into a loop for active polynomials of the shape OXXX1. However one does not have to fear this, because the procedure is only called for when

(apparently) best codes are found and they alwaors are of the shape 1XXX1

1XXX1

The procedure starts at state 1 and does every time a Ostep until the path (= cyclus) rearrives at stat:..I2IH1=1. If the hamming dis-tance staored unchanged (=0), the code must be catastrophic. After having controlled the cyclus starting at state 1, I controll the

cyclus starting at state 2 (if this were not in the first cyclus etc). The arry A (S) shows whether I have passed a state already

(27)

-'

230: NULPAD (=Opath)

80 = startposition; BITF = subtotal of bit errors; PA = ac-tive polynomial; PB = passive polynomal; HD = hamming

dis-( . N-2 11-1 )

tance till 80 ~.e. 0:=1, 1:= ••••• :=80 and 2 :=2 =0. DL = total number of bit errors (except 0:=1) in this traject, HD = the highest hamming distance of interst, N= constraint length, M = 2N-1•

270: if code is bad, don't calculate anything.

280: set local parameters; count bit error 0:=1 v~a 80M:=DL+1; I = counter of steps, necessary for controll of CEP for active polynomials of the shape OXXX1. When after (DH+1) *:M/2 steps the path has not been cut ofl"yet, the code must be catastrophic. 293: Ostep, followed by controll: + final state reached?

+ hamming distance not more than HD? + catastrophic code?

310-320 count bit errors and hamming distance. 330: reduction modulo M; count I.

332-334 CEP controll. 360: Bad co de

370: not interesting path, number of bit errors remains unchanged. 380: interesting path, increase total number of bit errors.

410: bad code stayes bad.

450: EEIIPAD (= 1path) = one 1step and a Opath. 490: if code is not bad then:

if the path has reached the terminal state, then stop. 500: by doing the 1step, the hamming distance increases by one. 510-530: if s: = 2*s turns out to be a Ostep, then 2*s+1 must be the

1step (so we will have a bit error: BF:=BF+1.)

620: EENPADEII (=lpaths) shifts the startposition of the "EENPAD" i.e.

do k times a Ostep and follow then the 1path.

660: k = 0 (so no Osteps are done before the 1path lS called)

670: setting of local cariables

680-690: while stopconditions are not satisfied do Ostep; stopconcitions are: - reaching terminal state

- hamming distance more than DH - code is catastrophic

(28)

715-718: controll for CEP (see Opath).

All the other procedures are the same (1 general procedure KPAD and 1 general procedure KPADEN could have been used).

So: TWEEPAD calls for EENPADEN, TWEEPADEN shifts the start position of the TWEEPAD etc.

2000: setting of variables:

if N=5 then H = 16 and DH = 8. X = some selecting parameter. .,) 1XXX1

In this particular case all codes of the shape 1XXX1

are investigated, so the hamming distance starts at the value 4. (HD: =4) •

Note that, the active polynomial must be of the shape XXX1. In this case FA (17, 19, •• 31) and PB (1719 •• 31) can change active and passive role.

A, B, C, D, E, are the subtotal of all biterrors after Opath PA, Opath PB, 1path FA etc.

'If' A> X 'then' A:= -1

2130: Here the twice counted contribution (PA contributes HD =2 and PB contributes HD =1) is substracted, of course only when the code lS good.

1740: \fuen an (apparently) best code is found, the CEP controll is called; if the code is catastrophic, one can see this in the third column of the printed output (A=-l )

*)

one could take X = the number of bit errors in Odenwalders best code; if a code had more errors it will be called bad.

(29)

tVALU~t~I,PI,M'#INTFGFR_SI,PI.N.~' tA€GINttINTEGFRtI.~OM,P.~,TI ~04:=OIS:=~T;P:=PI;T:=~. tF'OQtT:=ltC;TF.Ptl;tU~TTLtN;tOnt tp((;TN1r:=T/?: trF'tC;tG~O;tTtTHFNf tAE~T~tS:=S-T;.TFtPtGFQ;tTtTHFNt ~qFGINtP:=P-TI~O~:=504+llt~Nnt tF f'oll)t tFL C:Ft1IFtPtGF(')tT:tTH':::~I;tD :=o-T 1 tt:NO;t;

