On mobile sets in the binary hypercube

Ó ÄÊ 519.72 On mobile sets in the binary h yp erub e ∗ Y u. L. V asil'ev, S. V. A vgustino vi h, D. S. Kroto v Abstrat If t w o distane- 3 o des ha v e the same neigh b orho o d, then ea h of them is alled a mobile set. In the (4 k + 3) -dimensional binary h yp erub e, there exists a mobile set of ardinalit y 2 · 6 k that annot b e split in to mobile sets of smaller ardinalities or represen ted as a natural extension of a mobile set in a h yp erub e of smaller dimension. In tro dution By E n w e denote the metri spae of all length- n binary w ords with the Hamming metri. The spae E n is alled the binary , or unary , or Bo olean h yp erub e. The basis v etor with one in the i th o ordinate and zeros in the other is denoted b y e i . A subset M of E n is alled a 1 -o de if the radius- 1 balls with en ters in M are disjoin t. The union of the radius- 1 balls with the en ters in M is alled the neighb orho o d of M and denoted b y Ω( M ) , i.e., Ω( M ) = { x ∈ E n : d ( x, M ) ≤ 1 } . If a 1 -o de M satises Ω( M ) = E n , then it is alled p erfet, or a 1 -p erfe t  o de . 1 -P er- fet o des exist only when the dimension has the form n = 2 k − 1 . F or n = 7 , su h a o de is unique (up to isometries of the spae), the linear Hamming o de. F or n = 15 , the problem of  haraterization and en umeration of the 1 -p erfet o des is not solv ed y et, in spite of the inreasing omputation abilities (onsiderable results are obtained in [10, 2℄). In this on text, it is topial to study ob jets that generalize, in dieren t senses, the onept of 1 -p erfet o de and exist in in termediate dimensions, not only of t yp e n = 2 k − 1 . Examples of su h ob jets are the p erfet olorings (in partiular, with t w o olors [1℄), the en tered funtions [8℄, and the mobile sets, disussed in this pap er. A set M ⊆ E n is alled mobile ( m.s. ) i: 1) M is a 1 -o de; 2) there exists a 1 -o de M ′ disjoin t with M and with the same neigh b orho o d, i.e., M ∩ M ′ = ∅ and Ω( M ) = Ω( M ′ ) ; su h a set M ′ will b e alled the alternative of M . In other w ords, a 1 -o de is a m.s. i it has an alternativ e. F or ev ery o dd n = 2 m +1 , w e an onstrut a linear (losed with resp et to o ordinatewise mo dulo- 2 addition) m.s. in E n : M = { ( x, x, | x | ) : x ∈ E m } . (1) ∗ The seond author is partially supp orted b y the RFBR gran t 07-01-00248 1 (Here and b elo w | x | denotes the mo dulo- 2 sum of the o ordinates of x .) Resp etiv ely , M ′ = { ( x, x, | x | ⊕ 1) : x ∈ E m } . It is not diult to  he k the onditions 1 and 2 for these M and M ′ . Our main goal is to pro v e the follo wing: Theorem . F or all n ≥ 7 ongruen t to 3 mo dulo 4 , there exists an irreduible unsplittable mobile set in E n . A nonempt y m.s. M is alled splittable ( unsplittable ), i if an (resp etiv ely , annot) b e represen ted as the union of t w o nonempt y m.s. The onept of reduibilit y , whi h will b e dened in Setion 3, reets a natural reduibilit y of mobile sets to mobile sets in the h yp erub e of the t w o-less dimension. A simple w a y to onstrut a m.s. in a h yp erub e of a o de dimension n = 2 k − 1 is the follo wing. Let C and C ′ are 1 -p erfet o des in E n . Then M = C \ C ′ is a m.s. Indeed, w e an tak e C ′ \ C as M ′ . The ardinalit y of this m.s. is C − | C ∩ C ′ | . W e study the existene of m.s. that annot b e redued to o de dimensions. In Setion 1 w e dene extended mobile sets; that onept is on v enien t for the desription of our onstrution. In Setion 2 w e desrib e a onnetion b et w een the mobile sets and the i -omp onen ts, whi h w ere studied earlier. In Setion 3 w e desrib e a onstrution of inreasing dimension for mobile sets; that onstrution leads to the natural onept of a reduible m.s. In Setion 4 w e giv e the main onstrution and pro v e Theorem. In the nal setion, w e form ulate sev eral problems. 1. Extended mobile sets Lik e as with 1 -p erfet o des, it is sometimes on v enien t to w ork with mobile sets extending them b y the all-parit y  he k to the next dimension. In some ases w e get more symmetrial ob jets, whi h simplies pro ofs and form ulations of statemen ts. And. Some statemen ts b eome more simple and in tuitiv e while b eing form ulated for the extended ase, although geometrial in terpretations of extended ob jets an seem to b e not so elegan t and natural as for the original. Reall that the extension of the set M ⊆ E n is the set M ⊆ E n +1 obtained b y the addition of the all-parit y- he k bit to all the w ords of M : M = { ( x, | x | ) : x ∈ M ) } or M = { ( x, | x | ⊕ 1) : x ∈ M ) } . Punturing the i th o ordinate for some set of w ords in E n means remo ving the i th sym b ol from all the w ords of the set (the result is in E n − 1 ). Ob viously , the extension and punturing the last o ordinate lead to the original set; so, these op erations are opp osite to ea h other, in