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A k-extended Langford sequence of order n and defect d is a sequence (a1, a2, & , an) of n integers such that When k = 2n+1, it is simply called a Langford sequence of order n and defect d. ( 6, 5 ) is a 2-extended Langford sequence of order 2 and defect 2.;ZCCCCCCC$ 3 333$33 333Near-Skolem sequence( Let m and n be integers with n e" m. A near-Skolem sequence of order n and defect m is a sequence (a1,& , am-1, -, am+1,& , an) of n - 1 integers such that ( 3, 6, -, 5 ) is a near-Skolem sequence of order 4 and defect 3.CCCCcc""c "c"cckcckkcckkcckc"cc"g3$&3g3 &3g3&3g3,+#  :Theorem (Baker, JCD 1995) A k-extended Skolem sequence of order n exists if and only if k is odd and n a" 0, 1 (mod 4) , or k is even and n a" 2, 3 (mod 4)."YEvA#AC C  C c " c c " c c   lTheorem (Linek and Jiang, JCT(A) 1998) A k-extended Langford sequence of order n and defect 2 exists if and only if k is odd and n a" 0, 3 (mod 4) or k is even and n a" 1, 2 (mod 4).6tCvA%A AC C  C c " c c " c c    BTheorem (Shalaby, 1992) A near-Skolem sequence of order n and defect m exists if and only if n a" 0, 1 (mod 4) and m is odd, or n a" 2, 3 (mod 4) and m is even.@^Dv "A AC C  c " c " c c c " c " , }Cyclic Steiner Triple Systems(~A Steiner triple system of order v, STS(v), is cyclic if it has an automorphism which is a permutation of a single cycle of length v, for example, (1,2,3,& ,v). Theorem (Peltesohn, Compositio Math. 1939) For all v 1, 3 (mod 6) except 9, there exists a cyclic STS(v). Difference Method plays the key role!u!3N  303b$ Z  J3A couple of examples(.Cyclic STS(19) Generated by three difference triples: {1,4,5}, {2,6,8} and {3,7,9} which can be obtained by the sequence (5,8,10). Cyclic STS(21) Generated by three difference triples (full orbits) and one difference (short orbit): {1,4,5}, {2,8,10},{3,6,9} and {7}. Use (5,10,9).\3 3UCyclic m-cycle systems(2An m-cycle system of a graph G is a decomposition of G into m-cycles. A cyclic m-cycle system of order v is an m-cycle system of Kv which admits an automorphism (permutation) of length v. Skolem type sequences can be applied to construct cyclic m-cycle systems when m is odd. (See the following reference.)z316Z> q New Outcomes (@For each m 3 there exists a cyclic m-cycle system of order v 1 (mod 2m). (Fu and Wu; Buratti and Del Fra; Bryant, Gavlas and A. Ling.) Given an odd integer m 3, there exists a cyclic m-cycle system of order 2km + m for any admissible value of k with the only definite exceptions of (m,k) = (3,1), (15,0) and (pa,0) with p a prime and a > 1. (Buratti and Del Fra; Vietri) There exist cyclic m-cycle systems with m 32 or m = 2q where q is a prime power. (Fu and Wu, JCD 2006) Difference method and Skolem type sequences are utilized!(? 23 4333R3:Y  8  Conflict-avoiding code(A conflict-avoiding code (CAC) of length n and weight k is defined as a set C "{0,1}n of binary vectors, called codewords, all of Hamming weight k such that the distance of arbitrary cyclic shifts of two distinct codewords in C is at least 2k - 2. Differences again. 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. Go for optimal CACs!ZZ3 C CCCCK<C%%C CCC.C CCCCCCCCCCCG3G3G3?3P*r=>C ?@D!E"F#G$H%I&J'K(L)M*N+O,P-<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 u  /3 C C C C G 3 C C C    C C C A> J|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 !Q.BT1U2Skolem-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}.+ 9Question (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:0Odd sequences with doubly even integers latently11$HFor 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 with doubly even integers latently, 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. w#D3 30iii iD   $   iii iiiiiiiP$-;Examples $lAn Q4: {(7,11),(5,13),(3,9),(1,15)}. A 2-ext Q5 of defect {13,19}: {(7,11),(3,9),(1,17),(5,15)}. A {2,3}-ext Q8 of defect {3,5,29,31}: {(23,27),(9,25),(1,19),(11,13),(7,21),(15,17)}."! 1<+Joint works with Yi-Hean Lin and M. Mishima,, , Lemma There exists an Qn if n 0,1 (mod 4) and n 4. Lemma There exists an {n-1,n}-ext Qn of defect D if (1) n 3 (mod 4) and D = {1,11,4n-7,4n-5}, and (2) n 0,1 (mod 4) and D = {3,11,4n-7,4n-5}. Lemma There exists a 3-ext Qn of defect D if (1) n 0,1 (mod 4) and D = {1,3}, and (2) n 2,3 (mod 4) and D = {3,5}.4/[33  73  + =8More & to the end of Summer!