CLASSIFICATION DES CODES LINEAIRES TERTIAIRES OPTIMAUX [n, n/2]
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Texte intégral
(2) RESUME: Dans ce travail on considère les codes cycliques de rendement 1/2 sur les corps finis GF(3) et GF(5) et on accentue notre étude sur ceux iso-duaux. Le problème central dans la théorie du codage est trouver la meilleure distance minimum dq pour laquelle un code de paramètres [n, k, d] sur Fq existe. Dans ce contexte nous avons réussi à optimiser cette distance pour les codes cycliques de taux 1/2 sur GF(3) et GF(5) en allant jusqu'à la longueur 74 pour les codes ternaires et 42 pour ceux sur GF(5). Nous avons aussi réussi à construire sept classes de codes cycliques iso-duaux sur le corps fini à 3 éléments et trois classes de codes cycliques iso-duaux sur le corps fini à 5 éléments. En considérant les polynômes sur le corps fini de Galois à deux éléments GF(2), notre intention portait sur la divisibilité des trinômes x^am + x^bs + 1… (1), pour m > s ≥ 1 par un polynôme irréductible de degré r sur GF(2), pour cela, nous avons réalisé le résultat suivant: - S'il existe m, s des entiers positifs tels que le trinôme x^am + x^bs + 1 soit divisible par un polynôme irréductible T de degré r sur GF(2), alors a et b ne sont pas divisibles par (2^r-1). MOTS CLES: Corps finis, Codes cycliques, Distance minimum, Codes iso-duaux, Polynômes irréductibles, Divisibilité.. ABSTRACT: In this work we consider the cyclic codes of rate 1/2 over the finite fields GF(3) and GF(5) and we check our study over whose are isodual. The so-called fundamental problem in coding theory is finding the largest value of d for which a code of parameters [n, k, d] over Fq exists. In this context we have successfully optimize this distance for the cyclic codes of rate 1/2 over GF(3) et GF(5) up to length 74 for the ternary cyclic codes and 42 for whose over GF(5) . We have also successful to construct seven classes of isodual cyclic codes over the field of 3 elements and three classes over the field of 5 elements. Considering polynomials over the Galois finite fields for two elements GF(2). Our intention stand over the divisibility of the trinomials x^am + x^ bs + 1…(1), for m > s ≥ 1 by an irreducible polynomial T of degree r over GF(2). Finally, considering the polynomials of type (1) over GF(2), we contribute to the result: - If there exist positive integers m, s such that the trinomial x^am + x^bs + 1 is divisible by an irreducible polynomial T of degree r over GF(2), then a and b are not divisible by (2^r-1). KEY WORDS: Finite fields, Cyclic codes, Minimum distance, Isodual codes, Irreducible polynomials, Divisibility..
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(316) L?MJ >? "5 @B5=95B 3?45 4E =1D8P=1D9395> (- 1==9>7 ? =I>? a FYILCACH? OH <LC=IF;A? ?MN <;Md MOL F? N?MN >? J;LCNd (IHAO?OL . 4 %H@ILG;NCIH + 4 .?>IH>;H=? 2 4 ,IOL PICL MIH @IH=NCIHH?G?HN L?A;L>IHM OH ?R?GJF? CHMC JIOL 4 ?N >IH= . + 2 %F MY;ACN >YOH =I>? <CH;CL? >? FIHAO?OL =ILLCA?;HN ?LL?OL ?N ?H >dN?=N;HN . .
(317) !* + #!
(318)
(319)
(320)
(321)
(322)
(323)
(324)
(325)
(326) . =B;KO? FCAH? ?N =B;KO? =IFIHH? IHN ;FILM OH HIG<L? J;CL >? [[ (? N;OR >? NL;HMGCMMCIH ?MN J;MMd >? KOC dN;CN F? N;OR >? NL;HMGCMMCIH >O =I>? >? LdJdNCNCIH
(327) =ILL?=N?OL a (? >?ORCcG? =I>? >? $;GGCHA ;JJ?Fd =I>? >? $;GGCHA ;OL; OH N;OR >? F? G?CFF?OL JIMMC<F? JIOL OH =I>?
(328) =ILL?=N?OL NL;HMG?NN;HN >?M <FI=M >? <CNM. .
(329) $ ,%0.! %%. ?45C I3<9AE5C %@D9=1EH 9C?4E1EH 45 (5>45=5>D CEB .