:t T FtSOM;tDI Vt2*2=50~~THnJ:tOlJTPIJT: =0 tfL 5FtOUTPUT : = 11

tFNntf'UTDUTI 1'TNTF.r,~Pt.;tpPOCFnW~f"lfH.AMOJC;(,(I.X?t')IJTI,OUT:?)1 tV6.,.'JF. If X I. X?OIIT I,OIJT?; tT"-ITFGFPtXl.X?,f'llfTl,OIlT?: IfAFr,YNttY~TFGFQtT:T:=O' of I F:tJ( 1.#11:.0"t01 IT! :tTHFN"tT : =T + 1 ; ;t IFtX?'#IF:OtQUT2tTH!=:'N;tT: =T+ If Ha1oHHc;:=T; ;tFt-.Jf):tHAtJOrSf

;t P'IT~(jFP-ttPPOCF:DLlqf :tRtPt-AF:To I C (MOUD, XI, X 2, OUT 1 ,OllT2) 1

tVALI1F;ttJOUn,Xl,'(2.0UTl,QUT?;

;t T NTFGfQ"tMOUO, Xl, X2, OUT 1 ,('II)T:?' ;tBF:GTNttINTFGERtTf

T: =t-10IJO+HAMOYC; ex 1 ,x2,OUTI ,OUT2) •

qEPMETRIC:=TI tnl(h~~EPME"TR IC I tPEAL;tX,FOUTP,Y;X:=52I,P1?,;Y:=8,4413S. N:=4;M:=2**N-IIK:=?**(~-1)1 ~AE()Tp..lttTNTEG-=:Q:tt~PRAY-t()!JTlMOf'2r ~:M] ,OUT:?M002( 0:"'), O~ETOJCr~:K},N~FTqTC(O:K}, OPAn(o:~},MPAnr":Kl: ~:=4t~:=2 •• (N-1);OA:=11'PR:= 15tFOtJTP:=)/2St WRITf.."~OtJHSDACf:(2)]) ,

WCH TE (OUT, <Y 1 ,IIN". xs, tlPA It, xs, "PAll, '(5, "F"OUTPERC", X 2. "5T APTW"» ,

WOTTF(OtJT(SPACF.(2) 1';

~HHTE(OUT,<t?,Xl>,Mll

WRITE(OUT.<r6,Xl>,~A"

WQTTEfOUT.<16,Xl>,DRll

'itlPITE (n.UT ,<Fq. 3, y 1 ,FIt. n, Xl. I> ,FOIITP ,X) •

IJQTT£'(OUTrSPACt:(4) , , ;

't.F()Q;t~ : =OtSTFPt 1 #JNT r L t?*M-l tDO;t

:#qFG HJtOUT 1 L400~ [OS 1 : =OllTPUT (~, P A, N, "'*2) ,

nIJT?I~002 r s 1 : =OI1TPUT (S, o~, N, u*2) ,

;tEW);t : tF')Qf.S:~OtSTfPtl't.tj"lTTL""~-l;tnot n~FTPTCrS):=N~ETPlrrSl:=OPAnrC;l:=NPAD(~l:=Of :tFClRt J: = 1 t~ TFPt 1 ;tIINT I L of ;:u;tnOI :tP.FI';P";t tF()QtT:=It5T':PtltIlNTlLtQnnn:tnnl tRFGINt1IrtRAp../OO,",/XltLFotFOIJTP ;tTHFNtXl:=ltE,-'SFtX1:=OI

t J FtQ""'I)f'lM (X) Ifl !=,(J1TO! ITP tTHE" 1'-/1 X ?: = 1 tFL SEtX2: =0 ; fFO°;tC; : =1'!;t";TFPt 1 tlHJT T L t~-l tnOt

tREr,IN;tH~:=c;tf)IVf.?:U~:=M+S'HMS:=M~;tntV;t?1 ;tTFtT~nlV~2*?=T~TH~N~

;tAE~JN'II:~~~PMFTRIC(O~~TnTcrH51.xl,X?,O\JTlMOO~r~l,

OUT?~or)?rC;l)t

V: =REPl-olF:TRTt (00.4fTPTCr t-I!<4c; 1, Xl, X? nUT 1 I<40n?( Me; 1,

ollT~~f'n2r 1.4'; 1) I ~ J F";tt I<VtTHFN* ,f1:l:F"r. T"J~~NfTP l C { c; ) : ~I I. tTFtHS=C;/2tTHFNtNP6.nrSl:=OPAn(H~ ~fL~F~NPAnr~J:=OPhn(H~ 1+1' ~F.""'f)f' "I I F;lll>Vt THPI;t 4q~~T~lt~~~Tn,rrC;':=Vt

