On weak isometries of Preparata codes

On w eak isometries of Preparata co des Iv an Y u. Mogiln ykh Sob olev Institute of Mathematics, No v osibirs k , Russia e-mail: ivmo g84@gma il.c om Abstract Let C 1 and C 2 be c odes with co de distance d . Co des C 1 and C 2 are called we akly isometric , if there exists a mapping J : C 1 → C 2 , s uc h that for an y x, y from C 1 the eq ualit y d ( x, y ) = d holds if and only if d ( J ( x ) , J ( y )) = d . Ob viously tw o co des ar e weakly isometric if and only if the minimal distance graphs of these co des ar e isomorphic. In this pap er we prov e that Pr e parata co des of length n ≥ 2 12 are weakly isometric if and o nly if these co des are equiv alent. The ana logous result is obta ine d for punctured Prepa rata co des o f length not le ss than 2 10 − 1. Submitte d to Pr oblems of Information T r ansmission on 11th of January 2009. 1 In tro duction Let E n denote all b i n a r y ve ctors of length n. T h e Hamming distanc e b et w een t wo ve ctors fr o m E n is the num b er of places w h ere they differ. The weight of v ector x ∈ E n is the distance b et w een this vect or and the all-z ero v ector 0 n , and the supp or t of x is the set sup p ( x ) = { i ∈ { 1 , . . . , n } : x i = 1 } . A s et C, C ⊂ E n , is called a c o de with parameters ( n, M , d ), if | C | = M and the minimal distance b et ween t wo co dew ords f rom C equals d . W e sa y that a co de C is r e duc e d if it conta in s all-zero vect or. A collecti on of k -subsets (referr e d to as b loc ks) of a n -set suc h that an y t - su bset o ccurs in λ blo c ks precisely is called a ( λ, n , k , t )- design . The minimal distanc e gr aph of a co de C is defin e d as the graph w i th all co d ew ords of C as vertices, w it h tw o v ertices b eing connected if and only if the Hamming distance b et ween corresp onding co dew ords equals to the co de distance of the co de C . Tw o cod e word of C are called d - adjac ent if the Hamming distance equals co de distance d of the co de C . Tw o co des C 1 and C 2 of length n are called equiv alen t, if an automorphism F of E n exists suc h that F ( C 1 ) = C 2 . A mapping I : C 1 → C 2 of t wo co des C 1 and C 2 is called an isometry b et w een co des C 1 and C 2 , if the equalit y d ( x, y ) = d ( I ( x ) , I ( y )) holds for all x and y from C 1 . Then co d es C 1 and C 2 are called isometric . A mapping J : C 1 → C 2 is called a we ak isometry of co des C 1 and C 2 (and co des C 1 and C 2 w eakly isometrical), if f or an y x, y fr om C 1 the equalit y d ( x, y ) = d holds if and only if d ( J ( x ) , J ( y )) = d where d is the co de d i s tance of co de C 1 . Ob vious ly t w o co des are weakly isometric if and only if the minimal distance graph s of these co des are isomorphic. In [2] Avgustino vic h established that an y tw o w eakly isometric 1-p erfect co des are equiv alen t. In [5] it was prov ed that th is result also holds for extended 1-p erfect co des. In th is pap er any we ak isometry of t wo Preparata co des (punctur e d Pr ep a r ata co des) is pro ved to b e an isometry of these co des. Moreo ver, w eakly isometric Preparata co des (punctured Preparata co des) of length n ≥ 2 12 (of length n ≥ 2 10 − 1 resp ectiv ely) are p ro v ed to b e equiv alen t. This topic is closely related with problem of metrical rigidit y of co des. A co de C is called metric al ly rigid if an y isometry I : C → E n can b e extended to an isometry (automorphism) of the whole space E n . Ob vious ly any t w o m etrically rigid isometric co des are equiv alen t. In [4] it was established that any redu ce d bin a r y co de of length n con taining 2-( n, k , λ )-design is metrically rigid f o r any n ≥ k 4 . 