(330) >DB?4E3D9?> (; NBdILC? >O =I>;A? PCM? a =IHMNLOCL? >?M =I>?M =ILL?=N?OLM J?L@ILG;HNM IJdL;HN ;O JFOM JLI=B? >?M FCGCN?M NBdILCKO?M dN;<FC?M J;L F; NBdILC? >? FYCH@ILG;NCIH. ;HM =? =IHN?RN?. F; L?=B?L=B? >? =I>?M IJNCG;OR JL?H> ;FILM NION MIH M?HM JOCMKOYCF MY;ACN >? L?=B?L=B?L F? =I>? ;S;HN F; JFOM AL;H>? =;J;=CNd >? =ILL?=NCIH >Y?LL?OL JIMMC<F? JIOL OH? FIHAO?OL >? =I>? ?N OH? >CG?HMCIH WRd?.
(331)
(332) ?45C I3<9AE5C %@D9=1EH CEB *IOM ;>;JNIHM F?M HIN;NCIHM ?N >dWHCNCIHM CH>CKOd?M >;HM /ICN F? =ILJM WHC >? #;FICM a dFdG?HNM HINd
(333) 1H =I>? . + FCHd;CL? N?LH;CL? ?MN OH MIOM ?MJ;=? P?=NILC?F >? >CG?HMCIH + >? (? N;OR >YOH =I>? FCHd;CL? . + ?MN >dWHC J;L +.
(334). . ?OR =I>?M N?LH;CL?M ?N MIHN dKOCP;F?HNM MC FYOH ?MN I<N?HO a J;LNCL. >? FY;ONL? J;L OH? J?LGON;NCIH >?M =IIL>IHHd?M /ICN ' 8 F? JIFSHfG? AdHdL;N?OL >O =I>? =S=FCKO? ;FILM MIH =I>? >O;F=S=FCKO? ;>G?N JIOL JIFSHfG? AdHdL;N?OL F? JIFSHfG? Ld=CJLIKO? >? ( 8 . 8 ' 8. Ig F? JIFSHfG? Ld=CJLIKO? & 8 >YOH JIFSHfG? & 8 >? >?ALd 2 MOL ?MN >dWHC J;L. & 8 8 & . 8 1H =I>? ?MN >CN CMI
(335) >O;F MC ?N MIHN dKOCP;F?HNM (?M dFdG?HNM >? MIHN ;JJ?FdM >?M GINM >? =I>? ?N F? JIC>M 74 8 >YOH GIN >? =I>? 8 ?MN F? HIG<L? >? JIMCNCIHM HIH HOFM >;HM 8 (; >CMN;H=? $ 8 9 >? $;GGCHA ?HNL? >?OR GINM >? =I>?M ?MN >dWHC? J;L $ 8 9 74 8 9 (; >CMN;H=? GCHCG;F? >YOH =I>? FCHd;CL? ?MN $ -).$ 8 9 8 9 8 9
(336) .
(337) 1H =I>? FCHd;CL? . + $ MOL OH =ILJM WHC ?MN OH =I>? . + >? >CMN;H=? GCHCG;F? $ ,IOL OH =I>? FCHd;CL? F; >CMN;H=? GCHCG;F? ?MN dA;F? ;O JFOM J?NCN >?M JIC>M >? NIOM M?M GINM >? =I>?M HIH HOFM (? JLI<FcG? =?HNL;F >;HM F; NBdILC? >?M =I>?M ?MN >YIJNCGCM?L OH >?M J;L;GcNL?M . + ?N $ JIOL >?M P;F?OLM >IHHd?M >?M >?OR ;ONL?M 1 dN;HN WRd 1H? >?M >?OR P?LMCIHM ?MN . 0LIOP?L $ . + F; JFOM AL;H>? P;F?OL >? $ JIOL F;KO?FF? OH =I>? . + $ ?RCMN?. 1H =I>? KOC ;NN?CHN =?NN? P;F?OL ?MN ;JJ?Fd OH =I>? IJNCG;F (; L?=B?L=B? >?M =I>?M N?LH;CL?M IJNCG;OR ; dNd CHCNCd J;L $CFF ?N *?QNIH 78 %FM IHN NLIOPd F?M P;F?OLM >? . + $ JIOL + JIOL NION $ ?N F?M P;F?OLM >? . $ JIOL NION?M G;CM
