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We denote the class of all the CACs of length n and weight k by CAC(n, k). Note that a code C " CAC(n, k) can be viewed as an (n, k, 1) optical orthogonal code without the autocorrelation property. PP3 C CCCCK<C%%C CCC.C CCCCCCCCCCCG3G3G3?3>d=>VSet representation3We can use subsets of Zn to represent codewords by their natural correspondence. The three codewords mentioned above are three subsets of Z12: {0, 1, 2}, {0, 3, 6} and {1, 5, 9}. Corresponding differences are {-1, +1, -2, +2}, {-3, +3, -6, +6} and {-4, +4} repectively. 6s>& ,  C ?@D!E"F#G$H%I&J'K(L)M*N+O,P-Q.<Starts at Even n 2 (mod 4), 8$$3 ( Theorem (Levenshtein and Tonchev, 2005) If n 2 (mod 4), then M(n) = (n-2)/4. Furthermore, the optimal code obtained is an equi-difference (centered) code. Note: If n is odd, only partial results have been obtained. For odd prime n, M(n) n/4.(AE3AE Aq [ j #;k   n 0 (mod 4)? ,(  (Theorem (Jimbo et al., IEEE T. Information Theory 2007) Let n = 16m + 8. The maximum size M(n) of a code C "CAC(n) is w  /3 C C C C G 3 C C C    C C C A> L|5Use Skolem type sequences(JTheorem (Fu, Mishima and Uruno, DCC 2009) The maximum size M(n) = M(16m) of a code in CAC(n) is M(n) = 7n/32, if m 0 (mod 2); (7n-16)/32, if m 1, 5 (mod 6); and (7n+16)/32, if m 3 (mod 6), with the exceptions M(48) = 10 and M (64) = 13. R* 3I b !Kk-extended Skolem sequence&C((bLet k and n be integers with 1 k 2n+1. A k-extended Skolem sequence of order n is a sequence (a1, a2, & , an) of n integers such that When k = 2n + 1, it is simply called a Skolem sequence of order n. ( 2, 7, 6, 8 ) is a Skolem sequence of order 4 and ( 2, 5 ) is a 4-extended sequence of order 2.f2ZCCC       #                     1    3 3 3 3 3 3 3 3 3 3Skolem-type sequences!?( New sequences are needed!( ~6 Observation (,After pairing off differences for centered codewords, we are left with k doubly even integers and 2k odd integers. If we are  lucky enough, we can obtain k difference-triples. For example, if the differences are 1, 3, 5, 13, 4, 8, then we have two triples {1,3,4} and {5,13,8}.+ B9Question (Let A = {1,3,5, & , 4k-1} and B = {4,8,12, & ,4k}. Can we partition the union of A and B into k difference-triples? For example k = 4. Triples are: {11,15,4}, {3,5,8}, {1,13,12} and {7,9,16}. 15  11 = 4, 3 + 5 = 8, 13  1 = 12 and 7 + 9 = 16. (Odd sequence!) ZZ3 3 33 3A:Odd sequences " ($For positive integers k and n, let K be a k-subset of {1,2,& ,n} and D be a 2k-subset of {1,3,& ,4n-1}. A K-extended odd sequence of order n and defect D, denoted by K-ext Qn of defect D for short, is a collection of n - k ordered pairs of odd integers {(ai,bi): bi  ai = 4i or bi + ai = 4i, i{1,2, & , n} \ K} with the union of {ai,bi}, i{1,2, & , n} \ K, is equal to {1,3, & ,4n-1} \ D. If K = {t}, a K-ext Qn of defect D is simply denoted as t-ext Qn of defect D, and if K = , it is denoted just as Qn. #D3 3 i i i  iC                        $   i i i  i  i i i    i i i   P-;Examples $lAn Q4: {(7,11),(5,13),(3,9),(1,15)}. A 2-ext Q5 of defect {13,19}: {(7,11),(3,9),(1,17),(5,15)}. A {2,3}-ext Q8 of defect {3,5,29,31}: {(23,27),(9,25),(1,19),(11,13),(7,21),(15,17)}.  "      !        1  YSettle the remaining even case~Theorem (IEEE Transaction I.T., Nov. 2010) Let n = 8m + 4. Then M(n) = (7n+4)/32, if m a" 0 (mod 4), (7n+12)/32, if m a" 1 (mod 12), (7n-12)/32, if m a" 2,6 (mod 12), (7n-4)/32, if m a" 3 (mod 4), (7n-20)/32, if m a" 5,9 (mod 12), (7n+20)/32, if m a" 10 (mod 12). (*) Joint work with Yi-Hean Lin and Miwako Mishima. |+"v6vv 3 3 "  "" " $$("(,,%0"0 0&00"00&00"00&03XE >How about the case n is odd? ZHow many centered codewords? <G(n) = (V, E) V(G) = {1, 2, & , (n-1)/2} i is adjacent to j if and only if j a" 2i or j a" - 2i (mod n) G(n) is 