teo •• ,

8LOCK IS SEGMENl oHA IS Ot!, I 10 8LOCK IS ? 20 30 OUTPUT IS 1 40 50 70 4 71 80 5 90 6 1100 6 120 5 105 4 140 150 OUTPUT (004) J 200 210 220 ?3~~ HAMDIS T~ 1 240~ 250~ 260~ 270~ HAMDIS(005) J 280~ 290~ 300~ 310~ A[PMETOIC IS 3 320~ 330~ 340~ REP"ETRIC(OO~) 3 3 4 5 5 5 6 7

"

350 BLOCK I~ 354 355 360 BLOCK IS 370 380 395 860 8M 860 495 400 4]0 420 43. 440 450 4~6 460 4M 475 4AO 4QO 500 510 520 530 540 SSO 4M 570 SAO ~oo 'to ~f)2te' Sf"G"'EN" ~~3t" 483'" 5fGt.tEN'T '~4t~t '~4t~t ~f)"t~t

""n'

~44t'. "ft41't' Aft4ftt u"tt. U4tft' u"t~, ftt4t~, IS 0,] '''3t~' ftOJt~, 4'3to, .e~'·" SEGMENT ,'St a, ~1tC;1tt,. ~ft5t~f n~5t~~ IS O~f ~03t.' ~"t·~ ~f)3t~' '1t3t,!fI "fl3t ~fI SFGMFNl e'6t~' ~06t·1I) tft6t.r I~ 01'1'-flin:n,,' ~ftJ"·' ~EGMFNl tn7H,

""'7,14P

"''H1'_oIII; ~f:Gt.tENl ~~9U' U9 ... f'lftq,!"T "'C}t~, •• 9tO. 009'" '~9tr-· .,Cq, ... · '~9t~ '('91' M9t. '~9t~ .ft9t~ 8f9'. 8e9t~ f!lItCJt" tt9'4 ft~gto ~'9t~ ~Ogt~ "'Oqt" ~'9tO e~9t4 e.g . . U9tt IUl91e oo9t; .09F

.e9t-

.,qt;

'M'-,.9"

e.9t: .~

...

; ~e9t. ••

(30)

Clt-ttFfHC;=C;/?~THFNtNP~nr~J:;0P6n(HC;} ~F:LC;F:tf..lPA"rC;l:,=OPAnrHC;l+l' ;tf"fo,ln.:t. _FI_C;F.t~~FGr~;t~~FTPTr[C;l:=Vl ~lFt~~=5/?tTHFNt~PAn(S):=ODAnrH~Sl tfLC;FtNPAO(C;):=OPAn{~~Sl+1' t T F-t T;tn T Vt?* 2;f.~Ir:t)t t ;tTHnlf ;tAF.GrN;tU:;AFP~FTOTC(NM[TOIC{HC;J.Xl.X?nIITlMOn?{C;l. I1LJT?J.1nn?{S111 V:=RfPYFTQTr(~~FTRIC(HMS1.Xl.X?0UTl~0n~(Mc;1.