1 A maximal b i n a r y co de of length n = 2 m for ev en m , m ≥ 4 w i th cod e distance 6 is called a P r ep ar ata c o de P n . Punctur e d Pr ep a r ata c o de is a co de obtained from Preparata co de by deleting one co ordinate. By P n w e denote a pun ct u red Preparata co de of length n . Pr e p a r a ta co des and punctur e d Preparata co des ha ve some useful prop erties. All of them are distance in v arian t [1], s tr o n g ly distance inv ariant [3]. Also a punctured Pr e p a r a ta co de is con tained in the unique 1-p erfect co de [6]. An arbitrary p u nctured Preparata co de is un i f o r m ly pack ed [1]. As a consequence of this prop ert y , co dew ords of minimal wei ght of a Preparata co de (punctured Preparata co de) form a design. The last prop ert y is crucial in pro ving the main result of this pap er. 2 W eak isomery of pu nctured Preparata co des In this section we p ro v e that any tw o punctured Preparata co des of length n with isomorph i c minim u m d ista n ce graphs are isometric. Moreo v er, th e se co des are equiv alen t for n ≥ 2 10 − 1. First w e give some preliminary statemen ts. Lemma 1. [1]. L et P n b e an arbitr ary r e d uc e d punctur e d Pr ep a r ata c o de. Th e n c o dewor ds of weight 5 of the c o de P n form 2-(n,5,(n-3)/3) design. T aking in to accoun t a structure of the design f rom this lemma w e obtain Corollary 1. L et P n b e an arbitr a ry r e duc e d punctur e d Pr ep ar ata c o de an d r , s b e arbitr ary elements of the set { 1 , . . . , n } . Then ther e exists exactly one c o or dinate t such that al l c o d ewor ds of minimal weight of the c o de P n with ones in c o o r dinates r and s has zer o in the c o or dinate t . Let C b e a co de with co de distance d and x b e an arbitrary co dew ord of C of weigh t i . Denote by D i,j ( x ) the set of all co dew ords of C of weig ht j which are d -adjacen t with v ector x . In case when C is a pun c tu r ed Preparata co de w e giv e some prop erties of the set D i,j ( x ) that mak e the structur e of minimal distance graph of this co de more clear. Lemma 2. L et x b e an arbitr ar y c o dewor d of a punctur e d Pr ep ar ata c o de P n . Then any ve ctor fr om D i,i − 1 ( x ) ( D i,i − 3 ( x ) D i,i − 5 ( x ) r esp e ctively) has exactly 3 (4 and 5 r esp.) zer o c o or dinates fr om sup p ( x ) and exactly 2 (1 and 0 r esp.) nonzer o c o or d inates fr om { 1 , . . . , n }\ supp ( x ) . Pro o f. Supp ose a v ector y ∈ D i,i − k ( x ) h a s m k zero coord inat es from supp ( x ). Then it has exactly m k − k n o n z ero co ordinates from the set { 1 , . . . , n }\ supp ( x ). Since d ( x, y ) = 5 we ha ve m k = (5 + k ) / 2, which implies th e r e qu ired p roperty for k = 1 , 3 , 5 . N Let x b e a cod ew ord of w eight i from a P n ; m, l b e arb itrary co ordinates fr om supp ( x ). W e denote b y A m,l ( x ) ( B m,l ( x ) and C m,l ( x )) the sets D i,i − 1 ( x ) ( D i,i − 3 ( x ) and D i,i − 5 ( x ) resp ectiv