(338) P;F?OLM >? $ (YdN;N >? FY;LN ?N N;<F?M JIOL . $ ?N $ . ?MN >;HM 78 );LON; 78 ; JLIOPd FYCH?RCMN?H=? >? =?LN;CHM =I>?M >? >CG?HMCIH MOL ?N ; JLdM?HNd OH? HIOP?FF? N;<F? JIOL . $ a FY;>L?MM? MOCP;HN? BNNJQQQ A?I=CNC?M =IGG;LM A?IALC?MG?L BNGF #L;MMF 78 G;CHNC?HN a DIOL OH? N;<F? >?M <ILH?M MOJdLC?OL?M ?N CH
(339) @dLC?OL?M >? $ . + JIOL . . ?M <ILH?M JIOL >?M P;F?OLM HOGdLCKO?M >? $ . . IHN dNd ;GdFCILd?M IO dN;<FC?M a NL;P?LM F; =IHMNLO=NCIH >?M =I>?M KO;MC
(340) =S=FCKO?M J;L #OFFCP?L ?N +MN?LA;L> 78
(341)
(342) ?45C I3<9AE5C *INL? L?=B?L=B? ?MN ;Rd? MOL FYIJNCGCM;NCIH >? F; >CMN;H=? GCHCG;F? >?M =I>?M =S=FCKO?M N?LH;CL?M . Ig . ?MN J;CL HIH GOFNCJF? >? ?H J;LNC=OFC?L F? =I>? ,IOL F?M =I>?M 527H*2;.< =.;7*2;.< . F?M <ILH?M CH@dLC?OL?M ?N MOJdLC?OL?M >? $ JIOL . MIHN =IH@IH>O?M ,IOL . PICL 78 F?M <ILH?M MOJdLC?OL?M H? MIHN J;M NIODIOLM ;NN?CHN?M +H >IHH? C=C F; N;<F? >?M <ILH?M >? $ JIOL .
(343) H.
(344) . >.
(345)
(346)
(347)
(348)
(349)
(350)
(351) . H >. . . . . . . . . . . . . . . . .
(352)
(353)
(354)
(355)
(356)
(357)
(358)
(359)
(360) . .
(361) (YONCFCM;NCIH >? (Y;FAILCNBG? >? B?H JLdM?HNd >;HM FY;LNC=F? 78 & " 2IFI=B IG
(362) JONCHA NB? GCHCG;F >CMN;H=? I@ =S=FC= =I>?M IGJ ;H> JJFC?> );NB?G;NC=M 2IF JJ
(363) HIOM ; J?LGCM >YI<N?HCL NIOM F?M LdMOFN;NM MOL F?M =I>?M =S=FCKO?M
(364) !H ONCFCM;HN );NB?G;NC=; F; @;=NILCM;NCIH >O JIFSHfG? 8
(365) 8 8 8 8 8 8 8 8 8 8 8 . 8 8 8 8 8 8 8 8 8 8 8 . % ?N $%58 ?H @;=N?OLM CLLd>O=NC<F?M MOL F? =ILJM >IHH? (5)4 JIFSHIG?M >? $%'2 JIFSHfG?M >? $%'2 % CHMC JIOL ;PICL OH JIFSHfG? AdHdL;N?OL ' 8 >? >?ALd CF @;ON =BICMCL JIFSHfG?M J;LGCM F?M ?N ?H =BICMCL >? >?ALd =? KOC >IHH?
(366) =BICR JIMMC<F?M 0ION?M F?M =IG<CH;CMIHM JIMMC<F?M IHN dNd @;CN?M =? KOC ; >IHHd F?M JIC>M >?M GINM >? =I>? >O JIOL =B;KO? JIFSHfG? ' 8 =BICMC +H ?HL?ACMNL? >;HM F; N;<F? F?M LdMOFN;NM >? =;F=OF >O JIC>M >?M GINM >? =I>? JIOL =?LN;CHM JIFSHfG?M AdHdL;N?OLM 0;<F? AR. ". GIN >? =I>? ;. . ! ". "!". ! ". ". "!". "" . QN;. ' 8. . ' 8. . ! ". ". ". ". ' 8 . . ! "!". ". ". ' 8 . . . ./ )3/$. . . ./ )3/$. . . ./ )3/$. . . ./ )3/$. . . ./ )3/$. . ! "!". .