some sene. A set M ⊆ E n is alled extende d mobile (an e.m.s. ) i it an b e obtained as the extension of some m.s. W e will use the follo wing lemma, whi h giv es alternativ e denitions of an e.m.s. As a usual m.s., an e.m.s. M an b e dened together with some other e.m.s. M ′ , whi h an 2 also b e referred as an alternative of M (usually , it is lear from the on text what w e are talking ab out, mobile sets or extended mobile sets). F or the form ulation of the lemma and further using, it is on v enien t to dene the onept of the spheri al neighb orho o d Ω ∗ ( M ) = Ω( M ) \ M , whi h, for the extended mobile sets, pla ys the role similar to the role of the usual (ball) neigh b orho o d for the m.s. In partiular, part () of Lemma 1 denes an e.m.s. and an alternativ e similarly to the ase of a m.s. Lemma 1 (alternativ e denitions of an e.m.s.) . Let M and M ′ b e disjoin t 1 -o des in E n , and let their v etors ha v e the same parit y ( either all v etors are ev en, or o dd ) . Let i ∈ { 1 , . . . , n } . The follo wing onditions are equiv alen t and imply that M ( as lik e as M ′ ) is an e.m.s. (a) The sets M i and M ′ i obtained from M and M ′ b y punturing i th o ordinate are mobile and, moreo v er, are alternativ es of ea h other. (b) The ( bipartite ) distane- 2 graph G ( M ∪ M ′ ) of the union M ∪ M ′ has the degree n/ 2 . () Ω ∗ ( M ) = Ω ∗ ( M ′ ) . Pr o of . [The translation of the pro of to app ear℄ △ T aking in to aoun t (b) and the existene of a linear m.s., w e ha v e the follo wing imp ortan t orollary . Corollary 1 . Nonempt y m.s. ( e.m.s. ) exist in E n if and only if n is o dd ( resp etiv ely , ev en ) . 2. i -Comp onen ts A m.s. M is alled an i - omp onent i Ω( M ) = Ω( M ⊕ e i ) . Consider the set M i obtained from M b y punturing the i th o ordinate. Ïîñòðîèì íà M i , ê àê íà âåðøèíàõ, ò àê íà- çûâàåìûé ãðà ìèíèìàëüíûõ ðàññòî ÿíèé, ñîåäèíèâ ðåáðîì âåðøèíû íà ðàññòî ÿíèè 2 . The pro of of the follo wing lemma is similar to Lemma 1, and w e omit it. Lemma 2 . A 1 -o de M is an i -omp onen t if and only if the graph G ( M i ) is regular of degree ( n − 1) / 2 and bipartite. So, Lemmas 1 and 2 establish a orresp ondene b et w een pairs of alternativ e m.s. in E n − 1 and i -omp onen ts in E n +1 (for xed i , sa y , i = n + 1 ). This orresp ondene is eviden t as b oth ob jets orresp ond to a set in E n whose distane- 2 graph is bipartite and has the degree n/ 2 . In the rst ase, all the v erties of this set ha v e the same parit y . In the seond ase, this is not not neessary , but the subsets of dieren t parit y will orresp ond to a partition of the i -omp onen t in to indep enden t i -omp onen ts,  i -ev en and  i -o dd. F ormally , w e an form ulate the follo wing. Corollary 2 . Sets M , M ′ ⊆ E n − 1 are a m.s. and an alternativ e if and only if the set { ( x, | x | , 0 ) : x ∈ M } ∪ { ( x, | x | , 1) : x ∈ M ′ } 3 is an i -omp onen t with i = n + 1 . Corollary 3 . A set M ⊆ E n +1 is an i -omp onen t with i = n + 1 if and only if the sets M b a = { x : ( x, | x | ⊕ a, b ) ∈ M } , a, b ∈ { 0 , 1 } are m.s., where M 0 a and M 1 a are alternativ es to ea h other (the sets M 0 0 and M 1 0 orresp ond to the  i -ev en part of the i -omp onen t; M 0 1 and M 1 1 , to the  i -o dd; ea h of these parts an b e empt y; and if b oth are nonempt y , then the i -omp onen t is splittable). An example of i -omp onen t is the linear m.s. (1), i = n . F ormerly [4, 5 ℄ man y examples of nonlinear i -omp onen ts w ere onstruted. Ea h of them is em b eddable to a 1 - p erfet o des and has the ardinalit y , divisible b y the ardinalit y of the linear omp onen t. Moreo v er, it w as only pro v ed that these i -omp onen ts annot b e split in to smaller i - omp onen ts. Their splittabilit y on to mobile sets are still questionable. So, in spite of the fat that the resear hes are dev oted to ommon problems and a ommon approa h, the lines are sligh tly dieren t and the results do not o v erlap but omplemen t ea h other: w e giv e the em b eddabilit y to 1 -p erfet o des up (whi h is a w eak ening) but deal with a stronger splittabilit y and a wider sp eter of dimensions. 3. Reduibilit y Lemma 3 ( on the linear extension of a m.s. ) . Let M ⊆ E n b e an e.m.s. and let M ′ ⊆ E n b e an alternativ e of M . Then the set R = { ( x, 0 , 0 ) : x ∈ M } ∪ { ( x, 1 , 1) : x ∈ M ′ } (2) is an e.m.s. with an alternativ e R ′ = { ( x, 1 , 1) : x ∈ M } ∪ { ( x, 0 , 0 ) : x ∈ M ′ } . Pr o of . Condition (b) of Lemma 1 for M and M ′ implies the v alidit y of this ondition for R and R ′ . △ . An e.m.s. R ∈ E n is alled r e duible i it an b e obtained b y the onstrution (2 ) and applying some isometry of the spae (i.e., a o ordinate p erm utation and the in v ersion in some o ordinates). A m.s. is alled r e duible i the orresp onding e.m.s. is reduible. So, the existene of reduible m.s. is redued to the existene of m.s. in smaller dimensions. F rom this p oin t of view, the form ulation of the main theorem is natural. Remark . As w e an see from Corollary 3 , an y i -omp onen t is either reduible m.s. or an b e split in to t w o i -omp onen ts ( i -ev en and  i -o dd), whi h are reduible m.s. In partiular, the linear m.s. (1 ) is reduible. Moreo v er, the linear e.m.s., up to a o ordinate p erm utation, an b e obtained from the trivial e.m.s. { 00 } in E 2 b y sequen tial applying the onstrution from Lemma 3 . 