ely) with co ordinates m and l equal to zero. Lemma 3. L et x ∈ P n , m, l ∈ supp ( x ) and u, v b e arbitr ary c o d e w or ds of P n with zer os in c o or dinates m and l that ar e at distanc e 5 fr om x . Then u, v do not shar e zer o c o or dinates in supp ( x ) \ { m, l } and do not shar e c o or dinates e qual to one in the set { 1 , . . . , n }\ supp ( x ) . Pro o f. Let us supp ose the opp osite. Then the vecto r s x + u and x + v of weig ht five s h are at least three co ordinates with ones in them and therefore d ( u, v ) = d ( x + u, x + v ) ≤ 4 holds. S ince co de distance of th e co de P n equals 5 w e get a con tradiction. N 2 Lemma 4. L et x b e an arbitr ary c o dewor d of weight i fr om a punctur e d Pr ep ar ata c o de. Then the fol lowing i ne qualities hold: ( i − 3) C 2 i ≤ 3 | D i,i − 1 ( x ) | + 12 | D i,i − 3 ( x ) | + 30 | D i,i − 5 ( x ) | ≤ ( i − 2) C 2 i . (1) Pro o f. Fix t wo co o r d inate s m and l f r o m supp ( x ). By Lemma 2 an arb itrary v ector from A m,l ( x ) ( B m,l and C m,l ) has exact ly one zero co ordinate (t w o and three resp ectiv ely) from supp ( x ) \{ m, l } . Th e n ta kin g in to acc ount Le m m a 3 the num b er of co o r d inate s from s upp ( x ) \ { m, l } whic h are zero for v ectors from A m,l , B m,l and C m,l equals | A m,l ( x ) | , 2 | B m,l | and 3 | C m,l | r esp ec- tiv ely . Therefore the n u m b er of co ordinates from the supp ( x ) \{ m, l } w hic h are zero f o r vec tors from A m,l ∪ B m,l ∪ C m,l equals | A m,l ( x ) | + 2 | B m,l ( x ) | + 3 | C m,l ( x ) | . Since x is a v ector of we ight i an d m, l ∈ supp ( x ), this num b er do es not exceed i − 2. F rom the other hand by C orollary 1 there exists at most one co ordinate from supp ( x ) \{ m, l } such that all vec tors from A m,l ∪ B m,l ∪ C m,l ha ve one in it. Thus we hav e: i − 3 ≤ | A m,l ( x ) | + 2 | B m,l ( x ) | + 3 | C m,l ( x ) | ≤ i − 2 . Summing th e se in e qu a lities for all m, l ∈ supp ( x ) we obtain ( i − 3) C 2 i ≤ X m,l ∈ supp ( x ) | A m,l ( x ) | + 2 X m,l ∈ supp ( x ) | B m,l ( x ) | + 3 X m,l ∈ supp ( x ) | C m,l ( x ) | ≤ ( i − 2) C 2 i (2) As an arbitrary v ector from D i,i − 1 ( x ) h a s exactly 3 zero coord i n a tes fr o m supp ( x ), any such v ector is count ed C 2 3 times in the sum P m,l ∈ supp ( x ) | A m,l ( x ) | . Then X m,l ∈ supp ( x ) | A m,l ( x ) | = C 2 3 | D i,i − 1 ( x ) | . Analogously w e get: X m,l ∈ supp ( x ) | B m,l ( x ) | = C 2 4 | D i,i − 3 ( x ) | , X m,l ∈ supp ( x ) | C m,l ( x ) | = C 2 5 | D i,i − 5 ( x ) | . So fr o m (2) we get (1). N No w we prov e the main result using Lemmas 2 an d 4. Theorem 1. The minimal distanc e gr ap hs of two punctur e d Pr ep ar ata c o des ar e isomorphic i f and only if these c o des ar e isometric. Pro o f. It is obvi ous that if tw o p unctured Preparata co des are isometric then they are we akly isometric. Let J : P n 1 → P n 2 b e a w eak isometry of t wo p unctured Preparata co des P n 1 and P n 2 of length n . Without loss of generalit y supp ose that 0 n ∈ P n 1 , J (0 n ) = 0 n . W e n o w sho w that mappin g J is an isometry . F or proving th is it is su ffici ent to show that w t ( J ( x )) = w t ( x ) for all x ∈ P n 1 . Supp ose z is a co dew ord of the co de P n 1 , s uc h that wt ( J ( z )) 6 = w t ( z ) = i holds an d th e mapping J preserv es w eigh t of all