2-regular. O(n) = number of odd cycles in G(n). We have at least [(n-1)/2 - O(n)] /2 centered codewords. Example: G(21) = (1,2,4,8,5,10) + (3,6,9) + (7). O(21) = 2.xx9Z Z9ZC C C C C C C C C C C C C  C C C C C C C C C C C C C C C C C C C C C C C C C C C G G G G G G G C  * lh EX QuestionsFor which odd n, O(n) < 3? Is there an explicit formula for O(n)? For odd n, determine M(n), the maximum size of a CAC(n) with weight 3.vP(AC ( H(n) = (V,E) V(H) = {1, 2, & , (n-1)/2, & , n-1} i is adjacent to j if and only if j a" 2i (mod n) H(n) is 2-regular. Let en be the order of 2 in < Zn*, >. (1, 2, & , 2t) is a cycle where t = en  1.rxC C C C C C C C C C C C C  C C C C C C C C C C C C C C C C C K C K C C C K C K C K $ccB Lemma 2 For any cycle C = (a, 2a, 4a& ) in H(n), its length divides en and is equal to en whenever gcd (a, n)=1.|sxC C C C C C C C C C C C C C C K C C K C C C C C ,-5 C Theorem 3 Let p be an odd prime. (1) pa"1 (mod 4) O(p) = (2) pa"3 (mod 4) O(p) = ~x C A A A A A A A A A A A A A A A 4E.D ^Theorem Let n = , where p1 < p2 <& < pm are distinct odd prime factors and each ri is natural number. Then there exists a tight code C in CAC(n) if and only if one of the following holds: (a) O(pi) = 0, for all i or (b) p1 = 3, r1 = 1, and O(pi) = 0 for i > 1. J0xG 3 C C !C C K C K C K ,C C K 6C C C C 7C C C C K C C G 3C C K C C K  C K  C C C Xn;7/FWieferich primeE  ^ A prime p which satisfies 2p-1a"1 (mod p2).0xC  C C C K C C K C G &Lemma 5 Let n = , where each prime factors p1 < p2 < & < pm are non-Wieferich primes, and each ri is natural number. ThenxA C C 3C C K C K C K $C C K C ,^ H Theorem 6 (1) O(pr) = r" O(p) for p is a non-Wieferich prime.Ax A C C C C K C C CC C C C C C > I Theorem 7 The maximum size of a conflict-avoiding code of length pr is , where p is a non-Wieferich prime and r is a natural number. Conclusion There is a long way to go when n is odd.x/ C =A A I $A A A A A G i)G jw \8 We have to stop! Thank you for attending this talk. 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_rels/.relsPK-!fcWdrs/downrev.xmlPKr R r K <B  s *o#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!rdrs/downrev.xmlDъ0E߅0 5deQ6c[l&%Agν̗ič-+% k. ˤNSm;>-&STPЦR"~d[x;[g0ѕR;좹i8I&`1–K~5 fz?$>@}x_ ?]&E0PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!rdrs/downrev.xmlPKr K r D @B  s *#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!fcWdrs/downrev.xmlDn0H}k+qH@I1⯀z$jlǪ*3󍧍ę/-+x& 3K>g> k,+yNZcLh>EEu* 2軶&uC].K47%@,9&XӬg2 \6?i;iTy} V߮߮;&E0;PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!fcWdrs/downrev.xmlPKr D r = <B  s *o#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 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_rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!rdrs/downrev.xmlDъ0E߅0 5deQ6c[l&%Agν̗ič-+% k. ˤNSm;>-&STPЦR"~d[x;[g0ѕR;좹i8I&`1–K~5 fz?$>@}x_ ?]&E0PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!rdrs/downrev.xmlPK K D @B  s *#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!fcWdrs/downrev.xmlDn0H}k+qH@I1⯀z$jlǪ*3󍧍ę/-+x& 3K>g> k,+yNZcLh>EEu* 2軶&uC].K47%@,9&XӬg2 \6?i;iTy} V߮߮;&E0;PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!fcWdrs/downrev.xmlPK D = <B  s *o#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!rdrs/downrev.xmlDъ0E߅0 5deQ6c[l&%Agν̗ič-+% k. ˤNSm;>-&STPЦR"~d[x;[g0ѕR;좹i8I&`1–K~5 fz?$>@}x_ 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?]&E0PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!rdrs/downrev.xmlPK K D @B  s *#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!fcWdrs/downrev.xmlDn0H}k+qH@I1⯀z$jlǪ*3󍧍ę/-+x& 3K>g> k,+yNZcLh>EEu* 2軶&uC].K47%@,9&XӬg2 \6?i;iTy} V߮߮;&E0;PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!fcWdrs/downrev.xmlPK D = <B  s *o#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!rdrs/downrev.xmlDъ0E߅0 5deQ6c[l&%Agν̗ič-+% k. ˤNSm;>-&STPЦR"~d[x;[g0ѕR;좹i8I&`1–K~5 fz?