4. Pro of of Theorem 4 Let us x n divisible b y 4 : n = 4 k . P artition the o ordinate n um b ers in to k groups with 4 n um b ers in ea h group; rename the orresp onding orts as follo ws: e 1 0 , e 1 1 , e 1 2 , e 1 3 , e 2 0 , . . . , e k 3 . In ea h quadruple of t yp e { e i 0 , e i 1 , e i 2 , e i 3 } w e  hose arbitrarily (there exist 6 p o ibilities) a pair of dieren t orts e i j and e i t ; b y the index of the pair w e shell mean the n um b er p = j ⋆ t − 1 , where ⋆ is dened b y the v alue table ⋆ 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 (note that j ⋆ t = t ⋆ j and a ⋆ b = c ⋆ d for an y distint a , b , c , d ). Summarizing the  hosen pairs for all i = 1 , 2 , . . . , k , w e get a v etor of w eigh t 2 k , whi h will b e alled standar d . T otally , there exist 6 k standard v etors. By the index I ( v ) of a standard v etor v w e shell mean the mo dulo- 3 sum of the indexes of all the pairs of orts that onstitute v . Let us partition the set of standard v etors in to disjoin t subsets S 0 , S 1 , and S 2 in ompliane with the indexes of the v etors. Claim 1. L et i 6 = j , i, j ∈ { 0 , 1 , 2 } . Then the distan e-two gr aph G ( S i ∪ S j ) indu e d by the set S i ∪ S j is bip artite and r e gular of de gr e e 2 k . [The translation of the pro of to app ear℄ Claim 1 is pro v ed. So, S 0 (for example) is an e.m.s. of ardinalit y 2 · 6 k − 1 . Claim 2. The e.m.s. S 0 is unsplittable. [The translation of the pro of to app ear℄ Claim 2 is pro v ed. Claim 3. The e.m.s. S 0 is irr e duible. [The translation of the pro of to app ear℄ Claim 3 is pro v ed. The theorem is pro v ed. 5. Conlusion W e ha v e onstruted an innite lass of unsplittable irreduible m.s. Our onstrution generalizes the example men tioned in [ 7℄. In onlusion, w e form ulate sev eral problems, whi h are naturally onneted with the study of mobile sets and with the problem of  haraterization of their v ariet y . F or onstruting m.s., one an apply the generalized onatenation priniple whi h w orks for 1 -p erfet o des [ 9℄. In partiular, the onstrution from Setion 4 an b e treated in su h terms. Unsplittable m.s. onstruted in su h the w a y will ha v e non-full rank, i.e., for all the w ords of the set the o ordinates will satisfy some linear equation. Problem 1 . Construt an innite family of full-rank unsplittable m.s. Example . Consider the four w ords ( 100 110 010 ) , ( 011 110 000 ) , ( 101 001 011 ) , ( 001 100 111 ) , 5 in E 9 , listed, for on v eniene, as 3 × 3 arra ys, and all the w ords obtained from them b y yli p erm utations of ro ws and/or olumns of the arra y . W e get full-rank unsplittable m.s. of ardinalit y 36 . An alternativ e an b e obtained b y the in v ersion of all the w ords. Problem 2 . Construt a ri h lass of transitiv e unsplittable m.s., e.m.s. A set M ⊆ E n is alled transitiv e i the stabilizer Stab I ( M ) of M in the group I of isometries of the h yp erub e ats transitiv ely on the elemen ts of M ; i.e., for ev ery x, y from M there exists an isometry σ ∈ Stab I ( M ) su h that σ ( x ) = y . F or example, it is not diult to see that the m.s. onstruted in the urren t pap er are transitiv e. There are sev eral onstrutions of transitiv e 1 -p erfet and extended 1 -p erfet o des, see [ 6, 3℄ for the last results. Problem 3 . Study the em b eddabilit y of m.s. in to 1 -p erfet o des: the existene of nonem b eddable m.s. in the o de dimensions n = 2 k − 1 ; the existene of m.s. that annot b e em b edded with help of the linear extension (Lemma 3) in to a 1 -p erfet o de in a larger dimension. In partiular, for m.s. onstruted in Setion 4, the em b edding questions are op en pro vided n ≥ 11 . Problem 4 . Estimate the maximal ardinalit y of an unsplittable m.s. Problem 5 . Estimate the minimal ardinalit y of a nonlinear m.s. (the onstrution of Setion 4 together with Lemma 3 giv e the upp er b ound 1 , 5 L ( n ) , where L ( n ) = 2 ( n − 1) / 2 is the ardinalit y of the linear m.s.), of an irreduible unsplittable m.s. (the onstrution giv es the upp er b ound 1 , 5 ( n − 3) / 4 L ( n ) ), unsplittable m.s. of full rank. Referenes [1℄ D. G. F on-Der-Flaass. P erfet 2-olorings of a h yp erub e. Sib. Math. J. , 48(4):740 745, 2007. DOI: 10.1007/s11202-007-0075-4 translated from Sib. Mat. Zh. 48(4) (2007), 923-930. [2℄ S. A. Malyugin. On en umeration of nonequiv alen t p erfet binary o des of length 