&(3(Finally, the upper bounds of M(n) for n 0 (mod 4) are also lower bounds except for n = 48, 64..a9A>Question remains~For even n, an optimal CAC(n) with weight 3 can be constructed. How about the case when n is odd with weight 3? In my opinion, we need more infos from cyclotomic cosets of {1,2, & , (n-1)/2}.Pq 7 We need more sequences!(Z 8 We have to stop! Thank you for attending this talk. 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_rels/.relsPK-!fcWdrs/downrev.xmlPKr R r K <B  s *o#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!rdrs/downrev.xmlDъ0E߅0 5deQ6c[l&%Agν̗ič-+% k. ˤNSm;>-&STPЦR"~d[x;[g0ѕR;좹i8I&`1–K~5 fz?$>@}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 +8o0N0 CI;IH(     X0e0e?Rectangle 15,   QExampleC(    # `+0e0e?Rectangle 16T <4___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  H,?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 STS(13))D(2gPPgPPgPPgPP gPP"g4    ,    H81?Text Box 25"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!pV8drs/downrev.xmlDOK1G!&*6-"Z&IlAޏ7{p9R `vNj vIÑg'SL8܊"Ts_IjK(ą5!z匭4E;yXzt\,`|{ /]2ع;_jp"Ӑ!?z6e|=,љ5LQC+PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!pV8drs/downrev.xmlPK v  ,$0 F*{x, y} is a CAC of length 13 and weight 3.+ 2gggggg3fgwww ggPPg +  V} h S 0Group 35#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK]@,,$D0B  s FTDENFQRUVWfjJ? .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 17h ^ g3   H^?Text Box 29 e1  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 ,$D0B  s F4aDENFQRUVWfjJ? .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 20j]  ^ g3    Hq?Text Box 30+   e3  2g3   W~ }% T  S 0Group 37#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!R&drs/downrev.xmlDOk@&B-=BUo3 f߆6}Pq0hSj ERtxZpYccL`=l1v/ꏾ.Am"+*2%v}])uCF.U9,TRVQq?~*~-KR1݀4_C+x+PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!R&drs/downrev.xmlPKSl ,$D0B  s FsDENFQRUVWfjJ? .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 22}% T ^ g3   HĄ?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,$D0B   L,DENFQRUVWfjJ? .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 19   ^ g3   H ?Text Box 33 n  e5  2g3   ~ \,OE.7 S 00000S 26#"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Gdrs/downrev.xmlDOk@&J[B-=BUo3 f߆6}Pq0hSj ERtxZpYccL`=l1v/ꏾ.Am"+*2%v}])uCF.E9,TRVQq?~*~-KR1݀4_C+x+PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Gdrs/downrev.xmlPK1 ,$D0B   LPDENFQRUVWfjJ? .`T%`T`T.`T%`T`T`T @ @ ``T`TArc 21\,ZHE.7 ^ g3 *  H?Text Box 34n.1O#7 7 6 (mod 13)J 2g33 g3     X ?"0000juS0000000 24 f27 e   P "   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 +W&0N0  =(     ~0e0e?Rectangle 24   V Optimal Code C(     # *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___PPT10P___PPT92* V<A code of maximum size (maximum number of codewords) is said to be optimal. (Maximize the number of potential users) M(n) = M(n,3): the maximum size of a CAC of length n and weight 3. <v" "  wwwwww,PPwwwGGGGGGwwwGGwwwGgGwww*   A     -     X?"0000juS0000000 13 f26 e    "   r2ff=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& $NRPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!+Y-*drs/shapexml.xmlPK-!Fdrs/downrev.xmlPK$W $D04___PPT10 B___PPT9$  We want optimal CAC of weight 3.!ggPPggPPgg     H  0޽h ? BDVSTCg¨t:2___PPT10+&vD ' = @B DA ' = @BA?%,( < +O%,( < +D' =%(D' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-g6B fade*<3<*LD' =%(D' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*Lu%(D' =-g6B fade*<3<*LuD' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*v;%(D' =-g6B fade*<3<*v;+p+0+0 ++0+0 +1m 0N0 QQDQ(  +   ~tO0e0e?Rectangle 2,'g   eDifferences from a CodewordC(  "   # Pa0e0e?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!xdrs/downrev.xmlDQO0M5M:% \A ޯeV}<9'7ugv᱗`SXUaF | a6Pl 3BBh3)}Q&߳-btT.$TPQˋݷFhe>7iwn "pLJ]/M!?q|~Zmv/FSPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!xdrs/downrev.xmlPK+ $0 ph___PPT9JB $The set of differences from a codeword x = {0, i, j} is a multiset D(x) = {i, j, j"i, n"j+i, n"j, n"i}. The halved difference set of D(x): E" Z&" Z&" ZZ" Z'GgGGGGGG GGGGgGGGGGg G $g$(G(,g,0G04g48G8I)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!btdrs/downrev.xmlDAK0F!M Z7"ay̶uII_o7\Osz@th|"1i)zu~ܸ#oPF$ 51nbX8c:}#c^(u+-vZ詥z_VCy;>6r //w|WZ2b|g6"y ).