$>@}x_ ?]&E0PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!rdrs/downrev.xmlPK = 6 @B  s *#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ 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_rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!fcWdrs/downrev.xmlDn0H}k+qH@I1⯀z$jlǪ*3󍧍ę/-+x& 3K>g> k,+yNZcLh>EEu* 2軶&uC].K47%@,9&XӬg2 \6?i;iTy} V߮߮;&E0;PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!fcWdrs/downrev.xmlPK=D == <B  s *o#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!rdrs/downrev.xmlDъ0E߅0 5deQ6c[l&%Agν̗ič-+% k. ˤNSm;>-&STPЦR"~d[x;[g0ѕR;좹i8I&`1–K~5 fz?$>@}x_ ?]&E0PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!rdrs/downrev.xmlPK== =6 @B  s *#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!fcWdrs/downrev.xmlDn0H}k+qH@I1⯀z$jlǪ*3󍧍ę/-+x& 3K>g> k,+yNZcLh>EEu* 2軶&uC].K47%@,9&XӬg2 \6?i;iTy} 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_rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!fcWdrs/downrev.xmlDn0H}k+qH@I1⯀z$jlǪ*3󍧍ę/-+x& 3K>g> k,+yNZcLh>EEu* 2軶&uC].K47%@,9&XӬg2 \6?i;iTy} V߮߮;&E0;PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!fcWdrs/downrev.xmlPKR K <B  s *o#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!rdrs/downrev.xmlDъ0E߅0 5deQ6c[l&%Agν̗ič-+% k. ˤNSm;>-&STPЦR"~d[x;[g0ѕR;좹i8I&`1–K~5 fz?$>@}x_ ?]&E0PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!rdrs/downrev.xmlPKK D @B  s *#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!fcWdrs/downrev.xmlDn0H}k+qH@I1⯀z$jlǪ*3󍧍ę/-+x& 3K>g> k,+yNZcLh>EEu* 2軶&uC].K47%@,9&XӬg2 \6?i;iTy} 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_rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!fcWdrs/downrev.xmlDn0H}k+qH@I1⯀z$jlǪ*3󍧍ę/-+x& 3K>g> k,+yNZcLh>EEu* 2軶&uC].K47%@,9&XӬg2 \6?i;iTy} V߮߮;&E0;PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!fcWdrs/downrev.xmlPK`Y<B  s *o#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!rdrs/downrev.xmlDъ0E߅0 5deQ6c[l&%Agν̗ič-+% k. ˤNSm;>-&STPЦR"~d[x;[g0ѕR;좹i8I&`1–K~5 fz?$>@}x_ ?]&E0PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!rdrs/downrev.xmlPKYR @B  s *#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!fcWdrs/downrev.xmlDn0H}k+qH@I1⯀z$jlǪ*3󍧍ę/-+x& 3K>g> k,+yNZcLh>EEu* 2軶&uC].K47%@,9&XӬg2 \6?i;iTy} 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=A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 +Oo 0N0 ZIRIH(     90e0e?Rectangle 15,   QExampleC(    # S0e0e?Rectangle 16V >6___PPT9 @x = (1 1 0 0 1 0 0 0 0 0 0 0 0) y = (1 0 1 0 0 0 0 1 0 0 0 0 0)TA" EEEPPEE___EPPE___EEEEPPE___EE___EPPE___EEPPE___ E A E  HT?Text Box 23"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Qdrs/downrev.xmlDJ1EB!;Xѱi)Rv9y3 $cNop˽˙-۲B4I`*k$>^nńNaIP|t5RQO KSJcb\j,CUsۖO;zT'+a)`:kXa,ѐǼ?E) V/_DAB˶PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Qdrs/downrev.xmlPK  Ur ,$  0 9Csupp(x)={0,1,4}, supp(y)={0,2,7} (Base blocks of a cyclic 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2g3   U| j]   S 0Group 36#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!|drs/downrev.xmlDA0oVQEd]<ȂUXhmy)M,Yo{SWZVO"ę% x ydeR `M}.] DJdMlMm \7FB,9,XӾ_vYٞ29ŤhV <~Z  7OPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!|drs/downrev.xmlPK ,$D 0B  s F܎DENFQRUVWfjJ? .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 20j]  ^ g3    Hԑ?Text Box 30+   e3  2g3   W~ }% T  S 0Group 37#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!R&drs/downrev.xmlDOk@&B-=BUo3 f߆6}Pq0hSj ERtxZpYccL`=l1v/ꏾ.Am"+*2%v}])uCF.U9,TRVQq?~*~-KR1݀4_C+x+PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!R&drs/downrev.xmlPKSl ,$D 0B  s FTDENFQRUVWfjJ? .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 22}% T ^ g3   HH?Text Box 31& c e4  2g3   b} `n  S 0Group 38#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!