15 and rank 15. J. Appl. Ind. Math. , 1(1):7789, 2007. DOI: 10.1134/S1990478907010085 translated from Diskretn. Anal. Issled. Op er., Ser. 1 13(1) (2006), 77-98. [3℄ V. N. P otap o v. A lo w er b ound for the n um b er of transitiv e p erfet o des. J. Appl. Ind. Math. , 1(3):373379, 2007. DOI: 10.1134/S199047890703012X translated from Diskretn. Anal. Issled. Op er., Ser. 1, 13(4):49-59, 2006. [4℄ F. I. Solo v'ev a. On the fatorization of o de-generating d.n.f. In Meto dy diskr etno go analiza v issle dovanii funktsionalnyh sistem , v olume 47, pages 6688. Institute of Mathematis of SB AS USSR, No v osibirsk, 1988. In Russian. [5℄ F. I. Solo v'ev a. Struture of i -omp onen ts of p erfet binary o des. Disr ete Appl. Math. , 111(1-2):189197, 2001. DOI: 10.1016/S0166-218X(00)00352-8 . 6 [6℄ F. I. Solo v'ev a. On the onstrution of transitiv e o des. Pr obl. Inf. T r ansm. , 41(3):204211, 2005. DOI: 10.1007/s11122-005-0025-3 translated from Probl. P ereda hi Inf., 41(3):23-31, 2005. [7℄ Y u. L. V asil'ev and F. I. Solo v'ev a. Co degenerating fatorization on n -dimensional unite ub e and p erfet binary o des. Pr obl. Inf. T r ansm. , 33(1):6474, 1997. T ranslated from Probl. P ereda hi Inf. 33(1) (1997), 64-74. [8℄ A. Y u. V asil'ev a. A represen tation of p erfet binary o des. In Pr o . Seventh Int. W orkshop on A lgebr ai and Combinatorial Co ding The ory , pages 311315, Bansk o, Bulgaria, June 2000. [9℄ V. A. Zino viev and A. Lobstein. On generalized onatenated onstrutions of p erfet binary nonlinear o des. Pr obl. Inf. T r ansm. , 36(4):336348, 2000. [10℄ V. A. Zino viev and D. V. Zino viev. Binary extended p erfet o des of length 16 and rank 14. Pr obl. Inf. T r ansm. , 42(2):123138, 2006. DOI: 10.1134/S0032946006020062 translated from Probl. P ereda hi Inf. 42(2) (2006), 63-80. 7 Ó ÄÊ 519.72 Î ïî äâèæíûõ ìíî æ åñòâàõ â äâîè÷íîì ãèïåðêóáå ∗ Þ. Ë. Âàñèëüåâ, Ñ. Â. Àâãó ñòèíîâè÷, Ä. Ñ. Êðîòîâ Àííîò àöèÿ Åñëè äâà ê î äà ñ ðàññòî ÿíèåì òðè èìåþò î äèíàê îâóþ îêðåñòíîñòü, ê àæäûé èç íèõ íàçûâàåòñ ÿ ïî äâèæíûì ìíî æ åñòâîì.  äâîè÷íîì (4 k + 3) -ìåðíîì ãèïåðêóáå ñóùåñòâó åò ïî äâèæíîå ìíî æ åñòâî ìîùíîñòè 2 · 6 k , ê îòîðîå íåëüçÿ ðàçáèòü íà ïî- äâèæíûå ìíî æ åñòâà ìåíüøåé ìîùíîñòè èëè ïðåäñò àâèòü â âèäå åñòåñòâåííîãî ðàñøèðåíèÿ ïî äâèæíîãî ìíî æ åñòâà â ãèïåðêóáå ìåíüøåé ðàçìåðíîñòè. If t w o distane- 3 o des ha v e the same neigh b orho o d, then ea h of them is alled a mobile set. In the (4 k + 3) -dimensional binary h yp erub e, there exists a mobile set of ardinalit y 2 · 6 k that annot b e split in to mobile sets of smaller ardinalities or represen ted as a natural extension of a mobile set in a h yp erub e of smaller dimension. Ââåäåíèå ×åðåç E n îáîçíà ÷àåòñ ÿ ìåòðè÷åñê îå ïðîñòðàíñòâî íà ìíî æ åñòâå âñåõ äâîè÷íûõ ñëîâ äëèíû n ñ ìåòðèê îé Õåììèíã à. Ïðîñòðàíñòâî E n èíîã äà íàçûâàþò äâîè÷íûì, èëè åäèíè÷íûì, èëè áó ëåâûì êóáîì. Áàçèñíûé âåêòîð ñ åäèíèöåé â i -é ê îîð äèíàòå è íó ëÿìè â îñò àëüíûõ îáîçíà ÷àåòñ ÿ ÷åðåç e i . Ïî äìíî æ åñòâî M ⊆ E n áó äåì íàçû- âàòü 1 -ê î äîì, åñëè øàðû ðàäèó ñà 1 ñ öåíòðàìè èç M íå ïåðåñåê àþòñ ÿ ìåæäó ñîáîé. Îêðåñòíîñòüþ Ω( M ) ìíî æ åñòâà M íàçîâ¼ì îáúåäèíåíèå øàðîâ ðàäèó ñà 1 ñ öåíòðàìè èç M , ò . å. Ω( M ) = { x ∈ E n : d ( x, M ) ≤ 1 } . Åñëè 1 -ê î ä M îáëàäàåò ñâîéñòâîì Ω( M ) = E n , îí íàçûâàåòñ ÿ ñîâåðøåííûì, èëè 1 -ñîâåðøåííûì êîäî ì . 1 -Ñîâåðøåííûå ê î äû ñóùåñòâóþò ëèøü â ðàçìåðíîñò ÿõ âèäà n = 2 k − 1 . Äëÿ n = 7 ò àê îé ê î ä åäèíñòâåííûé (ñ òî÷íîñòüþ äî èçîìåòðèé ïðîñòðàí- ñòâà)  ëèíåéíûé ê î ä Õåììèíã à. Ïðè n = 1 5 ïðîáëåìà îïèñàíèÿ è ïåðå÷èñëåíèÿ 1 -ñîâåðøåííûõ ê î äîâ äî ñèõ ïîð íå ðåøåíà, íåñìîòð ÿ íà ïîñòî ÿííî ðàñòóùèå âîç- ìî æíîñòè âû÷èñëèòåëüíîé òåõíèêè (ñóùåñòâåííîå ïðî äâèæ åíèå ïîëó÷åíî â ðàáîò àõ [2, 8℄).  ê îíòåê ñòå óïîìÿíóòîé ïðîáëåìû ïðåäñò àâëÿåòñ ÿ àêòó àëüíûì èçó÷åíèå îáú- åêòîâ, îáîáùàþùèõ â ðàçíûõ ñìûñëàõ ïîíÿòèå 1 -ñîâåðøåííîãî ê î äà, è ñóùåñòâóþ- ùèõ íå òîëüê î ïðè n = 2 k − 1 , íî è â ïðîìåæóòî÷íûõ ðàçìåðíîñò ÿõ. Ò àêèìè îáúåê- ò àìè ÿâëÿþòñ ÿ ñîâåðøåííûå ðàñêðàñêè (â ÷àñòíîñòè, äâóõöâåòíûå [4 ℄), öåíòðèðîâàí- íûå óíêöèè [10℄, à ò àêæ å ïî äâèæíûå ìíî æ åñòâà, î ê îòîðûõ ïîéä¼ò ðå÷ü â äàííîé ñò àòüå. ∗ Èññëåäîâàíèå âòîðîãî àâòîðà âûïîëíåíî ïðè èíàíñîâîé ïî ääåð æê å  îññèéñê îãî îíäà óí- äàìåíò àëüíûõ èññëåäîâàíèé (ïðîåêò 07-01-00248) 1 Ìíî æ åñòâî M ⊆ E n íàçûâàåòñ ÿ ïîäâèæíûì ( ï. ì. ), åñëè: 1) M ÿâëÿåòñ ÿ 1 -ê î äîì, 2) ñóùåñòâó åò íåïåðåñåê àþùèéñ ÿ ñ M 1 -ê î ä M ′ ñ òîé æ å îêðåñòíîñòüþ, ò . å. M ∩ M ′ = ∅ è Ω( M ) = Ω( M ′ ) ; ò àê îå ìíî æ åñòâî M ′ áó äåì íàçûâàòü àëüòåðíàòèâîé ìíî æ åñòâà M . Äðóãèìè ñëîâàìè, 1 -ê î ä åñòü ï. ì., åñëè ó íåãî åñòü àëü òåðíàòèâà. Äëÿ âñ ÿê îãî íå÷¼òíîãî n = 2 m + 1 íåñëî æíî ïîñòðîèòü ëèíåéíîå (çàìêíóòîå îòíîñèòåëüíî ïîê îîð äèíàòíîãî ñëî æ åíèÿ ïî ìî äó ëþ 2 ) ïî äâèæíîå ìíî æ åñòâî â E n : M = { ( x, x, | x | ) : x ∈ E m } . (1) Çäåñü è äàëåå | x | åñòü ñóììà ê îîð