(367) AR. ". GIN >? =I>? ;. ! " ". ". ! "!" . ". "!". . ! ". ". "!". . ". ! "!" ! ". ". . "" . QN;. ' 8 . . ' 8. . ' 8. . ' 8. . ". ". ' 8. . ". ". ! "!". . ! "!". ' 8 . . . . ./ )3/$. . . . ./ )3/$. . . . . . . (5=1BAE5 %F S ; ?R;=N?G?HN =I>?M IJNCG;OR J;LGC F?M =I>?M =S=FCKO?M ?RCMN;HNM
(368) *INIHM $ F; G;RCG;F? >CMN;H=? GCHCGOG >YOH =I>? =S=FCKO? HIOM LdMOGIHM ;FILM HINL? JL?GC?L LdMOFN;N J;L &B?@?C9D9?> * -2<=*7,. 62726*5. 89=26*5. -.< ,8-.< ,B,52:>.< =.;7*2;.< (
(369) ) .<= -
(370)
(371)
(372)
(373) ?45C I3<9AE5C ;HM =? =;M IH ; =BICR M?OF?G?HN JIOL AR ?N F; @;=NILCM;NCIH >? 8 ?H @;=N?OLM CLLd>O=NC<F?M MOL HIOM >IHH?. 8 8 8 8 8 8
(374)
(375)
(376) 8 8
(377) . 8 8 8
(378)
(379)
(380) 8 8
(381) . CHMC IH ;OL; F?M GINM >? =I>?M ?N F?OLM JIC>M L?MJ?=NC@M. .
(382) 0;<F? . A R . G IN >? =I >? ; . ! . "" . Q N; . . ' 8. . . ' 8. . . ' 8. . ' 8. . . ! ! ! !. . . ! !. . . ! !. . (5=1BAE5 (?M =I>?M =S=FCKO?M >? J;L;GcNL?M MIHN 9C?4E1EH >IHN MIHN IJNCG;OR &B?@?C9D9?>
(383) * -2<=*7,. 62726*5. 89=26*5. -.< ,8-.< ,B,52:>.< =.;7*2;.< (
(384) ) .<= -
(385)
(386) ?45C I3<9AE5C ? GeG? C=C IH ; =BICR JIOL F? JIFSHfG? AdHdL;N?OL >O =I>? 7 8 8 8 8 8 8 8
(387)
(388)
(389) 8 8. 8 8 8
(390)
(391)
(392) 8 8. YIg F; N;<F? >?M GINM >? =I>?M ?N F?OLM JIC>M =ILL?MJIH>;HN 0;<F? . A R . G IN >? =I >? ;. ! . . . ! !. . . ! !. . . . "" . Q N; . ' 8. .
(393). ' 8. . . ' 8. . . ' 8. . ! ! ! !. (5=1BAE5 0IOM F?M =I>?M =S=FCKO?M N?LH;CL?M >? J;L;GcNL?M 7 8 MIHN 9C?4E1EH &B?@?C9D9?> * -2<=*7,. 62726*5. 89=26*5. -.< ,8-.< ,B,52:>.< =.;7*2;.< (
(394) ) .<= -
(395) .
(396)
(397)
(398) ?45C I3<9AE5C ,IOL F?M =I>?M =S=FCKO?M F; @;=NILCM;NCIH >? 8
(399) HIOM >IHH? =BICR JIMMC<F?M JIOL F? JIFSHfG? AdHdL;N?OL >? >?ALd 8
(400) 8 8 8 8 8 8 8
(401) 8 8. 8 8 8 8 8
(402) 8 8. 8 8 8 8 8 8 8. 8 8 8 8 8 8 8. !N J;L =IHMdKO?HN IH HIN? >;HM F; N;<F? NIOM F?M GINM >? =I>? ?N F?OLM JIC>M =ILL?
(403) MJIH>;HNM 0;<F? A R . G I N > ? = I > ? ; . . . . . ". "" . Q N; . "! ". ' 8. . ./ )3/$. . ' 8. . ' 8. . ./ )3/$. . "! ". ' 8. . "! ". ' 8. . ./ )3/$. . ' 8. . ' 8. . ./ )3/$. . ' 8. . ! " . "!". . . ! ". . . ! ". . . . . . . . . . . . . . . . . . ". "!". ". . "!". ! ". ". "!". ! ". ". "! ". .
(404) (5=1BAE5C
(405) ,IOL F?M =I>?M 9C?4E1EH >? J;L;GcNL?M IH ; MICN "!". ! ". ". "!". IO ! ". ". IO <C?H 5 8 6 8. . ' 8. !N >;HM F?M
(406) 31C IH ; "".