=ك(drs/downrev.xmlDAk@&-)% VzB g̾ 5+zffM ^D K?_>@8L [ϞVj;7 _a.ҕ5t b{>Ⱦ1M+_]l8,Ѧz-TĤ|ʗ I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK! _drs/downrev.xmlDA0 okZeEQDT<*GlKib[Y87|ٙR4T²x N.8Sp9o GXZ&/r\h/5'a*ҥ9tC[nk>:6M)GQ4  9V)}Fv57q_nR~ &w%xPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-! _drs/downrev.xmlPKX A,$D 0B   LDENFQRUVWfjJ? .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 19   ^ g3   H?Text Box 33 n  e5  2g3   ~ \,OE.7 S 00000S 26#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Gdrs/downrev.xmlDOk@&J[B-=BUo3 f߆6}Pq0hSj ERtxZpYccL`=l1v/ꏾ.Am"+*2%v}])uCF.E9,TRVQq?~*~-KR1݀4_C+x+PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Gdrs/downrev.xmlPK1 ,$D 0B   LDENFQRUVWfjJ? .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 21\,ZHE.7 ^ g3 *  H<?Text Box 34n.1O#7 7 6 (mod 13)J 2g3 3 g3     X?"0000juS0000000 24 f27 e   e "   rff=i=i?҉8NV҉b_ 25#"@PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!Rdrs/shapexml.xmlWn6;YqcŦ[ij%*֚"r^dzJgԻ0́Uok HrI5i_PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!R*drs/shapexml.xmlPK-!$drs/downrev.xmlPK"$D  0<___PPT10Z___PPT9<4 We are interested in k =3 case! ggPPgPPgg g .    H  0޽h ? BDVSTCg¨t%}%___PPT10]%+ jD$' = @B DT$' = @BA?%,( < +O%,( < +D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8 ' =%(D' =%(D' =4B BB@B%()))E5' =1B B`BPB,54*3>B ppt_c='`B@BPB<*D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =4B BB@B%()))E5' =1B B`BPB,54*3>B ppt_c='`B@BPB<* D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =4B BB@B%()))E5' =1B B`BPB,54*3>B ppt_c='`B@BPB<* D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D ' =%(D' =%(D' =4B BB@B%()))E5' =1B B`BPB,54*3>B ppt_c='`B@BPB<*D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =4B BB@B%()))E5' =1B B`BPB,54*3>B ppt_c='`B@BPB<*D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-g6B fade*<3<*"D' =%(D' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*"D%(D' =-g6B fade*<3<*"DD' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*++0+0 ++0+0 ++0+0 +" 0N0 G? (     ~0e0e?Rectangle 24   V Optimal Code C(  s   # 0e0e?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!6drs/downrev.xmlDN0HHܨC uP~TM`I ^Gۤo}o:L+N|cY(A\Zp`_n& |@Z&g0]^L1׶-v>Gu]./k2G#:!FWIpq`Ǝ5߻QeäXn]}e|(u}5,A8[ǗEi>@^ϟ[􁜂m)PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!6drs/downrev.xmlPK$ 0 8___PPT10R___PPT94, A code of maximum size (maximum number of codewords) is said to be optimal. M(n) = M(n,3): the maximum size of a CAC of length n and weight 3. M" "     www  www,  PP www ~GGGGGGwwwGGwwwGgG www*    }   -     X\7?"0000juS0000000 13 f26 e    "   r/ff=i=i?҉8NV҉b_ 14#"@PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!+Y-drs/shapexml.xmlWn7;VeZj*XN{ F\i#.rɏknr4@q ̟!g8?Ŷl#L]h3Rjח1g%J$|'j믞Wq]1Vu\%|emzu%t%rmJe2ʒRQt+P Wͼ7J_mflqο>?gcke1?V8A]mnT?f[VRm~.a4"gbkYGpSH';ӂ!N2^E 7QمH-6}{=nֲ&@,mH&zٔ?,?>/>/&PgXCs:!W9H^7V*{1; gYab>8ucf%gF_ C<څtxՊ7l=|1~`Ix>T7 66SEwxV5Nke,6w{σ@~ \g;i?tSkB;m!Q> ,L]|  !waK D~IU¿#6 svՂ3mYZBAxRͫYUW,!T"\^'k37ebX}wސkw hWp7+y/bi @"WPȮ("q…:z=wYN-8C4O& *]a5s?l-IPEPK ]F+X$"$Y\Q ף@\:08$P]><2'R]~p5Sbظӎ`vWR0)*lPYHFZtci}c&j+xTnp|#f%f*d2BKߟm}prݿ>g yߏ/}r z7zwҫݻWTG}k|V?PK!Fdrs/downrev.xmlD[K1FC7Uڴ^@}nf/,Il}p>d6N(ֳQf}v *&dg2ps*J$s4ФZDz!q{ba6^Y6[{kZ};|~YG6dp{*ѐWd Tx؆.1& $NRPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!+Y-*drs/shapexml.xmlPK-!Fdrs/downrev.xmlPKV T$D 04___PPT10 B___PPT9$  We want optimal CAC of weight 3.!ggPPggPPgg    H  0޽h ? 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Fdrs/shapexml.xmlXn7;;ʒ--jQ 9#.WژK.H"1zͭB.}ctH6\jE.g837?_6 ^t)ᒩ4˄ )1d BI-7߽,cST&.;VB\^tSjn`P!:nS@.%r݊ݮiB{芄m~>|zD#کe_KTEg,τxvq5?bed&?sbe4bx0 n\ 8RWvU%gc(\60zAh: A7]q甐mx#+,;g8qQ1]r<l uaf37'Tˋ7cXQQZAwwk4 ;sfjܪA鬔8cX 2diK.c@Up{~\6r$HsFRc*RPp=.+D n+ޠxӉmR^:|.ynxNM=uu>0v׆2Кpט F/Iek >j__)2 ! 