&r PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!btdrs/downrev.xmlPK mK,$@0 \g    H@?Text Box 11"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!Xçdrs/downrev.xmlDJ1E!Ģ"k"Z &v3Yn>g83 {p5R `Ow RF6& GJ0L2kor+Sl}%e-yLք13D<X[qYӣz^feV7yuwJ-BD!?_Ur Y?3kLWR 9PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Xçdrs/downrev.xmlPK ` :,$0  3 (mod 13)0 2& g     Hd|?Text Box 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!Tdrs/downrev.xmlDJ1E!M*6-E ~7mۿ7,Vʼnbr5&ǝ IÙ L>wH8UI)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!Jldrs/downrev.xmlDAK@FaBm)DPFy&iaw&zz>r=NȇֱۙA\9rD;Ǥa3μSk$2gR!az[4In;yBZl9-4SCձ0nRü|wGk}}5nADs^MjYS[kHq)5!PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Jldrs/downrev.xmlPK AK,$@0 \g  B  @  BCbDEF&QRUVW38c?Yp 2<b`Tb%b<2 ^Yp 2<b`Tb%b<2 ^`T2@    @    `bArc 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!idrs/downrev.xmlDN0HHܨuKK6&؎KXบ|bp:Q6x$E #(H ~C*I<h j!OBO>c%69]o^;l}Zh熪3PX1{ OҘa9%4|Umjyu\ P4RjB?PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!idrs/downrev.xmlPK lu,$@ 0 \g    Hɻ?Text Box 15"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!rtdrs/downrev.xmlDJ1@FfE֦Ed-fv$k;x o͗ĞRv1h)h\5ޟAi8Rl~[z+`K+)skcő.&Kr?Gő,_u6msk_Sjj//LԬOO6Ÿwgr\(i8Ner PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!rtdrs/downrev.xmlPKS i(,$ 0  4 (mod 13)0 2& g   } #--E|R  S 00000S 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!0\drs/downrev.xmlDOk@ouDĊP#Lٷ!͟o-zf76M#z\mYA@V\*?pYccL ͞h;'g_a6tKn;>Ȯ!M#WQ* *li_Q8lqO5I|6 i+nPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!0\drs/downrev.xmlPK d ,$D09A  P dIzR  C *҉8NV҉b_ 19"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! 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 $ػ"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!d~drs/downrev.xmlDj@E0<щ-jIEP5lIhMȌ}g˹]mzۈ;uv`6M@Nl~ dcRG6`v)*BhS)}YE?u-qd?b>6=Ibq–vכUQY}ˢ8s_Ro~"P^ec}%PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!d~drs/downrev.xmlPKBL ILzQ~v0___PPT106___PPT9 ,x = (1 1 0 0 1 0 0 0 0 0 0 0 0) -l( AgggPPggPPg - F  Jۻ?0000 0000 21Z -Eq!J > 0 1 2 3 4 5 6 7 8 9 10 11 124?g >g >     JH?0000 0000 22"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!E>Bdrs/downrev.xmlDN1EIfL|V"W 1Dm] m|ɝ{rmNND#xe\'~ߎbB$\(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 $D0 8___PPT10N___PPT90( >For any two distinct codewords x, y C, if D2(x) )" D2(y) = , C is a CAC. Packing {1, 2, & , n/2} with as many D2(x) as possible gives an optimal CAC.  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(u,v) E when either v = 2u if u n/4, or v = n"2u if u > n/4. (the set of all candidates for centered codewords possibly contained in a CAC.) B" ZGOGG GGGwwwGGCGGGGGGGGGGGGGGGGGG G*PP PP(www m       4  ?   )    FTJ?Text Box 8"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!*Ikdrs/downrev.xmlDQK0p_KnH]6D ksM{x86؉B4I `jk%?^nńNaIPjB;ЩJ- S cb\,CUsێυXr==ko+kJ/u{45Bz|zhLjE) Rkv`c !el |PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!*Ikdrs/downrev.xmlPK  S1 ,$ 0 2u or n"2u 2ggggggg&       HxD?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 gu" 2g     Hp(?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   ,$0 ~u 8 2ggf     XD)?"0000juS0000000 14 f18 e   W"  # ~$P 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.  $D08___PPT10N___PPT90( An edge (u,v) E corresponds to a possible centered codeword x s.t. 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For a non-centered codeword x in a CAC of length n and weight 3, let D2(x)={i, j, k}. 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BDVSTCg¨t___PPT10+Y:9D>' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* 8%(D' =-g6B fade*<3<* 8D' =%(D ' =%(DD' =ABBB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<* 8%(D' =-s6Bwipe(left)*<3<* 8DD' =ABBB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-s6Bwipe(left)*<3<*8DD' =ABBB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-s6Bwipe(left)*<3<*8DD' =ABBB@B0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-s6Bwipe(left)*<3<*8D ' =%(Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8Dh' =AB BB@B0B%()))D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8+0+0+80 ++0+ 80 ++0+ 80 ++0+80 ++0+80 ++0+80 ++0+80 ++0+80 ++0+80 ++0+80 +FZ0N0 G@?@ @?(  @; @  ~0e0e?Rectangle 200   uSpecial CodewordsC(&    & @  ~0e0e?Rectangle 31HHT <4___PPT9  C: a CAC of length n=4m !" 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