(407) %F S ; ?R;=N?G?HN =I>?M CMI
(408) >O;OR ?N =I>?M IJNCG;OR HIH CMI
(409) >O;OR &B?@?C9D9?> * -2<=*7,. 62726*5. 89=26*5. -.< ,8-.< ,B,52:>.< =.;7*2;.< (
(410) ) .<= -
(411)
(412)
(413)
(414) ?45C I3<9AE5C ,IOL F?M =I>?M =S=FCKO?M
(415) F; @;=NILCM;NCIH >? 8 HIOM >IHH? =BICR JIMMC<F?M JIOL F? JIFSHfG? AdHdL;N?OL >? >?ALd 8 8 8 8 8 8 8. 8 8 8 8. 8 8 8 8 8 8 8 8. !N J;L =IHMdKO?HN IH ; F; N;<F? KOC HIOM >IHH? NIOM F?M GINM >? =I>? ?N F?OLM JIC>M =ILL?MJIH>;HNM 0;<F? AR. GIN >? =I>? ;. . "" . QN;. ! !. ' 8. . ! !. ' 8. . ! !. ' 8. . ' 8. . . . . . . . . ! . . ! !.
(416) . ! . . "" . QN;. ! !. ' 8. . . ' 8. . ! !. . ' 8. . ! !. ' 8. . AR. GIN >? =I>? ;. . ! !. . (5=1BAE5 0IOM F?M =I>?M =S=FCKO?M
(417) MIHN 9C?4E1EH >IHN MIHN IJNCG;OR &B?@?C9D9?> * -2<=*7,. 62726*5. 89=26*5. -.< ,8-.< ,B,52:>.< =.;7*2;.< ( ) .<= -
(418)
(419)
(420) ?45C I3<9AE5C ,IOL F?M =I>?M =S=FCKO?M N?LH;CL?M F; @;=NILCM;NCIH >? 8 HIOM >IHH? =BICR JIMMC<F?M JIOL F? JIFSHfG? AdHdL;N?OL >? >?ALd 8 8 8 8 8
(421)
(422)
(423) 8 8. 8 8 8
(424)
(425)
(426) 8 8. !N J;L =IHMdKO?HN IH ; F; N;<F? KOC HIOM >IHH? NIOM F?M GINM >? =I>? ?N F?OLM JIC>M =ILL?MJIH>;HNM 0;<F?
(427). GIN >? =I>? ;. ! . . "" . QN;. . . . ! !. ' 8. . . . ! !. . ' 8. . . . ! !. . ' 8. . . ! !. ' 8. . AR. . . (5=1BAE5 (?M =I>?M =S=FCKO?M >? J;L;GcNL?M MIHN D?EC CMI
(428) >O;OR. .
(429) &B?@?C9D9?> * -2<=*7,. 62726*5. 89=26*5. -.< ,8-.< ,B,52:>.< =.;7*2;.< ( ) .<= -
(430)
(431)
(432) ?45C I3<9AE5C ? GeG? F; @;=NILCM;NCIH >O <CHfG? 8
(433) HIOM >IHH? =BICR >O JIFSHfG? AdHdL;N?OL >O =I>? =? KOC HIOM J?LG?N >? PICL ?RB;OMNCP?G?HN NIOM F?M JIC>M >?M GINM >O =I>? 8
(434) 8 8 8 8
(435)
(436)
(437) 8 8 . 8 8
(438)
(439)
(440) 8 8 . ,;L =IHMdKO?HN IH LdMOG? F?M J;L;GcNL?M >O =I>? >;HM F; N;<F? KOC MOCN 0;<F? . . GIN >? =I>? ;. ! . . "" . QN;. . . . ! !. ' 8. . . . ! !. . ' 8. . . . ! !. . ' 8. . . ! !. ' 8. . AR. . . .?G;LKO? 0IOM F?M =I>?M =S=FCKO?M MIHN 9C?4E1EH &B?@?C9D9?> * -2<=*7,. 62726*5. 89=26*5. -.< ,8-.< ,B,52:>.< =.;7*2;.< (
(441) ) .<= -
(442)
(443) ?45C I3<9AE5C ? GeG? F; @;=NILCM;NCIH >O <CHfG? 8
(444) HIOM >IHH? JIMMC<CFCNdM JIOL F? =BICR >O JIFSHfG? AdHdL;N?OL >O =I>? =? KOC HIOM J?LG?N >? PICL ?R;OMNCP?G?HN NIOM F?M JIC>M. .
(445) >?M GINM >O =I>? 8
(446) 8 8 8 8 8 8 8
(447) 8 8 8 8 8
(448) 8 8 8 8 8 8
(449) 8 8 8 8 8 8 8 8 8 8
(450) . 8 8 8 8
(451) 8 8 8 8 8
(452) . 8 8 8 8 8 8
(453) 8 8 8 8 8 8 8 8 8 8
(454) . CHMC IH LdMOG? F?M J;L;GcNL?M >O =I>? >;HM F; N;<F? 0;<F? A R . G I N > ? = I > ? ; . ". "" . Q N; . ! ".
