Reconstructing Extended Perfect Binary One-Error-Correcting Codes from Their Minimum Distance Graphs

1 Reconstructing Extended Perf ect Binary One-Err or -Corr ecting Codes from Their Minimum Distance Graphs Ivan Y u. Mogilnykh, Patric R. J. ¨ Osterg ˚ ard, Olli Po ttonen, Faina I. So lov’ev a Abstract — The minimum distance graph of a code has the codewords as vertices and edges exactly wh en the Hamming dis- tance between two codewo rds equals the minimum distance of the code. A constructive proof fo r reconstructibili ty of an extended perfect binary one-error -correcting code from its minimum distance graph is presented. Consequently , inequiva lent such codes hav e nonisomorphic minimu m d istance graphs. Moreov er , it is shown that the automo rphism group o f a minimum distance graph is isomorphic to that of the corresponding code. Index T erms — m inimum distance graph, extended perfect binary code, reconstructibility , weak isometry I . I N T RO D U C T I O N A binary code of length n is a subset of F n 2 , wher e F 2 = { 0 , 1 } is the field of two ele ments. T hrough out t his work, we use “code” in the m eaning of “binary code”. The support supp( x ) of a word x = ( x 1 , x 2 , . . . , x n ) is the set of its n onzero coord inates, the weight wt ( x ) o f x is the nu mber of nonzero coordinates, and the Ha mming distance d H ( x , y ) is the numbe r of coord inates in which th e words x and y d if fer . Formally , supp( x ) := { i : x i = 1 } , wt( x ) := | supp ( x ) | and d H ( x , y ) := wt( x − y ) . The minimu m d istance of a co de is the min imum Ha mming distance be tween any pair of distinct codew ords. For a co de with minimum distance d , the balls of radius r = ⌊ ( d − 1) / 2 ⌋ centered around the cod e words are nonintersectin g and such a code is c alled an r -err or-corr ecting code. If the balls cover the entire ambient space, th e code is called perfect , or more specifically , r -perfect . W ith one exception (the binary Go lay code), all nontrivial perfect binary codes h a ve d = 3 , n = 2 m − 1 . A permutation π acts on a c odew ord by permu ting th e coordin ates. A pair ( π, z ) acts on a co dew ord x as ( π , z )( x ) = z + π ( x ) . T wo cod es are equiva lent if th e a ction o f such a p air on the codewords of one co de produ ces the co dew ords of the other . The set o f all such pairs that m ap a cod e on to itself form the autom orphism gr oup of the co de. A Steiner system S ( t, k , v ) is a set o f v p oints together with a collection o f blocks , each consisting of k points, such tha t any t points o ccur in a unique block. Th e Steiner systems S (2 , 3 , v ) and S (3 , 4 , v ) are called Steiner triple systems and Steiner q uadruple systems , respec ti vely , or STS( v ) an d This work was supported in part by the Graduat e School in Electronics, T elecommunic ation and Automation and by the Academy of Finla nd, Grant Numbers 107493 and 110196. I. Y u. Mogilnykh and F . I. Solov’e v a are with the Sobole v Institute of Mathemat ics and Novosibi rsk State Uni ve rsity , Novosibirsk, Russia (e-mail: i vmog84@gmail .com, sol@math.nsc .ru). P . R. J. ¨ Osterg˚ ard and O. Pottone n are with the Depart ment of Com- municati ons and Networki ng, Helsi nki Uni versi ty of T echnology TKK, P . O. Box 3000, FI-02015 T KK, Finland (e-mail: patric.oste rgar d@tkk.fi, olli.pot tonen@t kk.fi). SQS( v ) fo r short. If C is a 1 -perf ect cod e of length n and x ∈ C , then th e blocks { s upp( x − y ) : d H ( x , y ) = 3 , y ∈ C } form an STS( n ) , called th e neig hborhoo d STS of x . Similar ly , if C is an extended 1 -p erfect cod e, th en each x ∈ C has an neighb orhood SQS with the blo ck set { supp( x − y ) : d H ( x , y ) = 4 , y ∈ C } . The block graph of an S ( t, k , v ) has the blocks of the design as vertices, with e dges incident to in tersecting blocks. The minimu m dista nce graph of a code with minimu m distance d has th e codewords as vertices and edg es between codewords with Ham ming distance d . In the re st of the p aper we co nsider such minimum distan ce graph s. No te that the distance between co dew ords is th en the distan ce betwee n the correspo nding vertices in the g raph; this is not to be confu sed with th e Hamming distance. Phelps and LeV an [1] asked whether 1-pe rfect co des with isomorph ic minimum distance grap hs ar e always equ i valent, and this q uestion was answered in the affirmati ve by A vgusti- novich [2], building on earlier work by A vgustinovich an d others [3], [4]; in fact, the r esult was annou nced already in [3] for lengths n ≥ 31 , but without details. W e star t o f f in Section II b y finalizing a proo f that extended 1-perf ect codes with isomorph ic minimum distance graphs are equiv alent for n ≥ 256 . The detailed treatment in th e r est o f the paper makes it po ssible to handle codes of shor ter leng ths. W e p rove i n Section III the strong er result that any e xtended 1- perfect