2;!4s /1ZxI)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!ثBdrs/downrev.xmlDj@EaxfFǠX~#y&̛03&w\˹g=YVD1” W Kl ?rm&okLK3W"@إKtEM]d:݌C,-[ǟRcƎk*C+9·5Y.^4t8>Sj6+FS?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!ثBdrs/downrev.xmlPKP dIzR  c $;"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!d~drs/downrev.xmlDj@E0<щ-jIEP5lIhMȌ}g˹]mzۈ;uv`6M@Nl~ dcRG6`v)*BhS)}YE?u-qd?b>6=Ibq–vכUQY}ˢ8s_Ro~"P^ec}%PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!d~drs/downrev.xmlPKBL ILzQ~v0___PPT106___PPT9 ,x = (1 1 0 0 1 0 0 0 0 0 0 0 0) -l( AgggPPggPPg - F  JR?0000 0000 21Z -Eq!J > 0 1 2 3 4 5 6 7 8 9 10 11 124?g >g >     Jpb?0000 0000 22"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!E>Bdrs/downrev.xmlDN1EIfL|V"W 1Dm] m|ɝ{rmNND#xe\'~ߎbB$\(zu5ZbtL++S*cb\Z,r= 1+ z|׾ݫL?oj3X#KlΏ:%az7[:`L$b[L~PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!E>Bdrs/downrev.xmlPK @t<$ 0v0___PPT106___PPT9 hx = {0,1,4}, D(x)={1,3,4,9,10,12}, D2(x)={1,3,4}5g g        4    H  0޽h ? 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PKP9s}|y&I ja<_oW{QMg`T_sPUpR>qEBTQ^W2j|(M*jc%1e]iJmWxÀp⻗,(FJP #@;87 \j8kDJq7dtk|bG8,XX!ѓ hcO'5CWh+>m#.m("t8729kfe A[lIjV硁8h6Dō놖ؙK01qrC4%}@$?#*Y@?9~Xk404mOu<ٚl g"ܣ:2N{z\$7SZ-'cF[th4'{]KwCkSyc#mҧp @#-q MAea7.A w )Z9@X)ik*>ǓjkGw5<΁UꈆP/ 6,bi'Иc8R1m5;١-P+aIfƩ[M@v޷7}:R H' EϖräIh4,4^I|IlJ!"[R]U>KEU 7wxM״Ӿfy]Whx~qC_2`ryL,pZfͤn0_y 4·>]*7{Os83HKK|LpIk88uq ;X%Q| 2Wjj .)0^( ~{j@n:dcću/h^?ci̫Or+0Hf Dp~vyxD_|HQYg\\^̩Pdv|܇ZD!4!􅔾jXٞMi 1Z*Gם$ɭԚP놫Fgۇ4+ץMloWD!1_(I>Z#!DUA.~PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!Mwc*drs/shapexml.xmlPK-!4ݼ drs/downrev.xmlPK T $D 0 8___PPT10N___PPT90( @For any two distinct codewords x, y C, if D2(x) )" D2(y) = , C is a CAC. Packing {1, 2, & , n/2} to obtain as many codewords as possible (optimal CAC).                        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(u,v) E when either v = 2u if u n/4, or v = n " 2u if u > n/4. " ZGOGG GGGwwwGG  CGGGGGGG GGGGGGGGGGG          G   www m         "      Hl?Text Box 13"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!m_drs/downrev.xmlDj1ECxjR˨F) 7$Tǿ7ta{93[e' x'y 2~ ;p KnK]jXXRWr+MwrW`13ܶ|(Ĉ[4.?hMSuZ _mVu?{Y#(R>>S`?K=O%aXD^]~Q[m)PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!m_drs/downrev.xmlPK   ,$ 0 ^" 2g     Hĝ?Text Box 14"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!sdrs/downrev.xmlDMK1E!/{Zjj>wN‹m7m3яb%VR^_ {`?z+QOJxR X>D2E ůSPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!sdrs/downrev.xmlPK   ,$ 0 g " 2gf     X@?"0000juS0000000 14 f18 e   "  # ~$ jJff=i=i?҉8NV҉b_ 15#"E @9 PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!:Adrs/shapexml.xmlZn7?;zr!rPp\ArڥAp9Cɒ?/dӿi~FX:heK#6ɹ$/},du+٥uJ Tˋ.}kS&C&]:}s7NtY'ұ1iV˂1OXB\[t jqiD5ZbI0Ӿt$!%%?~ӧ?IEk9P-1Yg$gVCͮ[\"2>`wikQoۖ3C|>h6[2E~|~i)jK|Ճ"v.j" d6=Ɍo1mΔ^,:p MLt) ȶ };\{t3sE3 )FrLZ\RE{<(Viëz8HLAs[u,0o7|Ewi]8>1qk3̯,Vj#K/W` W0j:Q-,3s=v&|oB Q.+EY}skWH0d9 3`=mmfwk Xf^J+i6yK= S+F1c-#;~eW͞0!0:w ڹ:k}(n,:aZ- \AL!ÎCK_5`l9;$oX)2.c&/x`xBA?0^R˵w2 m7_FKt^l`MEeSİ<4>C \/'ID`]?;fa9򸻿.Ҧ%:ӔjJ\rmD_$'L֐0_\Պ@6T}Ta B]2DJꖁY3Mɉ1F3Oy !46ǒX,ptƝ8W G7mj9 ɶŨν^LLf`Y vS=+a͘5=8"Z|4O~'<# T-r}Y=.R oķW_Y]uU~aٵwl*I? {yΓ6fV3ٶmFۍOK` g2Zr;=?̽ O՞?r~Ed2 T![ y'$Iu f_5Xde7jI.WI2nj3&,G/UF>OHu$˯Xp$^~VU_<9/\WpճyoJRʏ,~TW1%!TZNrVMO1+ r0&bW3tgЏM[.HK337mʹJ*Sds2nc| 4:J1 *52ykEXS۷;E/T4ΔPf"2h3 }2j}W}`nA$Fqc,@8@Ma[1 fPKduۊڱ5RP=ICMSr.yI [8jR.ϸ8#Eq1)}ϿEEAS<|)0+Yz#SFx`[LP'$πP}gPt^]uގR3*Kǟj?#=L/ywmŅ[ ?2;WPK!Yľdrs/downrev.xmlDMK1BCzXڴ,Z-z臂fn&K7x 3'r`\M1:mC%} LJ_dЏlGYiOWI妕wIJ Dž;ZTFASz ˛^tlIA5z>SbjD g?PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!:A*drs/shapexml.xmlPK-!Yľ( drs/downrev.xmlPK. ) $D 08___PPT10N___PPT90( An edge (u,v) E corresponds to a possible centered codeword x s.t. 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BDVSTCg¨t$___PPT10+K/vD' = @B D' = @BA?