äèíàò âåêòîðà x ïî ìî äó ëþ 2 . Ñîîòâåòñòâåííî M ′ = { ( x, x, | x | ⊕ 1) : x ∈ E m } . Óáåäèòüñ ÿ â âûïîëíåíèè ó ñëîâèé 1 è 2 äëÿ M è M ′ âïîëíå íåòðó äíî. Îñíîâíîé öåëüþ íàøåé ðàáîòû ÿâëÿåòñ ÿ äîê àçàòåëüñòâî ñëåäóþùåãî àêò à: Ò åîðåìà . Äëÿ âñåõ n ≥ 7 , ñðàâíèìûõ ñ 3 ïî ìî äó ëþ 4 , â E n ñóùåñòâó åò íåðåäó- öèðó åìîå íåäåëèìîå ïî äâèæíîå ìíî æ åñòâî. Íåïó ñòîå ï. ì. M íàçûâàåòñ ÿ ð àçäå ëèìûì ( íåäå ëèìûì ), åñëè åãî ìî æíî (ñîîò- âåòñòâåííî, íåëüçÿ) ïðåäñò àâèòü â âèäå îáúåäèíåíèÿ äâóõ íåïó ñòûõ ï. ì. Ïîíÿòèå ðåäóöèðó åìîñòè, ê îòîðîå áó äåò ñîðìó ëèðîâàíî â ðàçäåëå 3 , îòðàæ àåò åñòåñòâåí- íóþ ñâî äèìîñòü ïî äâèæíûõ ìíî æ åñòâ ê ïî äâèæíûì ìíî æ åñòâàì â ðàçìåðíîñòè íà 2 ìåíüøåé. Ïðîñòîé ñïîñîá ïîñòðîåíèÿ ï. ì. â ãèïåðêóáàõ ê î äîâîé ðàçìåðíîñòè n = 2 k − 1 çàêëþ÷àåòñ ÿ â ñëåäóþùåì. Ïó ñòü C è C ′ ÿâëÿþòñ ÿ 1 -ñîâåðøåííûìè ê î äàìè â E n . Ò îã äà M = C \ C ′ åñòü ï. ì. Äåéñòâèòåëüíî, â ê à ÷åñòâå M ′ ìî æíî âçÿòü C ′ \ C . Ìîùíîñòü ò àê îãî ï. ì. ðàâíà C − | C ∩ C ′ | . Ìû æ å èññëåäó åì âîïðîñ ñóùåñòâîâàíèÿ ï. ì., ê îòîðûå íå ñâî äÿòñ ÿ ê ê î äîâûì ðàçìåðíîñò ÿì.  ðàçäåëå 1 ìû îïðåäåëÿåì ðàñøèðåííûå ïî äâèæíûå ìíî æ åñòâà, â òåðìèíàõ ê îòî- ðûõ ó äîáíî îïèñûâàòü ê îíñòðóêöèþ.  ðàçäåëå 2 îïèñàíà ñâÿçü ïî äâèæíûõ ìíî æ åñòâ è i -ê îìïîíåíò , ê îòîðûå àêòèâíî èçó÷àëèñü ðàíåå.  ðàçäåëå 3 îïèñûâàåòñ ÿ ê îíñòðóê- öèÿ óâåëè÷åíèÿ ðàçìåðíîñòè ïî äâèæíîãî ìíî æ åñòâà, ïðèâî äÿùÿÿ ê åñòåñòâåííîìó ïîíÿòèþ ðåäóöèðó åìîãî ï. ì.  ðàçäåëå 4 ïðèâî äèòñ ÿ îñíîâíàÿ ê îíñòðóêöèÿ è äîê à- çàòåëüñòâî òåîðåìû.  çàêëþ÷èòåëüíîì ðàçäåëå ìû îðìó ëèðó åì íåñê îëüê î çàäà ÷. 1.  àñøèðåííûå ïî äâèæíûå ìíî æ åñòâà Ñ ïî äâèæíûìè ìíî æ åñòâàìè, ê àê è ñ 1 -ñîâåðøåííûìè ê î äàìè, ÷àñòî áûâàåò ó äîá- íî ðàáîò àòü, ðàñøèðèâ èõ â ñëåäóþùóþ ðàçìåðíîñòü ïðîâåðê îé íà ÷¼òíîñòü. Ïðè ýòîì â íåê îòîðûõ ñëó÷àÿõ ïîëó÷àþòñ ÿ áîëåå ñèììåòðè÷íûå îáúåêòû, ÷òî óïðîùàåò äîê àçàòåëüñòâà è îðìó ëèðîâêè óòâåð æäåíèé. È, õ îò ÿ ãåîìåòðè÷åñê àÿ èíòåðïðåò à- öèÿ ðàñøèðåííûõ îáúåêòîâ ìî æ åò ê àçàòüñ ÿ íå ñòîëü èçÿùíîé è åñòåñòâåííîé, ê àê 2 â îðèãèíàëå, è ïåðåõ î ä ñ íåé òðåáó åò íåê îòîðîãî ïðèâûê àíèÿ, ìíîãèå óòâåð æäåíèÿ ñò àíîâÿòñ ÿ áîëåå ïðîñòûìè è íàã ëÿäíûìè, áó äó÷è ñîðìó ëèðîâàííûìè äëÿ ðàñøè- ðåííîãî ñëó÷àÿ. Íàïîìíèì, ÷òî ð àñøèðåíèå ì ìíî æ åñòâà M ⊆ E n íàçûâàåòñ ÿ ìíî æ åñòâî M ⊆ E n +1 , ïîëó÷åííîå äîáàâëåíèåì ïðîâåðêè íà ÷¼òíîñòü (íå÷¼òíîñòü) ê î âñåì ñëîâàì ìíî æ åñòâà M : M = { ( x, | x | ) : x ∈ M ) } èëè M = { ( x, | x | ⊕ 1 ) : x ∈ M ) } . Âûêàëûâàíèå i -é ê îîð äèíàòû â íåê îòîðîì ìíî æ åñòâå ñëîâ èç E n îçíà ÷àåò ó äàëåíèå i -ãî ñèìâîëà âî âñåõ ñëîâàõ ìíî æ åñòâà (ðåçó ëü ò àò áó äåò ëåæ àòü â E n − 1 ). Î÷åâèäíî, ÷òî ðàñøèðåíèå è çàòåì âûê àëûâàíèå ïîñëåäíåé ê îîð äèíàòû ïðèâî äèò ê èñ õ î äíîìó ìíî æ åñòâó , òî åñòü ýòè îïåðàöèè â îïðåäåë¼ííîì ñìûñëå îáðàòíûå äðóã äðóãó . Ìíî æ åñòâî M ⊆ E n íàçîâ¼ì ð àñøèðåííûì ïîäâèæíûì ( ð. ï. ì. ), åñëè îíî ïîëó- ÷àåòñ ÿ ðàñøèðåíèåì íåê îòîðîãî ï. ì. Íàì áó äåò ïîëåçíîé ñëåäóþùàÿ ëåììà, ê îòîðàÿ äà¼ò àëü òåðíàòèâíûå îïðåäåëåíèÿ ð. ï. ì. Êàê è îáû÷íîå ï. ì., ð. ï. ì. M ìî æíî îïðåäåëèòü â ïàðå ñ äðóãèì ð. ï. ì. M ′ , ê îòîðîå ò àêæ å åñòåñòâåííî íàçûâàòü àëüòåðíàòèâîé ð. ï. ì. M (èç ê îíòåê ñò à îáû÷- íî ÿñíî, ðå÷ü èä¼ò î ïî äâèæíûõ ìíî æ åñòâàõ èëè ðàñøèðåííûõ ïî äâèæíûõ ìíî æ å- ñòâàõ). Äëÿ îðìó ëèðîâêè ëåììû è äàëüíåéøåãî èñïîëüçîâàíèÿ ó äîáíî îïðåäåëèòü ïîíÿòèå ñ åðè÷åñêîé îêðåñòíîñòè Ω ∗ ( M ) = Ω( M ) \ M , ê îòîðîå äëÿ ðàñøèðåííûõ ïî äâèæíûõ ìíî æ åñòâ âûïîëíÿåò ðîëü, àíàëîãè÷íóþ ðîëè îáû÷íîé (øàðîâîé) îêðåñòíîñòè äëÿ ï. ì.  ÷àñòíîñòè, ó ñëîâèå () ëåììû 1 îïðå- äåëÿåò ð. ï. ì. è åãî àëü òåðíàòèâó àíàëîãè÷íî ñëó÷àþ ñ ïî äâèæíûì ìíî æ åñòâîì. Ëåììà 1 (îá àëü òåðíàòèâíûõ îïðåäåëåíèÿõ ð. ï. ì.) . Ïó ñòü M è M ′ åñòü íåïå- ðåñåê àþùèåñ ÿ 1 -ê î äû â E n , âåêòîðû ê îòîðûõ èìåþò î äèíàê îâóþ ÷¼òíîñòü ( ëèáî âñå ÷¼òíîâåñîâûå, ëèáî íå÷¼òíîâåñîâûå ) . Ïó ñòü i ∈ { 1 , . . . , n } . Ñëåäóþùèå ó ñëîâèÿ ýêâè- âàëåíòíû è âëåêóò òîò àêò , ÷òî M ( ê àê è M ′ ) åñòü ð. ï. ì. (a) Ìíî æ åñòâà M i è M ′ i , ïîëó÷åííûå èç M è M ′ âûê àëûâàíèåì i -é ê îîð äèíàòû, ïî äâèæíûå è ÿâëÿþòñ ÿ àëü òåðíàòèâîé äðóã äðóã à. (b) ðà ( äâó äîëüíûé ) ðàññòî ÿíèé 2 G ( M ∪ M ′ ) îáúåäèíåíèÿ M ∪ M ′ èìååò ñòåïåíü n/ 2 . () Ω ∗ ( M ) = Ω ∗ ( M ′ ) . Äîêàçàòå ëüñòâî . Ïðè i = n èç (a) ñëåäó åò , ÷òî M åñòü ð. ï. ì., ïî îïðåäåëåíèþ. Ïîñê îëüêó ó ñëîâèÿ (b) è () íå çàâèñ ÿò îò âûáîðà i , äîñò àòî÷íî ïîê àçàòü ýêâèâàëåíò- íîñòü (a), (b) è (). () ⇒ (b).  àññìîòðèì âåêòîð v èç M .  àññìîòðèì ìíî æ åñòâî ïàð i, j ∈ { 1 , . . . , n } ò àêèõ, ÷òî v ⊕ e k ⊕ e j ∈ M ′ . (2) 3 Ïîñê îëüêó M è M ′ íå ïåðåñåê àþòñ ÿ, k è j â ò àê îé ïàðå âñåã äà ðàçëè÷íû. Ïîñê îëüêó M ′ åñòü 1 -ê î ä, äâå ðàçëè÷íûå ïàðû íå ïåðåñåê àþòñ ÿ. È èç ó ñëîâèÿ Ω ∗ ( M ) = Ω ∗ ( M ′ ) ñëåäó åò , ÷òî äëÿ ëþáîé ýëåìåíò èç { 1 , . . . , n } ïðèíàäëåæèò íåê îòîðîé ïàðå. Ò àêèì îáðàçîì, ìû èìååì ðàçáèåíèå { 1 , . . . , n } íà ïàðû k , j , ó äîâëåòâîð ÿþùèå (2). Îòñþ äà ñëåäó åò , ÷òî ñòåïåíü âåðøèíû v â ãðàå G ( M ∪ M ′ ) ðàâíà n/ 2 . Ò î æ å âåðíî äëÿ ëþáîãî v ′ èç M ′ . (a) ⇒ ().  