I"!lJT?~lOn~r Me; III

f! F;tU<VtTI .. jj':Nt tRFG IN:tn''''fTPT C{ <; J : =IH tIF;tHc;.=C;/?~THf."'J:tOPtl.n{Sl:=NDAr"jfH<;l tFLC;Ft~PAn(S):=NP~~(HC;).ll tTF1:lI>V:tTHnJ-t fRfGTN1:~~fTPTr{C;l:=VI ;trF~HS=S/?;tTHF~toPAnrSl:=NPAnrHYSl :tFLC;Ft0Dtl.n(Sl:=NPA~(HMC;1·11 tF!oJf"Ifl ;tJFtlJ=V~TH~~-tflFtR~~n0~tYl<1/2;tTH~~;t tENDt: tRfr; 1 "'Jfr)"AFTP TI": r S 1: :11 I ;tJF~HC;=C;/?:tTHFNtODAnrc;l:=NPAD(HC;] -tEl <;F'1nPAnrC;1:=NPAn[H&I.)I ;tF:,If)t fFlC;fttRFGIN;tO~~TPJ~rC;l:=V: tIF;t~S~C;/?:tTHFNt~p~n{51:=~P6DrH~Sl ~flSftOPAnrSl:=NPAnrHMS}+ll tE:NOtl 1E",D;t.: WRIT~(OltT,<X?"'S".XS."YFTRIC".X2.'fqtTr~UTEN"»' ~PITF(OlJT[SPACFP) J)' U:=HMJ:TRTCrOl' :tFOR;tS:=OtSTfPtltUNTIL~t.t-l:tDOt f:FlFGtNt.tIF'tN).AfTRICrC;}tLFQ;t.Ut.THENt tRfGINtWRTTE (O'JH C;PACF' ( t ) 1) I

WPITr.(nUT.<J1.Xl>,~) I

WRITE WilT, < 17. xl>, MMFTRIC( S J' •

'rIRITE (OllT,<l7.xt >,"'lPAD(S} J •

"'F.'tJOt.I 1!~Nn;t1 I<fRJTE (OllT15PACE (4) ) I tE"NO:i1 tE''''Ot.I WRITF(OUT[SPAr.J:C4l1); WRTTFfOIJT,<II .. • .. r ... J ... II>ll

\oIRJ TE (oUT. <" (I-I)" (J-l) =~ET"AANT AL.A ITPARE","OAT"c;fTFST .. r C;u» I

WPITF{OIJT(SPACF.'(2) 1) C WPJTE(QUT,<JS.xl>,T) :1 WPITE(OIJT.<13.X}>,.Jll WPITF(I)UTfSPACE'(21 J ) ;

~R 1 TF {OllT • <X: 1. "Of ."N JFIIWF.. x .. wmlDT .. : "> J I

WRIT~(nUT.<tl?,Xl>,Xl;

I4QITECOIIT(SDtl.CF.{;:?} 1);

WRtTE (oUT. <x 1. uOE .... "'IEUw~ .. Y .. WOJ:WT .. : ">1 •

WPITECOUT.<TI4.Xl>.Y'.

:I;~Nnt'

WQITF.' (OlJT.<IIII/F8.?"TTJOINC)ECON!">ENII>.ARSfMYSE'LF.DqOC ESSTTMF/416667))1

tFNO;U OF EPP0PS nETFCTfD = 0000. 9 9 a 9 9 9 9 9 9 9 a 7 6 6 7 5 4 3 643 644 64~ 646 647 64R 649 640 6~0 660 670 6AD 690 700 710 720 730 740 750 760 770 7AO 790 641 642 643 644 645 646 64 64R 649 BOO AI5 810 395 815 820 825 830 640 A~O 860 AAO 673 A75 ABO A8A RLOl':K(OOq, 900 90S 860 860 aA9 BLOCK/On7) 890 BLOCK/003' 910 ALOC~(OO~) O~TA ~ftQ'4"

0091"-'.9U'

~f9U'

"9'

fit .,91., ~09'" ~~91" .n91~' f'n9t~f ~091~' R09H'

n"9.'"

""9tf'f' ~tlqt" M91o, ft.,q • .,~ "~9t4V U91., t09.M ft~91~' ~~9IN eft9t lllfl "~9'~' "~91ft1 !t'q'l'e" ft,Q.tt, 809t~' r:"9t~t' ftt'91fv

"9'"

&~9t'~ .'91,,1 '~9t~' &e9'~' O~91'" ~09"J U9tAJ .'91~1 U9t~J Oft9U] " 9 r ' ] rft9t~J 'f,9"1 "(191"1 '~9t1' ~'Qt'" 1t~9t~1 4091.] T~ 017 tHI 7 t fll' 1;37.,." tlft7'''~

"'71'"

t')f'7t~,. ~~7tM l'''1t,r· lte7t!" 3"7'''~ ~ft71" M7f" ~ft7t"f ~tl71'" 15 0._ 'U'3"" eii3tftf JS O~' eti?Uf IS

0.'"