%,( < +O%,( < +D' =%(D0' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =-g6B fade*<3<*>Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*>}%(D' =-g6B fade*<3<*>}D' =%(D' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*~%(D' =-g6B fade*<3<*~D"' =%(D' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =K@BB BBPB0B%(/%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*++0+0 ++0+0 ++0+0 ++0+0 + 0N0 @RR@(   2  P33?Oval 2"OIPK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!*rj drs/shapexml.xmlWn6;^[v,Eݢf8k1XkH/-my~d;ɩp^:#jfUBZWU3 }OΜ'2Z|-w7o6kׯR>NLeIƻؒ<UIPN]R ҋq5A狑eE,3M%~\b]ntju3$w00mOI_Skr*)s# : pr6Y~0`xDUn 9,$D 0 $V = {1, 2, & , n/2}L(2ggg   2   n0ffԔ?Oval 3 Q  \g  2  \1faa?Oval 4"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!_Ddrs/downrev.xmlD1O0Fw$uHlIVTMv!);OoMNvV@>ɀmԶ[= Yp+vK:,IlP8"az7[.=tȲ)7mZPӳסj3+u6 q{3>=4?/W%/)rZn1DR`MPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!_Ddrs/downrev.xmlPK  " ,$D 0 \g     ~0e0e?Rectangle 5g   Example of Gc(n)l C(C(K(C(C(C(&    %  F?Text Box 6"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!3drs/downrev.xmlDJ1@a U6-"ZJ/0nf7M&>p3suŁb+(G uַ {(RF (|v~6Jaۂ%>UWBڐ4 =yfM3:7R ==?_Nk׬jl{;+S.&Pdږ?zRJYޣL\ f?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!3drs/downrev.xmlPK _ ,$ 0 Gc(14)H 2eme     F?Text Box 7(Y f24  2g     F'?Text Box 83   f12  2g      F(?Text Box 9  N  u30ge      H*?Text Box 10 Ib v270ge      HL?Text Box 11 E  u60ge      HU?Text Box 12 ( k v110ge      H^?Text Box 13 $d  f20  2g     HD`?Text Box 14 ^ u50ge     Hlr?Text Box 15 E  v100ge     Hc?Text Box 163 q  v250ge     H?Text Box 17n @ } B  e8  2g     H?Text Box 18 & f13  2g     Hd?Text Box 19F  v170ge     H?Text Box 20,H  v260ge     Hȝ?Text Box 21,b  f21  2g     HL?Text Box 22o  f28  2g     H?Text Box 23; y v140ge     H?Text Box 24, u70ge     Ht?Text Box 25S '  v230ge     HP?Text Box 269 'f  f16  2g     H?Text Box 27 w  v220ge   2  X?Oval 28   \g    HL?Text Box 29 a  v180ge     H?Text Box 30   u90ge   2  ^t?Oval 319hW \g  l~ _   S 0Group 32" B ! j?Line 33" " B " j-?Line 34 _ B # j?Line 35F  $ H?Text Box 368   v190ge    % H!?Text Box 37  e4  2g    & H#?Text Box 38A w ^ e2  2g    ' H<-?Text Box 39j  e1  2g    ( H6?Text Box 40b @6  f15  2g   ~~   )S 0Group 41"q n { ^ )  *S 0Group 42" B + d?Line 43u ) B , j?Line 44^ ^ B - j?Line 45{ { z n   .S 0Group 46"~  B /  p<?Line 47n " B 0 jZ?Line 48< < B 1 jj?Line 49l lR 2 2 ^tI?Oval 50  \g  ~~   3S 0rGroup 51"  n { ^ )  4S 0Group 52" B 5 d?Line 53u ) B 6 j?Line 54^ ^ B 7 j?Line 55{ { z n   8S 0Group 56"~  B 9  p<?Line 57n " B : jZ?Line 58< < B ; jj?Line 59l lR X~  F : <S 0Group 60" ) B = jrn?Line 61 ( B > jr?Line 62 %B ? jr?Line 63  B @  pr?Line 64 F B A jr`?Line 65  :B B jr?Line 66{ { t C HP?Text Box 67"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!ϴXSdrs/downrev.xmlDOK1G!&jumZDZkA=x7`Ilw7oeG/a:7AI8mn^ $`N‹mwu7Ob%R^^ 2 َ_ճ*-|~ݎѨS(܂/PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!ϴXSdrs/downrev.xmlPK  ,$ 0 x(1, 2) ! {0, 1, 2} (27, 2) ! {0,27,54} (2, 4) ! {0, 2, 4}==g = B D@ X3o?Line 68"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!odrs/downrev.xmlDn1 +VDA RJ=T@xY%),o_*h3cj3^])γ|"1u~{Y b )fzJhS cՒ8Ih&,{; -o~r;|rOlW_R4n_~~>/A}1& d# ^PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!odrs/downrev.xmlPK " ,$D  0B E ^T3o?Line 69"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDOK0  \ZEݲ^uzּ6&)I\w<~|L!)r׽|~TwK18xG .aZcj:S'8B~$\0tR8>˞qUPXh߫Qfޭ@$ӿy~A/Su1QP! 7PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!drs/downrev.xmlPK !,$@  0B F@ ^3o?Line 70"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!!drs/downrev.xmlDO[k0~"3"Q @t9mʚD[0/VmŅ|h+ȧk_ۇ95I!"p(P+ !a:U[ Zj} eٳpj0P{8[I[ߎIWx?t?g oC$ }J rPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!!drs/downrev.xmlPK ,$@  0 G xA??menu_i2Picture 71menu_i2 B H@ ^h3o?Line 