àññìîòðèì âåêòîð w íà ðàññòî ÿíèè 1 îò M . Íàì íóæíî ïîê àçàòü, ÷òî îí ðàñïîëî æ åí íà ðàññòî ÿíèè 1 îò M ′ . Äåéñòâèòåëüíî, â ïðîòèâíîì ñëó÷àå ðàññòî- ÿíèå îò w äî M ′ ê àê ìèíèìóì 3 (ó÷èòûâàÿ î äèíàê îâóþ ÷¼òíîñòü M è M ′ ), è ïî- ñëå âûê àëûâàíèÿ i -é ê îîð äèíàòû îí íå ïîïàä¼ò â Ω( M ′ i ) , ÷òî ïðîòèâîðå÷èò ó ñëîâèþ Ω( M i ) = Ω( M ′ i ) . Ò àêèì îáðàçîì, Ω ∗ ( M ) ⊆ Ω ∗ ( M ′ ) ; àíàëîãè÷íî, Ω ∗ ( M ′ ) ⊆ Ω ∗ ( M ) \ M . (b) ⇒ (a).  àññìîòðèì âåêòîð v èç M . Ïîñê îëüêó ñòåïåíü v â G ( M ∪ M ′ ) ðàâíà n/ 2 è â M ′ íåò äâóõ âåêòîðîâ íà ðàññòî ÿíèè 2 , âñå ê îîð äèíàòû äåëÿòñ ÿ íà ïàðû k , j , ó äîâëåòâîð ÿþùèå 2. Îòñþ äà, ëþáîé âåêòîð âèäà v + e j , 1 ≤ j ≤ n , ëåæèò â Ω( M ′ ) , à ïîñëå âûê àëûâàíèÿ i -é ê îîð äèíàòû  â Ω( M ′ i ) . Íî âñå ò àêèå âåêòîðà ïîñëå âûê àëûâàíèÿ ñîñò àâëÿþò Ω( M i ) , îòêó äà èìååì Ω( M i ) ⊆ Ω( M ′ i ) . Àíàëîãè÷íî, Ω( M ′ i ) ⊆ Ω( M i ) . Îñò àëîñü çàìåòèòü, ÷òî M i ∩ M ′ i = ∅ , ïîñê îëüêó M è M ′ íåïåðåñåê àþòñ ÿ è èìåþò î äíó ÷¼òíîñòü. △ Ó÷èòûâàÿ ó ñëîâèå (b) è ñóùåñòâîâàíèå ëèíåéíîãî ï. ì., èìååì ñëåäóþùåå âàæíîå ñëåäñòâèå. Ñëåäñòâèå 1 . Íåîá õ î äèìûì è äîñò àòî÷íûì ó ñëîâèåì ñóùåñòâîâàíèÿ íåïó ñòûõ ï. ì. ( ð. ï. ì. ) â E n ÿâëÿåòñ ÿ íå÷¼òíîñòü ( ñîîòâåòñòâåííî, ÷¼òíîñòü ) n . 2. i -Êîìïîíåíòû Ñî äåð æ àíèå äàííîãî ðàçäåëà íå èñïîëüçó åòñ ÿ ïðè äîê àçàòåëüñòâå îñíîâíîãî ðå- çó ëü ò àò à. Îäíàê î, îíî íåîá õ î äèìî äëÿ ïîíèìàíèÿ ñâÿçåé ñ ïðåäøåñòâóþùèìè èññëå- äîâàíèÿìè, îðèåíòèðîâàííûìè íà ÷àñòíûé ñëó÷àé ï. ì., ò àê íàçûâàåìûå i -ê îìïîíåíòû. Ï. ì. M áó äåì íàçûâàòü i -êî ìïîíåíòîé , åñëè Ω( M ) = Ω( M ⊕ e i ) .  àññìîòðèì ìíî æ åñòâî M i , ïîëó÷åííîå èç M âûê àëûâàíèåì i -é ê îîð äèíàòû. Ïîñòðîèì íà M i , ê àê íà âåðøèíàõ, ò àê íàçûâàåìûé ãðà ìèíèìàëüíûõ ðàññòî ÿíèé, ñîåäèíèâ ðåáðîì âåðøèíû íà ðàññòî ÿíèè 2 . Äîê àçàòåëüñòâî ñëåäóþùåé ëåììû àíàëîãè÷íî ëåììå 1, è ìû îïó ñê àåì åãî. Ëåììà 2 . 1 -Êî ä M ÿâëÿåòñ ÿ i -ê îìïîíåíòîé òîã äà è òîëüê î òîã äà, ê îã äà ãðà G ( M i ) ÿâëÿåòñ ÿ î äíîðî äíûì ñòåïåíè ( n − 1) / 2 è äâó äîëüíûì. Ò àêèì îáðàçîì, ëåììû 1 è 2 ó ñò àíàâëèâàþò ñîîòâåòñòâèå ìåæäó ïàðàìè àëü òåð- íàòèâíûõ ï. ì. â E n − 1 è i -ê îìïîíåíò àìè â E n +1 (ïðè èê ñèðîâàííîì i , íàïðèìåð, n + 1 ). Ýòî ñâÿçü ïðî ÿâëÿåòñ ÿ â òîì, ÷òî îáîèì îáúåêò àì ñîîòâåòñòâó åò ìíî æ åñòâî â E n , ãðà ðàññòî ÿíèé äâà ê îòîðîãî ÿâëÿåòñ ÿ äâó äîëüíûì è èìååò ñòåïåíü n/ 2 .  ïåð- âîì ñëó÷àå âñå âåðøèíû ìíî æ åñòâà áó äóò èìåòü î äèíàê îâóþ ÷¼òíîñòü. Âî âòîðîì  íå îá ÿçàòåëüíî, î äíàê î ìíî æ åñòâà ðàçíîé ÷¼òíîñòè áó äóò ñîîòâåòñòâîâàòü ðàçáèåíèþ 4 i -ê îìïîíåíòû íà íåçàâèñèìûå i -ê îìïîíåíòû,  i -÷¼òíóþ è  i -íå÷¼òíóþ. Ôîðìàëüíî, ìû ìî æ åì ñîðìó ëèðîâàòü ñëåäóþùåå. Ñëåäñòâèå 2 . Ìíî æ åñòâà M , M ′ ⊆ E n − 1 åñòü ï. ì. è åãî àëü òåðíàòèâà åñëè è òîëüê î åñëè ìíî æ åñòâî { ( x, | x | , 0 ) : x ∈ M } ∪ { ( x, | x | , 1) : x ∈ M ′ } åñòü i -ê îìïîíåíò à ïðè i = n + 1 . Ñëåäñòâèå 3 . Ìíî æ åñòâî M ⊆ E n +1 åñòü i -ê îìïîíåíò à, i = n + 1 , åñëè è òîëüê î åñëè ìíî æ åñòâà M b a = { x : ( x, | x | ⊕ a, b ) ∈ M } , a, b ∈ { 0 , 1 } åñòü ï. ì., ïðè÷¼ì M 0 a è M 1 a ÿâëÿþòñ ÿ àëü òåðíàòèâàìè äðóã äðóãó (ìíî æ åñòâà M 0 0 è M 1 0 ñîîòâåòñòâóþò  i -÷¼òíîé ÷àñòè i -ê îìïîíåíòû, M 0 1 è M 1 1   i -íå÷¼òíîé; ê àæ- äàÿ èç ýòèõ ÷àñòåé ìî æ åò áûòü ïó ñòîé, ïðè÷¼ì, åñëè îáå íåïó ñòû, òî i -ê îìïîíåíò à ðàçäåëèìà). Ïðèìåðîì i -ê îìïîíåíòû ÿâëÿåòñ ÿ ëèíåéíîå ï. ì. (1 ), i = n .  àíåå [ 5, 9℄ óæ å ñòðî- èëèñü ìíîãî÷èñëåííûå ïðèìåðû íåëèíåéíûõ ïî äâèæíûõ ìíî æ åñòâ, ÿâëÿþùèõ ñ ÿ i - ê îìïîíåíò àìè. Âñå îíè áûëè âëî æèìû â 1 -ñîâåðøåííûå ê î äû, ê àæäûé èç óïîìÿíó- òûõ ïðèìåðîâ èìåë ìîùíîñòü, êðàòíóþ ìîùíîñòè ëèíåéíîé ê îìïîíåíòû. Êðîìå òî- ãî, äîê àçàííîé áûëà ëèøü íåäåëèìîñòü ýòèõ i -ê îìïîíåíò íà ìåíüøèå i -ê îìïîíåíòû. Âîïðîñ îá èõ íåäåëèìîñòè ê àê ïî äâèæíûõ ìíî æ åñòâ îñò à¼òñ ÿ îòêðûòûì. Ïîýòîìó , íåñìîòð ÿ íà òî ÷òî èññëåäîâàíèÿ ïîñâÿùåíû îáùåé ïðîáëåìå è äàæ å îáùåìó ïî äõ î äó ê ðåøåíèþ ýòèõ ïðîáëåì, íàïðàâëåíèÿ íåñê îëüê î ðàçëè÷íû è ðåçó ëü ò àòû íå ïåðåêðû- âàþòñ ÿ, à äîïîëíÿþò äðóã äðóã à: ìû îòê àçûâàåìñ ÿ îò âëî æèìîñòè â 1 -ñîâåðøåííûå ê î äû (÷òî ÿâëÿåòñ ÿ îñëàáëåíèåì), çàòî èìååì äåëî ñ áîëåå ñèëüíîé íåäåëèìîñòüþ è ñ á îëüøèì ñïåêòðîì ðàçìåðíîñòåé. 3.  åäóöèðó åìîñòü Ëåììà 3 ( î ëèíåéíîì ðàñøèðåíèè ï. ì. ) . Ïó ñòü M , M ′ ⊆ E n åñòü ð. ï. ì. è åãî àëü òåðíàòèâà. Ò îã äà ìíî æ åñòâà R = { ( x, 0 , 0 ) : x ∈ M } ∪ { ( x, 1 , 1) : x ∈ M ′ } (3) R ′ = { ( x, 1 , 1) : x ∈ M } ∪ { ( x, 0 , 0 ) : x ∈ M ′ } åñòü ð. ï. ì. è åãî àëü òåðíàòèâà â E n +2 . Äîêàçàòå ëüñòâî . Âûïîëíåíèå ó ñëîâèÿ (b) ëåììû 1 äëÿ M è M ′ íåïîñðåäñòâåííî âëå÷¼ò ñïðàâåäëèâîñòü ýòîãî ó ñëîâèÿ äëÿ R è R ′ . △ .  . ï. ì. R ∈ E n íàçîâ¼ì ðåäóöèðóå ìûì , åñëè îíî ìî æ åò áûòü ïîëó÷åíî ê îíñòðóê- öèåé (3), à ò àêæ å ïåðåñò àíîâê îé ê îîð äèíàò è èíâåðñèåé íåê îòîðûõ ñèìâîëîâ, ïðèìå- í¼ííûìè ê î âñåì âåêòîðàì ìíî æ åñòâà î äíîâðåìåííî. Ï. ì. íàçîâ¼ì ðåäóöèðóå ìûì , åñëè ñîîòâåòñòâóþùåå åìó ð. ï. ì. ðåäóöèðó åìî. 5 Ò àêèì îáðàçîì, âîïðîñ ñóùåñòâîâàíèÿ ðåäóöèðó åìûõ ï. ì. ñâî äèòñ ÿ ê ñóùåñòâî- âàíèþ ï. ì. â ìåíüøèõ ðàçìåðíîñò ÿõ. Ñ ýòîé òî÷êè çðåíèÿ îðìó ëèðîâê à îñíîâíîé òåîðåìû åñòåñòâåííà. Çàìå÷àíèå . Êàê âèäíî èç Ñëåäñòâèÿ 3, ëþáàÿ i -ê îìïîíåíò à ëèáî ÿâëÿåòñ ÿ ðåäó- öèðó åìûì ï. ì., ëèáî ðàçáèâàåòñ ÿ íà äâå i -ê îìïîíåíòû ( i -÷¼òíóþ è  i -íå÷¼òíóþ), ê àæäàÿ èç ê îòîðûõ åñòü ðåäóöèðó åìîå ï. ì.  ÷àñòíîñòè, ëèíåéíîå ï. ì. ( 1) ðåäóöè- ðó åìîå. Áîëåå òîãî, ëèíåéíîå ð. ï. ì., ñ òî÷íîñòüþ äî ïåðåñò àíîâêè ê îîð äèíàò , ìî æ åò áûòü ïîëó÷åíî èç òðèâèàëüíîãî ð. ï. ì. { 00 } â E 2 ïîñëåäîâàòåëüíûì ïðèìåíåíèåì ê îíñòðóêöèè èç ëåììû 3 . 4. Äîê àçàòåëüñòâî òåîðåìû Çàèê ñèðó åì n , êðàòíîå ÷åòûð¼ì: n = 4 k .  àçîáü¼ì íîìåðà ê îîð äèíàò íà k ãðóïï ïî 4 â ê àæäîé è ïåðåîáîçíà ÷èì ñîîòâåòñòâóþùèå îðòû ñëåäóþùèì îáðàçîì: e 1 0 , e 1 1 , e 1 2 , e 1 3 , e 2 0 , . . . , e k 3 .  ê àæäîé ÷åòâ¼ðê å âèäà { e i 0 , e i 1 , e i 2 , e i 3 } âûáåðåì ïðîèçâîëüíî (âñåãî 6 âîçìî æíîñòåé) ïàðó íåñîâïàäàþùèõ îðòîâ e i j è e i t è íàçîâ¼ì å¼ èíäåê ñîì ÷èñëî p = j ⋆ t − 1 , ã äå ⋆ îïðåäåëÿåòñ ÿ ò àáëèöåé çíà ÷åíèé ⋆ 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 (çàìåòèì, ÷òî j ⋆ t = t ⋆ j è a ⋆ b = c ⋆ d äëÿ ëþáûõ ïîïàðíî ðàçëè÷íûõ a , b , c , d ). Ïðîñóììèðîâàâ âûáðàííûå ïàðû ïî âñåì i = 1 , 2 , . . . , k , ìû ïîëó÷èì âåêòîð âåñà 2 k , ê îòîðûé áó äåì íàçûâàòü ñòàíäàðòíûì . Âñåãî ïîëó÷èòñ ÿ 6 k ñò àíäàðòíûõ âåêòîðîâ. Èíäåê ñîì I ( V ) ñò àíäàðòíîãî âåêòîðà v ìû áó äåì íàçûâàòü ñóììó ïî ìî äó ëþ 3 âñåõ èíäåê ñîâ ñîñò àâëÿþùèõ åãî ïàð îðòîâ.  àçîáü¼ì ìíî æ åñòâî ñò àíäàðòíûõ âåêòîðîâ íà íåïåðåñåê àþùèåñ ÿ ïî äìíî æ åñòâà S 0 , S 1 è S 2 â ñîîòâåòñòâèè ñ èõ èíäåê ñàìè. Óòâåð æäåíèå 1. Ïóñòü i 6 = j , i, j ∈ { 0 , 1 , 2 } . Ò îãäà ãð à G ( S i ∪ S j ) ð àññòîÿ- íèé äâà, èíäóöèðîâàííûé ìíîæåñòâî ì âåêòîðîâ S i ∪ S j , ÿâëÿåòñÿ äâóäî ëüíûì è îäíîðîäíûì ñòåïåíè 2 k . Äëÿ íà ÷àëà çàìåòèì, ÷òî ãðàû G ( S i ) è G ( S j ) ïó ñòû. Äåéñòâèòåëüíî, ðàññìîò- ðèì ïàðó âåêòîðîâ v , u ∈ S i . Ëèáî v è u ðàçëè÷àþòñ ÿ â î äíîé ÷åòâ¼ðê å ê îîð äèíàò , òîã äà d ( v , u ) = 4 , ïîñê îëüêó ó íèõ èíäåê ñû î äèíàê îâû, ëèáî v è u ðàçëè÷àþòñ ÿ â áîëüøåì, ÷åì î äíà, ÷èñëå ÷åòâ¼ðîê, òîã äà d ( v , u ) ≥ 4 , ïîñê îëüêó ïî ê àæäîé ÷åòâ¼ð- ê å ðàññòî ÿíèå ìåæäó ñò àíäàðòíûìè âåêòîðàìè ÷¼òíî. Ò àêèì îáðàçîì, äâó äîëüíîñòü ãðàà G ( S i ∪ S j ) îáåñïå÷åíà. Ò àêæ å ëåãê î ïîíÿòü, ÷òî âñ ÿêèé âåêòîð èíäåê ñà i èìååò ðîâíî äâóõ ñîñåäåé íà ðàñ- ñòî ÿíèè 2 èç S j , ðàçëè÷àþùèõ ñ ÿ ñ íèì â î äíîé èê ñèðîâàííîé ÷åòâ¼ðê å ê îîð äèíàò . Ýòî îçíà ÷àåò , ÷òî ñòåïåíü ãðàà åñòü 2 k . Óòâåð æäåíèå 1 äîê àçàíî. 6 Ò àêèì îáðàçîì, ìíî æ åñòâî S 0 (íàïðèìåð) ÿâëÿåòñ ÿ ðàñøèðåííûì ïî äâèæíûì è èìååò ìîùíîñòü 2 · 6 k − 1 . Óòâåð æäåíèå 2.  . ï. ì. S 0 íåäå ëèìîå. Ïðåäïîëî æèì, ÷òî P ⊆ S 0 è S 0 \ P åñòü ð. ï. ì. è P íåïó ñòî. Ò îã äà ñóùåñòâó åò P èìååò àëü òåðíàòèâó P ′ . Ñíà ÷àëà óáåäèìñ ÿ, ÷òî (*) P ′ ñîñòîèò èç ñòàíäàðòíûõ âåêòîðîâ , òî åñòü ò àêèõ, ÷òî â ê àæäîé ÷åòâ¼ðê å ê îîð äèíàò ñî äåð æèòñ ÿ ðîâíî äâå åäèíèöû. Äåéñòâèòåëüíî, â ïðîòèâíîì ñëó÷àå P ′ ñî äåð æèò âåêòîð ñ íåñò àíäàðòíîé ÷åòâ¼ðê îé, è, ê àê ñëåäñòâèå, Ω ∗ ( P ′ ) ñî äåð æèò âåê- òîð ñ äâóìÿ íåñò àíäàðòíûìè ÷åòâ¼ðê àìè.  òî æ å âðåìÿ Ω ∗ ( P ) ñîñòîèò èç âåêòîðîâ ñ î äíîé íåñò àíäàðòíîé ÷åòâ¼ðê îé è, ñëåäîâàòåëüíî, íå ìî æ åò ñîâïàäàòü ñ Ω ∗ ( P ′ ) , ÷òî ïðîòèâîðå÷èò ëåììå 1. (*) äîê àçàíî.  àññìîòðèì ïðîèçâîëüíûé âåêòîð p èç P è ïîê àæ åì, ÷òî (**) âñå âåêòîðû èç S 0 , îòëè÷àþùèåñÿ îò p íå áî ëåå ÷å ì â äâóõ ÷åòâ¼ðêàõ, òàêæå ïðèíàäëåæàò P . Áåç ïîòåðè îáùíîñòè ðàññìîòðèì äâå ïåðâûå ÷åòâ¼ðêè. Ïîëî æèì p = ( h, t ) , ã äå h è t  âåêòîðû äëèíû 8 è n − 8 ñîîòâåòñòâåííî.  àññìîòðèì âåêòîð p ⊕ e i j èç Ω ∗ ( P ) , ã äå i ∈ { 1 , 2 } è j ∈ { 0 , 1 , 2 , 3 } . Ñîã ëàñíî ëåììû 1 , p ⊕ e i j ∈ Ω ∗ ( p ′ ) äëÿ íåê îòîðîãî p ′ èç P ′ . Êàê äîê àçàíî âûøå, âåêòîð p ′ ñò àíäàðòíûé, ïîýòîìó p ′ = p ⊕ e i j ⊕ e i j ′ äëÿ íåê îòîðîãî j ′ ∈ { 0 , 1 , 2 , 3 } , îòêó äà ñëåäó åò , ÷òî p ′ ñîâïàäàåò ñ p â ïîñëåäíèõ n − 8 ê îîð äèíàò àõ. Èç ýòèõ ðàññóæäåíèé ñëåäó åò , ÷òî Ω ∗ ( P 8 ) = Ω ∗ ( P ′ 8 ) , ã äå P 8 = { b ∈ E 8 : ( b, t ) ∈ P } P ′ 8 = { b ∈ E 8 : ( b, t ) ∈ P ′ } è, ïî óòâåð æäåíèþ () ëåììû 1, ìíî æ åñòâî P 8 åñòü ð. ï. ì. â E 8 . Ëåãê î ó ñò àíîâèòü (íàïðèìåð, ïîëüçó ÿñü óòâåð æäåíèåì (b) ëåììû 1), ÷òî ìîùíîñòü ð. ï. ì. â E 8 áîëü- øå 6 . Ñ äðóãîé ñòîðîíû, ïî ïîñòðîåíèþ, ðîâíî 12 âåêòîðîâ èç S 0 èìåþò âèä ( b, t ) , b ∈ E 8 . Ñëåäîâàòåëüíî, áîëüøå ïîëîâèíû ò àêèõ âåêòîðîâ ïðèíàäëåæ àò P . Åñëè áû íå âñå ïðèíàäëåæ àëè P , òî ê îñò àâøèìñ ÿ âåêòîðàì (èç S 0 \ P ) áûëè áû ïðèìåíèìû àíàëîãè÷íûå ðàññóæäåíèÿ, ÷òî ïðèâåëî áû ê ïðîòèâîðå÷èþ. Ñëåäîâàòåëüíî, âñå 12 âåêòîðîâ èç S 0 , ñîâïàäàþùèõ ñ p âî âñåõ ê îîð äèíàò àõ êðîìå ïåðâûõ âîñüìè, ïðèíàä- ëåæ àò P ′ , ÷òî äîê àçûâàåò (**). Ò àêèì îáðàçîì, ëþáûå äâà âåêòîðà èç S 0 íà ðàññòî ÿíèè 4 äðóã îò äðóã à î äíîâðå- ìåííî ëèáî ïðèíàäëåæ àò P , ëèáî íåò . Ïîñê îëüêó S 0 , î÷åâèäíî, ñâÿçíî ïî ðàññòî ÿíèþ 4 , ïîëó÷àåì P = S 0 . Óòâåð æäåíèå 2 äîê àçàíî. Óòâåð æäåíèå 3.  . ï. ì. S 0 íå ðåäóöèðóå ìî. Çàìåòèì, ÷òî â ê îíñòðóêöèè (3) ñóììà ïîñëåäíèõ äâóõ ê îîð äèíàò ðàâíà 0 äëÿ ëþáîãî ñëîâà èç R . Ó÷èòûâàÿ ïåðåñò àíîâêó ê îîð äèíàò è èíâåðñèþ ñèìâîëîâ, ìî æíî óòâåð æäàòü, ÷òî ó ðåäóöèðó åìîãî ð. ï. ì. ñóùåñòâóþò äâå ê îîð äèíàòû, ñóììà ê îòî- ðûõ ðàâíà 0 ëèáî 1 î äíîâðåìåííî äëÿ âñåõ ñëîâ ìíî æ åñòâà. Ëåãê î ïðîâåðèòü, ÷òî S 0 íå ó äîâëåòâîð ÿåò ýòîìó ó ñëîâèþ: ëþáûå äâå ê îîð äèíàòû ñî äåð æ àò âñå ÷åòûðå ê îìáèíàöèè èç 0 è 1 . Óòâåð æäåíèå 3 äîê àçàíî. Ò åîðåìà äîê àçàíà. 7 5. Çàêëþ÷åíèå Ìû ïîñòðîèëè áåñê îíå÷íûé êëàññ íåäåëèìûõ íåðåäóöèðó åìûõ ï. ì. Êîíñòðóêöèÿ îáîáùàåò ïðèìåð, óïîìÿíóòûé â ê îíöå ðàáîòû [1 ℄.  çàêëþ÷åíèå ìû ñîðìó ëèðó åì íåñê îëüê î çàäà ÷, åñòåñòâåííî ñâÿçàííûõ ñ èññëåäîâàíèåì ïî äâèæíûõ ìíî æ åñòâ è ñ ïðîáëåìîé õ àðàêòåðèçàöèè èõ ìíîãîîáðàçèÿ. Äëÿ ïîñòðîåíèÿ ï. ì. ìî æíî ïðèìåíÿòü ïðèíöèï îáîáù¼ííîé ê àñê àäíîé ê îíñòðóê- öèè äëÿ 1 -ñîâåðøåííûõ ê î äîâ [3℄.  ÷àñòíîñòè, ê îíñòðóêöèÿ èç ðàçäåëà 4 ìî æ åò áûòü èíòåðïðåòèðîâàíà â ò àêèõ òåðìèíàõ. Íåäåëèìûå ï. ì., ïîñòðîåííûå ò àêèì îáðàçîì, áó äóò èìåòü íåïîëíûé ðàíã , òî åñòü äëÿ âñåõ ñëîâ ìíî æ åñòâà ê îîð äèíàòû áó äóò ó äî- âëåòâîð ÿòü íåê îòîðîìó ëèíåéíîìó óðàâíåíèþ (íåïîëíîé ïðîâåðê å íà ÷¼òíîñòü èëè íå÷¼òíîñòü). Ïðîáëåìà 1 . Ïîñòðîèòü áåñê îíå÷íûé êëàññ íåäåëèìûõ ï. ì. ïîëíîãî ðàíã à. Ïðèìåð .  àññìîòðèì ÷åòûðå ñëîâà ( 100 110 010 ) , ( 011 110 000 ) , ( 101 001 011 ) , ( 001 100 111 ) , èç E 9 , çàïèñàííûå äëÿ ó äîáñòâà â âèäå ìàññèâà 3 × 3 , à ò àêæ å âñå ñëîâà, ïîëó÷åííûå èç íèõ öèêëè÷åñêèìè ïåðåñò àíîâê àìè ñòðîê è/èëè ñòîëáöîâ ìàññèâà. Ïîëó÷èì íåäå- ëèìîå ï. ì. ïîëíîãî ðàíã à ìîùíîñòè 36 . Àëü òåðíàòèâà ïîëó÷àåòñ ÿ èíâåðñèåé âñåõ ñëîâ. Ïðîáëåìà 2 . Ïîñòðîèòü áîã àòûé êëàññ òðàíçèòèâíûõ íåäåëèìûõ ï. ì., ð. ï. ì. Ìíî æ åñòâî M ⊆ E n íàçûâàåòñ ÿ òðàíçèòèâíûì, åñëè ñò àáèëèçàòîð Stab I ( M ) ìíî- æ åñòâà M â ãðóïïå I èçîìåòðèé ãèïåðêóáà äåéñòâó åò òðàíçèòèâíî íà ýëåìåíò àõ M , ò . å. äëÿ ëþáûõ x, y èç M íàéä¼òñ ÿ èçîìåòðèÿ σ ∈ Stab I ( M ) ò àê àÿ, ÷òî σ ( x ) = y . Íàïðèìåð, íåòðó äíî ïîê àçàòü, ÷òî ï. ì., ïîñòðîåííûå â äàííîé ðàáîòå, ÿâëÿþòñ ÿ òðàíçèòèâíûìè. Èçâåñòíî íåñê îëüê î ê îíñòðóêöèé òðàíçèòèâíûõ 1 -ñîâåðøåííûõ è ðàñøèðåííûõ 1 -ñîâåðøåííûõ ê î äîâ, ïîñëåäíèå ðåçó ëü ò àòû ñìîòðè â [6, 7 ℄. Ïðîáëåìà 3 . Èññëåäîâàòü âîïðîñ âëî æèìîñòè ï. ì. â 1 -ñîâåðøåííûé ê î ä: ñó- ùåñòâîâàíèå íåâëî æèìûõ ï. ì. â ê î äîâûõ ðàçìåðíîñò ÿõ n = 2 k − 1 ; ñóùåñòâîâàíèå ï. ì., íåâëî æèìûõ ïðè ïîìîùè ëèíåéíîãî ðàñøèðåíèÿ (ëåììà 3) â 1 -ñîâåðøåííûé ê î ä íè â î äíîé áîëüøåé ðàçìåðíîñòè.  ÷àñòíîñòè, äëÿ ï. ì., ïîñòðîåííûõ â ðàçäåëå 4, âîïðîñû âëî æèìîñòè îòêðûòû ïðè n ≥ 11 . Ïðîáëåìà 4 . Îöåíèòü ìàê ñèìàëüíûé ðàçìåð íåäåëèìîãî ï. ì. Ïðîáëåìà 5 . Îöåíèòü ìèíèìàëüíûé ðàçìåð íåëèíåéíîãî ï. ì. (ê îíñòðóêöèÿ ðàç- äåëà 4 âìåñòå ñ ëåììîé 3 äà¼ò âåð õíþþ îöåíêó 1 , 5 L ( n ) , ã äå L ( n ) = 2 ( n − 1) / 2  ìîù- íîñòü ëèíåéíîãî ï. ì.), íåðåäóöèðó åìîãî íåäåëèìîãî ï. ì. (ê îíñòðóêöèÿ äà¼ò âåð õ- íþþ îöåíêó 1 , 5 ( n − 3) / 4 L ( n ) ), íåäåëèìîãî ï. ì. ïîëíîãî ðàíã à. 8 Ëèòåðàòóðà [1℄ Âàñèëüåâ Þ. Ë., Ñîëîâüåâà Ô. È. Êî äîîáðàçóþùèå àêòîðèçàöèè n - ìåðíîãî åäèíè÷íîãî êóáà è ñîâåðøåííûõ äâîè÷íûõ ê î äîâ // Ïðîáëåìû ïåðåäà ÷è èíîðìàöèè. 1997. Ò . 33,  1. Ñ. 6474. [2℄ Çèíîâüåâ Â. À., Çèíîâüåâ Ä. Â. Äâîè÷íûå ðàñøèðåííûå ñîâåðøåííûå ê î äû äëèíû 16 ðàíã à 14 // Ïðîáëåìû ïåðåäà ÷è èíîðìàöèè. 2006. Ò . 42,  2. Ñ. 6380. [3℄ Çèíîâüåâ Â. À., Ëîáñòåéí À. Îá îáîáù¼ííûõ ê àñê àäíûõ ê îíñòðóêöèÿõ ñî- âåðøåííûõ äâîè÷íûõ íåëèíåéíûõ ê î äîâ // Ïðîáëåìû ïåðåäà ÷è èíîðìàöèè. 2000. Ò . 36,  4. Ñ. 5973. [4℄ Ôîí-Äåð-Ôëààññ Ä. . Ñîâåðøåííûå 2-ðàñêðàñêè ãèïåðêóáà // Ñèáèðñêèé ìàòåìàòè÷åñêèé æóðíàë. 2007. Ò . 48,  4. Ñ. 923930. [5℄ Ñîëîâüåâà Ô. È. Î àêòîðèçàöèè ê î äîîáðàçóþùèõ ä.í.. // Ìåòî äû äèñ- êðåòíîãî àíàëèçà â èññëåäîâàíèè óíêöèîíàëüíûõ ñèñòåì. Íîâîñèáèðñê: Èí-ò ìàòåìàòèêè ÑÎ ÀÍ ÑÑÑ , 1988. Ò . 47. Ñ. 6688. [6℄ Ñîëîâüåâà Ô. È. Î ïîñòðîåíèè òðàíçèòèâíûõ ê î äîâ // Ïðîáëåìû ïåðåäà ÷è èíîðìàöèè. 2005. Ò . 41,  3. Ñ. 2331. [7℄ Ïîò àïîâ Â. Â. Î íèæíåé îöåíê å ÷èñëà òðàíçèòèâíûõ ñîâåðøåííûõ ê î äîâ // Äèñêðåò . àíàëèç è èññëåä. îïåðàöèé. Ñåð. 1. 2006. Ò . 13,  4. Ñ. 4959. [8℄ Ìàëþãèí Ñ. À. Î ïåðå÷èñëåíèè íåýêâèâàëåíòíûõ ñîâåðøåííûõ äâîè÷íûõ ê î- äîâ äëèíû 15 è ðàíã à 15 // Äèñêðåò . àíàëèç è èññëåä. îïåðàöèé. Ñåð. 1. 2006. Ò . 13,  1. Ñ. 7798. [9℄ Solo v'ev a F. I. Struture of i -omp onen ts of p erfet binary o des // Disrete Appl. Math. 2001. V ol. 111, no. 1-2. Pp. 189197. DOI: 10.1016/S0166-218X(00)00352-8 . [10℄ V asil'ev a A. Y u. A represen tation of p erfet binary o des // Pro . Sev en th In t. W orkshop on Algebrai and Com binatorial Co ding Theory . Bansk o, Bulgaria: 2000. June. Pp. 311315. 9