IS 0,.

or

SFr,~F'NTC;

=

01). TOTAL SFGMF'NT 5J7.E

=

nnO~60 W0ROS. COQr ESTtMATF

=

002132 WORDS. STACK E5Tl~ATr=O~f

SJ~~

=

on0141 CA~ns. 00117] SYNTACTIC ITfMS, 00on30 DISK SEG~ENTS.

FH F ~H\tIE: P.JEOftK I04~/RIJN/E/fC~,/vnpOF.l/F'O'JTS T"'/vroS IE 1 TION TtMF

=

noon07 ~fr.0NOS EL~PSFO. 000001 S~CONnS pQOCESSING.

(31)

Comments to the simulationprogramme o~ the Viterbi decoding

algo-rithm (~or progr_ see page B.l.

21

We draw a state out o~ the trellis diagram and. the two states it can be originated ~rom:

-MS

We know: s:" sK2 or s:= 2.os+1 this means that s has been

ori-ginated ~rom eiter:

HS-S + 2 (the "hal~ o~ S") or S has been originated ~rom HMS Q MS + 2 (M+S=S (mod-M). is

called MS. HMS

=

the "Hal~ o~ MS") The weights at the branches are stored in two arrays. OUT1MOD2 and OUT2MJD2. In this example the arrll¥S are ~illed like:

OtrrlMOD2 OtJr2MOD2

••• 1 •••••••••••• 0 •••••••••• ••• 0 •••••••••••• 1 ••••••••••

5 M MS

The new metric NMETRIC{S) ~ollows ~om:

NMETRIC{S) : _ min

{U:-

OMETRIC{HS) + HAMDIS V:- OMETRIC(HMS)+ HAMDIS

(Xl. X2. 1.0); (Xl.X2.0.1);

in this ~ormula HAMDIS determines the hamming distance

be-tween the received bi tpair Xl. X2 and the values that are stored in OUTiMlD2. i-l, 2.

We do not have to recall the exact path, but only the ntunber of biterrors in it. (i.e. the number o~ odd states).

So: i~ U

<

V then NPAD{S):- OPAD{HS) +

°

i~ S is even + 1 i~ S is odd etc.

Line nr.

101 Output (see page A.4) is needed to determine the weights

that are to be stored in OtJriMOD2.

280 The procedure BEPMETRIC is used to determine the two

pos-sible new METRIC values (U and V) without making a chOice . between them.

(32)

354

x

is the starting value for the procedure "RANOON"; if one has many experiments to do, one could take the last value of X to start a next experiment.

Y is the starting value (for "RANOOM") for tossing in the case U=V.

355 determines the required storage room for the arrays

380 determines all important variables: constraint length N, the

subgeneratorpolynomials, PA and PB, and the transition pro-bability FOUTP.

410-450 all arrays are filled.

450-460 I and J determine the number of bitpairs that are to be in-vestigated; the subresults are printed J times.

460-475 RANOOM produces a real number between 0 and 1; this is com-pared with FOUTP; in this way it is decided whether X1 and X2 were 0 or 1.

480 the following block is examined for all (trellis) states.

490-810 the "block" is spli t into two parts:

If the first part is used for the k-th bitpair, the second part is used for the k+1st bitpair; in this way the arrays OMETRIC and NMETRIC (for the old and new metric) change roles every time. (and OPATH and NPATH too of course).

510-649 U and V are determined by the procedure BEPMETRIC, see intro-duction. The case U=V requires tossing by means of "RAl'iOOM"

(641).

570, 620, 643 If the new state 1S even, the number of bit errors

re-mains unchanged.

820 Only the states that have a metric that is less (or equal) than the metric of state 0, are printed.

(33)

1. A.J. Viterbi, Convolutional codes and their performance in Communication systems. IEEE, Trans. Comm. Techn. Oct. '71.

2. J.P. Odenwalder, Optimal Decoding of Convolutional codes. Dissertation, University of California, Los Angelas, Ph.D., 1970.

3. L. v.d. Meeberg, Een Viterbi decoder, afstudeerverslag groep ECA Technical University Eindhoven, May 1973.

4.

K.J. Larsen, Short convolutional codes with maximal free dis-tance for rates 1/2, 1/3 and 1/4. IEEE, Trans. Inf. Theory, May 1973.

5.