co dew ords of weig ht smaller that i . T he v ector z satisfying these conditions w e call critic al Since J (0 n ) = 0 n and the mapping J pr e serves the distance 3 b et w een all co dew ords at distance 5, we ha ve i ≥ 6. W e pro ve that ther e is no critica l co dew ords in P n 1 . F rom 0 n ∈ P n 1 holds that the weak isometry J preserv es a parit y of w eight of a vecto r and th e r e f ore w t ( J ( z )) equals either i + 2 or i + 4. Supp ose w t ( J ( z )) = i + 2. Since J is a weak isometry and z is a critical vect or we ha ve the follo wing: | D i +2 ,i − 1 ( J ( z )) | = | D i,i − 1 ( z ) | , | D i +2 ,i − 3 ( J ( z )) | = | D i,i − 3 ( z ) | , | D i,i − 5 ( z ) | = 0. T a kin g in to accoun t these equalities, from the inequalities of Lemma 4 for ve ctors z and J ( z ) we get ( i − 3) C 2 i ≤ 3 | D i,i − 1 ( z ) | + 12 | D i,i − 3 ( z ) | , (3) 3 | D i +2 ,i +1 ( J ( z )) | + 12 | D i,i − 1 ( z ) | + 30 | D i,i − 3 ( z ) | ≤ iC 2 i +2 . (4) Multiplying b oth sides of in e qu a lity (3) by − 4 w e get − 12 | D i,i − 1 ( z ) | − 48 | D i,i − 3 ( z ) | ≤ − 4( i − 3) C 2 i . Summing th i s inequ ality with (4) we get 3 | D i +2 ,i +1( J ( z )) | − 18 | D i,i − 3 ( z ) | ≤ iC 2 i +2 − 4( i − 3) C 2 i , and th e r e f ore | D i,i − 3 ( z ) | ≥ 4( i − 3) C 2 i − iC 2 i +2 18 . (5) In particular, from the inequ ality (5) we ha ve | D i,i − 3 ( z ) | ≥ 1 for i = 6 and i = 7. But there is no co dew ords of wei ght 3 and 4 in the P 1 since P 1 is reduced co de with co de d ista n c e 5. Th erefore i ≥ 8. F rom Lemma 4 we hav e the follo win g | D i,i − 3 ( z ) | ≤ ( i − 2) C 2 i 12 . (6) But for i ≥ 10 the inequ a lity 3( i − 2) C 2 i < 2(4( i − 3) C 2 i − iC 2 i +2 ) h ol d s. This con tradicts with (5) and (6). So it is only remains to pro ve that there are no co dew ords of weig ht 8 and 9, su c h that their images under the mapping J hav e w eigh ts 10 and 11 resp ecti vely . Obviously the Hammin g distance b et we en any tw o ve ctors from D i +2 ,i − 3 ( J ( z )) is n ot less than 6. By Lemma 2 all ones co o r d inate s of eac h v ector from D i +2 ,i − 3 ( J ( z )) are in set sup p ( J ( z )). So | D i +2 ,i − 3 ( J ( z )) | do es not exceed th e cardinalit y of maximal constant we ight co de of length i + 2, with all co de w ord s of w eight b eing equal i − 3 and b eing at d ista n ce not less than 6 pairwise. F or i = 8 and i = 9 the cardinalities of su c h co des equal to 6 and 11 r e sp ectiv ely , bu t from (5 ) w e ha ve | D 10 , 5 ( J ( z )) | = | D 8 , 5 ( z ) | ≥ 12 , | D 11 , 6 ( J ( z )) | = | D 9 , 6 ( z ) | ≥ 21 , a contradictio n . Therefore there is n o critical ve ctors z in P n 1 , w t ( z ) = i , suc h that wt ( J ( z )) = i + 2. Supp ose w t ( J ( z )) = i +4. In th is case w e hav e | D i,i − 3 ( z ) | = | D i,i − 5 ( z ) | = 0, | D i +4 ,i − 1 ( J ( z )) | = | D i,i − 1 ( z ) | . Using these equalities we ha ve fr o m the inequalities of Lemma 4 for the vect ors z and J ( z ) the follo wing: ( i − 3) C 2 i ≤ 3 | D i,i − 1 ( z ) | , 30 | D i,i − 1 ( z ) | ≤ ( i + 2) C 2 i +4 . F rom these last t wo inequalities we obtain 4 ( i − 3) C 2 i 3 ≤ ( i + 2) C 2 i +4 30 , and th e r e f ore 10 i ( i − 1)( i − 3) ≤ ( i + 4)( i + 3)( i + 2) that imp lie s 10 i ( i − 1)( i − 3) ≤ 2 i ( i + 3)( i + 2) . Since last inequalit y do es not hold f o r i ≥ 6 th e r e is no