(455). . . . ./. )3/$. . . . . ./. )3/$. . . . . ./. )3/$. . . . . ./. )3/$. . . . . ./. )3/$. . . . . ./. )3/$. . . . . ./. )3/$. . . . . ./. )3/$. . . . . ./. )3/$. . . . . ./. )3/$. . . . . ./. )3/$. . . . . ./. )3/$. . ,IOL =?M =I>?M F; @;=NILCM;NCIH >? 8
(456) 8 8 MYd=LCN. .
(457) 8 8 8 5 8 5 8. 8 8 6 8 6 8. /C IH JL?H> NIOM F?M ' 8 KOC >CPCM?HN 8
(458) =IGG? =Y?MN CH>CKOd >;HM N;<F? IH ; D?E:?EBC ?N . 8
(459) ' 8. ' 8. 8
(460) ' 8 ' 8. ? KOC GIHNL? KOYCF >L5H9CD5 @1C 45 3?45C 3I3<9AE5C 9C?4E1EH >? J;L;GcNL?M
(461) (5=1BAE5 %F S ; ?R;=N?G?HN =I>?M =S=FCKO?M IJNCG;OR &B?@?C9D9?> * -2<=*7,. 62726*5. 89=26*5. -.< ,8-.< ,B,52:>.< =.;7*2;.< (
(462) ) .<= -
(463)
(464) ?45C I3<9AE5C (; @;=NILCM;NCIH >? 8 HIOM >IHH? =BICR JIMMC<F?M JIOL ' 8 >? >?ALd 8 8 8 8 8 8 8. 8 8 8 8. 8 8 8 8 8 8
(465) 8 8 8 8 8 8
(466) . 8 8 8 8 8 8 8 8 8. 8 8 8 8 8 8 8 8 8. 8 8 8 8 8 8 8 8 8. 8 8 8 8 8 8 8 8 8. ,IOL =?M =I>?M IH ; =;M .
(467) MICN ". ". ! "!". ". ". IO ! "!" ! . ;P?=. . 8 ' 8. ' 8. MICN ". " ! "!". 8 ' 8 ' 8. ;P?=. (?M JL?GC?LM =;M MIHN PdLCWdM J;L F?M =I>?M >? >CMN;H=? GCHCGOG $
(468) (? cG? =;M ?MN PdLCWd J;L F?M =I>?M ;S;HN $
(469) +H HIN? KO?
(470) >?M =I>?M ;S;HN $
(471) PdLCW?HN F?M ?L ?N cG? =;M ?N KO? F?M. . ;S;HN $ PdLCW?HN F? cG? =;M. ;HM F;. >?LHCcL? =IFIHH? >? F; N;<F? MOCP;HN? IH HIN? F?M JIC>M GCHCGOM >?M GINM =ILL?MJIH>;HNM ;OR AdHdL;N?OLM >?M =I>?M =S=FCKO?M >YIg F?M J;L;GcNL?M =;L;=NdLCM;HN F?M =I>?M =S=FCKO?M 7 8
(472). .
(473) A R . ". ". ' 8. . ". ". ' 8 . . ". ". ' 8. . ". ". ' 8. . ". ". ' 8 . . ". ". ' 8. . ". ". ' 8. . ". ". ' 8 . . ". ". ' 8. . ". ". ' 8. . ' 8 . . . ! "!". . . ! "!". . ! "!". . . ! "!". . . ! "!". . . ! "!". . . ! "!". . . ! "!". . . ! "!" ! "!" ". " ! "!" ". ". ' 8. . ". ". ' 8. . ". ". ' 8 . . ". ". ' 8. . ". ". ' 8. . ". ". ' 8 . . ". ". ' 8. . ". ". ' 8. . ". ". ' 8 . . ". ". ' 8. . " ! "!". ' 8. . ! "!" ! "!" ! "!". . . ! "!". . . ! "!". . . ! "!". . . ! "!". . . ! "!". . . . Q. ". . . "" . ! ". G IN >? =I >? ;. ! "!" ! "!" ". ". ". ' 8 . . ". ". ' 8. . ' 8. . ! "!" ! "!" ". " ! "!" ". ". ' 8 . . ". ". ' 8. . ! "!" ! "!".
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