G.D. Forney, Jr. Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference. IEEE Trans. Inf. Theory, May 1972.

6.

L.R. Bahl and F. Jelinek, Rate-~ convolutional codes with com-plementory generators. IEEE Trans. Inf. Theory, Nov. 1971.

7. L.R. Bahl, C.D. Cullum, W.D. Frazer, F. Jelinek, An efficient algorithm for computing free distance. IEEE Trans. Inf. Theory,

1972.

8. K.J. Larsen, Comments on "an efficient algorithm for computing free distance". IEEE Trans Info Theory July 1973.

9. S. Golomb, Shift Register Sequences. 10. E.R. Berlekamp, Algebraic Coding Theory.

(34)

Reports:

1). Dijk, J., M. Jeuken and E.J. Maanders

AN ANTENNA FOR A SATELLITE COMMUNICATION GROUND STATION

(PROVISIONAL ELECTRICAL DESIGN). TH-report 68-E-Ol. March 1968. ISBN 90 6144 001 7

2) Veefkind, A., J.H. Blom and L.Th. Rietjens

THEORETICAL AND EXPERIMENTAL INVESTIGATION OF A NON-EQUILIBRIUM PLASMA IN A MHD CHANNEL. TH-report 68-E-2. March 1968. Submitted to the Symposium on a Magnetohydrodynamic Electrical Power

Generation, Warsaw, Poland, 24-30 July, 1968. ISBN 90 6144 002 5 3) Boom, A.J.W. van den and J.H.A.M. Melis

A COMPARISON OF SOME PROCESS PARAMETER ESTIMATING SCHEMES. TH-report 68-E-03. September 1968. ISBN 90 6144 003 3 4) Eykhoff, P., P.J.M. Ophey, J. Severs and J.O.M. OOme

AN ELECTROLYTIC TANK FOR INSTRUCTIONAL PURPOSES REPRESENTING THE COMPLEX-FREQUENCY PLANE. TH-report 68-E-04. September 1968. ISBN 90 6144 004 1

5) Vermij, L. and J.E. Daalder

ENERGY BALANCE OF FUSING SILVER WIRES SURROUNDED BY AIR. TH-report 68-E-05. November 1968. ISBN 90 6144 005 X 6) Houben, J.W.M.A. and P. Massee

MHD POWER CONVERSION EMPLOYING LIQUID METALS. TH-report 69-E-06. February 1969. ISBN 90 6144 006 8

7) Heuvel, W.M.C. van den and W.F.J. Kersten

VOLTAGE MEASUREMENT IN CURRENT ZERO INVESTIGATIONS. TH-report 69-E-07. September 1969. ISBN 90 6144 007 6

8) Vermij, L.

SELECTED BIBLIOGRAPHY OF FUSES. TH-report 69-E-08. September 1969. ISBN 90 6144 008 4

9) Westenberg, J.Z.

SOME IDENTIFICATION SCHEMES FOR NON-LINEAR NOISY PROCESSES. TH-report 69-E-09. December 1969. ISBN 90 6144 009 2

10) KeOp, H.E.M., J. Dijk and E.J. Maanders

ON GONICAL HORN ANTENNAS. TH-report 70-E-l0. February 1970. ISBN 90 6144 010 6

11) Veefkind, A.

NON-EQUILIBRIUM PHENOMENA IN A DISC-SHAPED MAGNETOHYDRODYNAMIC GENERATOR. TH-report 70-E-ll. March 1970. ISBN 90 6144 011 4 12) Jansen, J.K.M., M.E.J. Jeuken and C.W. Lambrechtse

THE SCALAR FEED. TH-report 70 E 12. December 1969. ISBN 90 6144 012 2 13) Teuling, D.J.A.

ELECTRONIC IMAGE MOTION COMPENSATION IN A PORTABLE TELEVISION CAMERA. . TH-report 70-E-13. 1970. ISBN 90 6144 013 0

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15) Smets, A.J.

THE INSTRUMENTAL VARIABLE METHOD AND RELATED IDENTIFICATION SCHEMES. TH-report 70-E-15. NQvember 1970. ISBN 90 6144 015 7

16) White, Jr., R.C.

A SURVEY OF RANDOM METHODS FOR PARAMETER OPTIMIZATION. TH-report 70-E-16. February 1971. ISBN 90 6144 016 5

17) Talmon, J.L.