critical vect ors in P n 1 and th e r e f ore th e mapping J is an isometry . N In [4] the f ollo w i n g theorem w as p ro v ed Theorem 2. Any r e d uc e d c o d e of length n , that c ontains a 2 − ( n, k , λ ) -design is metric al ly rigid for n ≥ k 4 . T aking into accoun t that by Lemma 1 an y punctured reduced Preparata co de cont ains 2- ( n, 5 , ( n − 3) / 4)-design applying T heorems 1 and 2 we get Corollary 2. L et n ≥ 2 10 − 1 . Tw o punctur e d Pr ep ar ata c o des of length n ar e e quivalent i f and only if the minimal distanc e gr aphs of these c o des ar e isomorphic. 3 W eak isometry of Preparata co des Using the analogous considerations, Theorems 1,2 and Corollary 2 can easily b e extended for extended Pr e p a r a ta co des. W e no w giv e the analogues of Lemmas 1-4 omitting their p roofs. Lemma 5. ( [1]) L et P n b e an arbitr ary r e d uc e d Pr ep ar ata c o de. Then c o dewor ds of weight 6 of c o de P n form 3-(n,6,(n-4)/3)-design. Lemma 6. L et x b e an arbitr ary c o dewor d of a Pr ep ar ata c o de P n , w t ( x ) = i . Then any ve ctor fr om D i,i − 2 ( x ) ( D i,i − 4 ( x ) D i,i − 6 ( x ) r esp e ctively) has exa c t ly 4 (5 and 6 r esp e ctively) zer o c o o r dinates fr o m supp ( x ) and exactly 2 (1 and 0 r esp e ctively) no nzer o c o or dinates fr om { 1 , . . . , n }\ supp ( x ) . Lemma 7. L et x ∈ P n , m, l, k ∈ supp ( x ) , and u, v b e arbitr ary c o dewor ds of P n at dis- tanc e 6 fr om the ve ctor x with zer o c o or dinates in p ositions m , l , k . Then ther e is no c o or- dinate fr om supp ( x ) \ { m, l , k } such that u, v have zer o s in it and ther e is no c o or dinate fr om { 1 , . . . , n }\ supp ( x ) such that u, v have ones in i t . Lemma 8. L et x b e an arbitr ary c o dewo r d of weight i f r om a Pr ep ar ata c o de. Then the fol lowing ine qualities hold: C 3 i ( i − 4) ≤ 4 | D i,i − 2 ( x ) | + 20 | D i,i − 4 ( x ) | + 60 | D i,i − 6 ( x ) | ≤ C 3 i ( i − 3) . (7) Using Lemm a s 5-8 and the same argu m e nts as in the pro of of Theorem 1 the follo wing theorem it is not d iffi cu l t to prov e Theorem 3. The minimal distanc e gr aphs of two Pr ep ar ata c o des ar e isomorphic i f and only if the c o des ar e isometric. F rom this theorem, Lemma 5 and Theorem 2 we get Corollary 3. L et n ≥ 2 12 . Two Pr ep ar ata c o des of length n ar e e quivalent if and only if the minimal distanc e g r aphs of these c o d es ar e i s omorphic. The Author is d e epf uly grateful to F aina Iv ano vna S o lov ev a for introdu ci n g in to the topic, problem statemen t and all around supp ort of this wo r k. 5 References [1] Semakov N.V., Z ino vi ev V.A., Zaitsev G.V. Uniformly pack ed co des // Probl. Inf. T r a n s. 1971. V. 7. 1. P . 30–39. [2] Avgustinovich S.V. Perfect bin a r y (n,3) co des: the stru c tur e of graphs of m i n im um d ista n ces // Discrete App l. Math. 2001. V. 114. P . 9–11. [3] V asil’eva, A.Y u. Strong distance inv ariance of p erfect binary co des// Diskr . Anal. Issled. Op er., 2002. Iss. 1. V. 9. 4. P . 33–40. [4] Avgustinovich S.V., Soloveva F.I. T o th e Metrical Rigidit y of Binary Co des // Pr ob lems of Inf o r m . T ransm. 2003. V. 39. 2. P . 23–28. [5] I. Y. Mogiln ykh, P . R. J. ¨ Osterg ˚ ard, O. Po ttonen and F. I. Solo vev a, accepted to IEEE Inform. 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