72"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Q"Yudrs/downrev.xmlDON1ݛ5qc3 BF #[F_j{;i0]<9ztV 񬠜  v_ 1!kI"WWK?:"pPAR_IƉ3wa0RI)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!6drs/downrev.xmlDOO0?4ět0A!p|ma}]=x/ӊ39XV0' K|3> kl-+yX.Fws̵P>Gu]./k2Ƕ#:!BWIM+$ 5v4PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!6drs/downrev.xmlPK c ,$@ 0B J@ ^3o?Line 74"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK![drs/downrev.xmlDOMK@ e/nXk춈"PzfI4;vm_=U{ l fKo _cP1![l=#EM&XX%WR±@uJ]u,kr#n$0TI)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!'drs/downrev.xmlDO[K0~B  }<6gMIJ{?r=^b睂| xݹVj"&t{H"WgK?-ujXPIC;`11 n{ye7b@*?bSgE}Fˋv"єW󢜗 /-DAL\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!'drs/downrev.xmlPK ,$D 0B L X* vB?Line 76"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!8-drs/downrev.xmlDO]K0} ts( 2TzmnMR0|/Viʼn|hU0$ ȖN7Rr7"Z`3z.VClPAcIʚ 2w`d+=nZy$3iPcGyMqgaL,SJ 'xoOI)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK! 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TIfM+22N6HIlOZE2ơ JTbFf:4i9R+X;̿tr9Y- b[OxC:is-cB[k\TA QeiSpW)^ !e!2r2*l+eޭnThe8tRZ9IoDzpSmD^Eaie[F*l^ jȁ#ڡm v,5ʪk((~ ;ug?d]#F1[K-ZzxZD,.v{ y=힘Dݭ`S_j[;oӪz /[mk0ǿ9ڱ:^,'j$Yr$ .58'$V$O=K('_'tP/)%}B^,xm եvV,/MEɂ[Æ. ̶a.)P̠Rhej˲v.[ykxˑ2G"]YIZr2YG̍+>뱾Y6gmKpB2sW ץMG9?P`+hZ(}¡qj,mjpoЅ 5%dQ+'4j@.x Y*:wQT9WP- +<v~=S2BlG qoPtv@:ܣsC!39KI΋&׺=|Qjw:y}_SN(5?H=1,fX\Y?n~MʰHGx[cڭY`mxs[i>hI>h8+L`U&ûN-zSG6qYSv4b-䑸Cw~}[|7)ۜLd [!>J?eٝ_+Z)'+~VD3K vNHgZpaltN3)Wxb2xPK!R<drs/downrev.xmlDMO@&͘xP+ i?oCwhuwx;y󌧽k՞Bl<dKo _\ b )trz2iI )uֱaXPi paÆeƎf5og Y==[(޷qq4/nA%zW/nq܆Ʈ1& " zPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!PSK*drs/shapexml.xmlPK-!R< drs/downrev.xmlPK Lv 8___PPT10N___PPT90( HLemma 1. A CAC of length n = 4m can contain at most one centered codeword among {0, i, 2i}, {0, 2i, 4i}, {0, n/2 " i, n " 2i} for each odd integer 1 i n/4.S0"ggggggwww2ggPPgPPgPPgPPgPPgPPgPPgPPgPPgPPgPPgPPgPPgPPgPPgPPgPPgPPg     ls    (     J9?0000 0000 18"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!,drs/downrev.xmlDOK1G!ElRAѵiBA&IlAޏ7 0)u0[ = vIÉ̦gL8܊"Ts_IjK(ą5!z匭4E;yԝ,XRV]4[:߹;_j"Ӑ4V|Uo+B#style.visibility<* %(D' =-g6B fade*<3<* D[' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D[' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 + 0N0 TT((  (2 ( j33Ԕ?Oval 80}   b$ g  2 ( X?Oval 81l6 b$ g  2 ( X?Oval 82 6  b$ g  2 ( Xd?Oval 83 -  b$ g  2 ( X?Oval 84 -  b$ g   ( Hl?Text Box 85 ~v0___PPT106___PPT9 Gc(16)H 2eme    ( H?Text Box 86W f32  2g     ( H?Text Box 87!:G e4  2g     ( H8?Text Box 88  e8  2g     ( H8?Text Box 89? 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BDVSTCg¨t___PPT10+*^pD' = @B D`' = @BA?%,( < +O%,( < +D ' =%(D' =%(D[' =4@BB BB%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =-g6B fade*<3<* 0D' =%(D' =%(DD' =ABBB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-s6Bwipe(left)*<3<*0D' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =%(D' =%(DD' =ABBB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-s6Bwipe(left)*<3<*0D' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =%(D' =%(DD' =ABBB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-s6Bwipe(left)*<3<*0D' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0++0+00 ++0+ 00 ++0+00 ++0+00 ++0+00 ++0+00 ++0+00 ++0+00 +u 0N0 WW8W(  8R 8  ~ B0e0e?Rectangle 2RL   (Classification of Non-Centered Codewords))C$&    5 8 ZԔ?Rectangle 4"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!(drs/downrev.xmlDN1E&CLIG0#u Ǵ2mǶ6.tyso͙-zӲp?ȀԶw`!:K.`1a.ٖtbĆP+2#c9MˇY jv>VF@{,)_>KT_zFx8vX]^C$/ %d |PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!(drs/downrev.xmlPK k,$ 0 O ={i | i a" 1 (mod 2)} E ={i | i a" 2 (mod 4)} D ={i | i a" 0 (mod 4)}H~ F                   H  ~ aVqtL/ 8S 00000S 25" U 8 F?Text Box 5aV1!t)8___PPT10N___PPT90(  {0, i, j}v 2ggggg   B 8 s DDENFQRUVWPPjJ?  .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 6>]v&@b + \g  B 8 s DDENFQRUVWPPjJ?  .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 7dF&i'W+ \g  B 8 s DDENFQRUVWPPjJ?  .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 8;^xhP' \g    8 F?Text Box 97])>cL/ gi" 2gPP     8 Hl?Text Box 11bq9i hj " 2gPP     8 X?"0000juS0000000 19 f13 e   "  8 # ~ijJff=i=i?҉8NV҉b_ 20#" @ PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!^/L drs/shapexml.xml]nF? -=lŎ%H 8`92#pH G6n o{g(TDLɣǐ{}||qrdƢ5=„y/ON<**cz-KϾiMRIH1h(N}XFTj$L(BoǍ{7/q?K=\߿ӧ?$azYa(+HFT2S $}0KڍG#2y 7Gq:˦v>99xć'p&2U?xY^Ł\n{]O?m)g'3梊oЃ t3.iDG˜ aGFS9?Vwu+2R;e"X-HgW=/ətD;\:hV¿U!z)u:h[Q-me|Z2h=N3OgoZ1 ˁ0"hۨCna RHZ2<7v61qq/ ,X)caJJ, { :pc˝@?ɞH%:+mjW[0gQ9]T}䑷k PUp1==2Pf6qmKXol;6o_Ѧv [ky.cƎd.WGBeZ:3O?37hb 9)V~|G*jR#Ӄ/KJS c6uhn{gWP#,XFe$myPK!e)I"drs/downrev.xmlDKK1!܂I+2jZ.;:II{ ];|`Zѳpqw"DZgaՊ2.6+ 6dPeRfCa:+7SԞ. nZ9Wj! 56=Sm˜ݩ׷ϟKěɰ}ycY;W^4|܏_9!%դ rPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!^/L *drs/shapexml.xmlPK-!e)I" drs/downrev.xmlPK  q*$D 0P___PPT100(___PPT9xp Lemma 3. For a non-centered codeword x in a CAC of length n and weight 3, let D2(x)={i, j, k}. Then, (iv) two of i, j, k are in O and one is in E, (v) two of i, j, k are in O and one is in D, (vi) two of i, j, k are in E and one is in D, (vii) i, j, k D.Phgg g ggggg PP  PP PP PP PP PPgPPgPPgPPgPPgPP g PP g PP g  g  g  g  g  g    g  g  g  g  g  g  g  g      g  g    g  g  g  g  g  g  g  g    g  g    g  g  g  g  g  g  g  g  g    g  g  g  g             "  8@ b?AutoShape 12"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Hfdrs/downrev.xmlDMO1&fLJ!1밝mie_ow3_v-P9`4L@͜lu0YRSnvgvυRTP27!+`lr r<}!sZd" V66к5 V)fϴ~0[R3o'kxAkI2+DhMA.PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Hfdrs/downrev.xmlPKz A ,$@  0 \g   " 8@ b, ?AutoShape 14"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!:Pdrs/downrev.xmlDMO1&fH BDb41@agCˮy'ϛgt-ZvFx`YL2 am +jy{݇BD)*(ChR)}VF? i 1BSZ>$DjL\(璳Vnu1׌]oi#[D.??}\ L'c_h>S/6\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!:Pdrs/downrev.xmlPK8 A ,$@  0 \g   " 8@ b)?AutoShape 16"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!o(drs/downrev.xmlDMk1E ݔт(RZn|$t7t˽˙-:]Q0$ d*S(|{ am +b~{3Ôl݅BD)*(ChR)}VF? i 1BSZd,5V&>kq,[]Lw3%zX r] "ppR>Hd<@UE)~6PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!o(drs/downrev.xmlPK A ,$@  0 \g   8 H,2?Text Box 11k)@e tk = j i or i j 2gPPgPPgPPgPPgPPgPPgPPgPPgPPgPP@        8 H?Text Box 14"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Dkdrs/downrev.xmlDN1MxfH,B $&vv۰zyLomNx$2p|{2:w$J  +/nG}nYTs8O&i5>Z%Ɩ-W4zTV{lv_Wb%VRX>?}/MIf}=Fv2E RT&PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Dkdrs/downrev.xmlPK > 8 ,$ 0 Noe: 2go    8 Hx?Text Box 16"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDj1E!<8RF)B^'o&I2&E]^\l1؎)D㝄D#W{e\+PńNaIR|t7RyZ!.(AԗZ8=5>XL9 /x{Zj+kkV[#$VRޏ`tOa$fsF0& \V͚PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!drs/downrev.xmlPK   ,$  0 Nod: 2go    8 H8?Text Box 18"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!4drs/downrev.xmlDJ1E&V26-Z H|sfIlop˽˙/Gߋk(M0; DD 9V&yG}DpPyL%iҵ!z%N*u'=:.ZYj>^kn>k7xKAdWu+Q/FD9}Dgv2E EMPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!4drs/downrev.xmlPK   ,$  0 Ne: 2go     8 H?Text Box 20"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!"drs/downrev.xmlDJ1EB!;XTشv$8I$mop˽˙/۱hp;^J}̀ńNaIPbt5BiZ!.(AZ8=5>XL9  -4ԺׯX!_ԛ0֬.`c !eլ |PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!"drs/downrev.xmlPK  ,$  0 Nd: 2go    " 8@ b?AutoShape 16"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK![?drs/downrev.xmlDMKADahA$DkgCwfvwǢWj9;$Ůq ( 1VӓݎT*A\ rcQ8ԕ>XCMcVϲZ[j\zEXO˷}nųݚ^oφ[Pƒ[6/ ̮@!4fGQ8 $4A/PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-![?drs/downrev.xmlPK  Al,$D  0 \g  H 8 0޽h ? 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