APPROXIMATED GAUSS-MARKOV ESTIMATIONS AND RELATED SCHEMES. TH-report 71-E-17. February 1971. ISBN 90 6144 017 3

18) Kalasek, V.

MEASUREMENT OF TIME CONSTANTS ON CASCADE D.C. ARC IN NITROGEN. TH-report 71-E-18. February 1971. ISBN 90 6144 018 1

19) Hosselet, L.M.L.F.

OZONBILDUNG MITTELS ELEKTRISCHER ENTLADUNGEN. TH-report 71-E-19. March 1971. ISBN 90 6144 019 X

20) Arts, M.G.J.

ON THE INSTANTANEOUS MEASUREMENT OF BLOODFLOW BY ULTRASONIC! MEANS. TH-report 71-E-20. May 1971. ISBN 90 6144 020 3

21) Roer, Th.G. van de

NON-ISO THERMAL ANALYSIS OF CARRIER WAVES IN A SEMICONDUCTOR. TH-report 71-E-21. August 1971. ISBN 90 6144 021 1

22) Jeuken, P.J., C. Huber and C.E. Mulders

SENSING INERTIAL ROTATION WITH TUNING FORKS. TH-report 71-E-22. September 1971. ISBN 90 6144 022 X

23) Dijk, J. and E.J. Maanders

APERTURE BLOCKING IN CASSEGRAIN ANTENNA SYSTEMS. A REVIEW. TH-report 71-E-23. September 1971. ISBN 90 6144 023 8

24) Kregting, J. and R.C. White, Jr.

ADAPTIVE RANDOM SEARCH. TH-report 71-E-24. October 1971. ISBN 90 6144 024 6

25) Darnen, A.A.H. and H.A.L. Piceni

THE MULTIPLE DIPOLE MODEL OF THE VENTRICULAR DEPOLARISATION.

TH-report 71-E-25. October 1971. ISBN 90 6144 025 4 (In preparation).

26) Bremmer, H.

A MATHEMATICAL THEORY CONNECTING SCATTERING AND DIFFRACTION PHENOMENA, INCLUDING BRAGG-TYPE INTERFERENCES. TH-report 71-E-26. December 1971. ISBN 90 6144 026 2

27) Bokhoven, W.M.G. van

METHODS AND ASPECTS OF ACTIVE-RC FILTERS SYNTHESIS. TH-report 71-E-27. 10 December 1970. ISBN 90 6144 027 0

28) Boeschoten, F.

TWO FLUIDS MODEL REEXAMINED. TH-report 72-E-28. March 1972. ISBN 90 6144 028 9

(36)

L.H.Th. Rietjens.

TH-report 72-E-29. April 1972. ISBN 90 6144 029 7 30) Kessel, C.G.M. van and J.W.M.A. Houben

LOSS MECHANISMS IN AN MHD GENERATOR. TH-report 72-E-30. June 1972. ISBN 90 6144 030 0

31) Veefkind, A.

CONDUCTING GRIDS TO STABILIZE MHD GENERATOR PLASMAS AGAINST IONIZATION INSTABILITIES. TH-report 72-E-31. September 1972. ISBN 90 6144 031 9

32) Daalder, J.E. and C.W.M, Vos

DISTRIBUTION FUNCTIONS OF THE SPOT DIAMETER FOR SINGLE- AND MULTI-CATHODE DISCHARGES IN VACUUM. TH-report 73-E-32. January 1973. ISBN 90 6144 032 7

33) Daalder, J.E.

JOULE HEATING AND DIAMETER OF THE CATHODE SPOT IN A VACUUM ARC. TH-report 73-E-33. January 1973. ISBN 90 6144 033 5

34) Huber, C.

BEHAVIOUR OF THE SPINNING GYRO ROTOR. TH-report 73-E-34. February 1973. ISBN 90 6144 034 3

35) Bastian, C. et al.

THE VACUUM ARC AS A FACILITY FOR RELEVANT EXPERIMENTS IN FUSION

RESEARCH. Annual Report 1972. EURATOM-T.H.E. Group "Rotating Plasma". TH-report 73-E-35. February 1973. ISBN 90 6144 035 1

36) Blom, J .A.

ANALYSIS OF PHYSIOLOGICAL SYSTEMS BY PARAMETER ESTIMATION TECHNIQUES. 73-E-36. May 1973. ISBN 90 6144 036 X

37) Lier, M.C. van and R.H.J.M. Otten

AUTOMATIC WIRING DESIGN. TH-report 73-E-37. May 1973. ISBN 90 6144 037 8 (vervalt zie 74-E-44)

38) Andriessen, F.J., W. Boerman and I.F.E.M. Holtz

CALCULATION OF RADIATION LOSSES IN CYLINDRICAL SYMMETRICAL HIGH

PRESSURE DISCHARGES BY MEANS OF A DIGITAL COMPUTER. TH-report 73-E-38. October 1973. ISBN 90 6144 038 6

39) Dijk, J., C.T.W. van Diepenbeek, E.J. Maanders and L.F.G. Thurlings

THE POLARIZATION LOSSES OF OFFSET ANTENNAS. TH-report 73-E-39. June 1973. ISBN 90 6144 039 4 (in preparation)

40) Goes, W.P.

SEPARATION OF SIGNALS DUE TO ARTERIAL AND VENOUS BLOOD FLOW IN THE DOPPLES SYSTEM THAT USES CONTINUOUS ULTRASOUND. TH-report 73-E-40. September 1973. ISBN 90 6144 040 8

41) Darnen, A.A.H.

COMPARATIVE ANALYSIS OF SEVERAL MODELS OF THE VENTRICULAR DE-POLARISATION; INTRODUCTION OF A STRING-MODEL. TH-report 73-E-41. October 1973.

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43) Breimer, A.J.

ON THE IDENTIFICATION OF CONTINUOUS LINEAR PROCESSES. TH-report 74-E-43, January 1974. ISBN 90 6144 043 2

44) Lier, M.C. van and R.H.J.M. Otten

CAD OF MASKS AND WIRING. TH-report 74-E-44. February 1974. ISBN 90 6144 044 0

45) Bastian, C. et al.

EXPERIMENTS WITH A LARGE SIZED HOLLOW CATHODE DISCHARGE FED WITH ARGON. Annual Report 1973. EURATOM-T.H.E. GRoup "Rotating Plasma". TH-report 74-E-45. April 1974. ISBN 90 6144 045 9

46) Roer, Th.G. van de

ANALYTICAL SMALL-SIGNAL THEORY OF BARITT DIODES. TH-report 74-E-46. May 1974. ISBN 90 6144 046 7

47) Leliveld, W.H.

THE DESIGN OF A MOCK CIRCULATION SYSTEM. TH-report 74-E-47. June 1974. ISBN 90 6144 047 5

48) Damen, A.A.H.

SOME NOTES ON THE INVERSE PROBLEM IN ELECTRO CARDIOGRAPHY. TH-report 74-E-48. July 1974. ISBN 90 6144 048 3

49) Meeberg, L. van de

A VITERBI DECODER. TH-report 74-E-49. October 1974. IScN 90 6144 049 1 50) poel, A.P.M. van der

A COMPUTER SEARCH FOR GOOD CONVOLUTIONAL CODES. TH-report 74-E-50. October 1974. ISBN 90 6144 050 3

51) Sampic, G.

THE BIT ERROR PROBABILITY AS A FUNCTION PATH REGISTER LENGTH IN THE VITERBI DECODER. TH-report 74-E-51. October 1974. ISBN 90 6144 051 3 52) Schalkwijk, J.P.M.

CODING FOR A COMPUTER NETWORK. TH-report 74-E-52. October 1974. ISBN 90 6144 052 1

53) stapper, M.

MEASUREMENT OF THE INTENSITY OF PROGRESSIVE ULTRASONIC WAVES BY MEANS OF RAMAN-NATH DIFRACTION. TH-report 74-E-53. November 1974.

ISBN 90 6144 053 X

54) Schalkwijk, J.P.M. and A.J. Vinck

SYNDROME DECODING OF CONVOLUTIONAL CODES. TH-report 74-E-54. November 1974. ISBN 90 6144 054 8

55) Yakimov, A.

FLUCTUATIONS IN I11PATT-DIODE OSCILLATORS WITH LOW q-SECTORS. TH-report 7--E-55. November 1974. ISBN 90 6144 054 6