c ode can be reco nstructed from its minimum distance graph, and , in Section IV, show how this im plies an analogou s result for 1 -perfect codes. In Section V we prove that the automor phism g roups o f these codes ar e isomo rphic to the automor phism gr oups of their minimu m distance graph s for lengths n ≥ 15 . Section VI concludes the paper . I I . C O D E I S O M E T RY A N D E QU I V A L E N C E A bijection I : C 1 → C 2 is called an iso metry if d H ( x , y ) = d H ( I ( x ) , I ( y )) for all x , y ∈ C 1 . Mo reover , such a m apping is a wea k isometry if d H ( x , y ) = d iff d H ( I ( x ) , I ( y )) = d , where d is the minim um distance of the codes C 1 and C 2 . W e m ay now rephr ase the qu estion b y Phelps and L eV an [1] in the defined terms: Are wea kly isom etric 1-perfect codes always equivalent? Th e idea of the proof completed in [2] is to comb ine a proof that weak ly isometric such code s are isometric with a pr oof (from [3], [4]) that isometr ic such codes are eq uiv alen t. W e may act analogo usly for exten ded 1-perf ect codes, and use a r esult from [5] that isometr ic such codes are equiv alent for leng ths n ≥ 256 . Th en it only rem ains to prove that weakly isometric codes are isometric, wh ich can b e done for arbitrar y lengths. Theor em 1: W eak ly isome tric extended 1 -perfect codes are isometric. Pr oo f: W e show that one is able to dedu ce the Hammin g distance between any two codewords, given the minimu m dis- tance graph. Consider an arbitrary code word x . The codewords y with d H ( x , y ) = 4 are giv en b y the minimum distance graph. Having iden tified all co dew ords y with d H ( x , y ) ≤ i , we nee d to d istinguish between the cases d H ( x , z ) = i + 2 and d H ( x , z ) = i + 4 for a codeword z in order to proceed with 2 induction . If z has a neighbo ur v with d H ( x , v ) = i − 2 , then d H ( x , z ) = i + 2 . All re maining co dew ords z with d H ( x , z ) = i + 2 ha ve  i +2 3  neighbo urs that ar e at Hammin g distan ce i from x , whereas those codewords z with d H ( x , z ) = i + 4 have at most  i +4 3  / 4 such neigh bors (con sider respectively the triples and qu adruples of supp( x − z ) in the neighb orhoo d SQS of z ). For i ≥ 4 we h a ve  i +2 3  >  i +4 3  / 4 . Theor em 2: W eak ly isome tric extended 1 -perfect codes are equiv alent for leng ths n ≥ 256 . Pr oo f: Follo ws f rom Th eorem 1 an d [5]. I I I . R E C O N S T R U C T I N G E X T E N D E D 1 - P E R F E C T C O D E S A clique in a graph is a set o f mutually adjacen t vertices. The idea o f utilizing maximu m cliques in r econstruction has earlier been u sed by Spielma n [6]; see also [7]. It follows from a resu lt by Rands [ 8] that the max imum cliques in the block graph of a Steiner system can be used to identify th e points of the design wh enever the n umber of points ( v ) exceeds a certain value that d epends only on the par ameters k a nd t . Unfortu nately , the b ound d eriv ed in [8] for the threshold value is too la rge f or th e smallest cases that we want to hand le, so we need to carry ou t a mo re d etailed tr eatment. In the p reparation fo r a recon structibility proof for exten ded 1-perf ect codes, Theo rem 3, we prove three lemma ta. Lemma 1: The codew ords with Hammin g distance 6 can be recogn ized from the min imum distance graph of an extended 1 -perf ect code. Pr oo f: Follo ws f rom the pr oof of Theorem 1 . Lemma 2: If Q is a cliq ue in the b lock graph of an SQ S( v ) , v ≥ 16 , such that th ere is no poin t that occu rs in ev ery blo ck of Q , then | Q | < ( v − 1)( v − 2) / 6 . Pr oo f: Consider a cliqu e Q such that no po int occurs in ev ery block of Q . First note that any pair of po ints is contained in (v -2)/2 block s of an SQS(v) and ther efore in at mo st (v-2) /2 blocks of Q. W e consider the size o f a no nempty Q in three separate cases. 1) Ther e is a point x th at occu rs in every block of Q except one: Assume that x 6∈ { a, b, c, d } ∈ Q . Since Q is a cliq ue in the block g raph, e very b lock of Q co ntaining x contains at least one of the pair s { x, a } , { x, b } , { x, c } , { x, d } . From the fact that each pair occurs in at most ( v − 2) / 2 blocks, it follows that | Q | ≤ 4( v − 2) / 2 + 1 = 2 v − 3 . 2) Ther e is a pair of points { x, y } that inter sects e very block of Q , but no point occur s in | Q | − 1 b locks: There are at least tw o blocks th at do no t co ntain x ; let B 1 and B 2 be two such blocks. Sin ce x 6∈ B 1 and x 6∈ B 2 , by the assumption y ∈ B 1 ∩ B 2 . If | B 1 ∩ B 2 | = 2 , B 1 = { y , a, b, c } and B 2 = { y , a, d, e } with d istinct elements a, b, c, d, e . Any block th at contain s x but no t y must co ntain either a (there are at most ( v − 2) / 2 such blocks), o r b an d d (at most 1), or b and e (at most 1), o r c and d (at mo st 1) , or c an d e (at mo st 1) , so ther e a re at most ( v − 2) / 2 + 4 blocks that con tain x b ut no t y . On the o ther han d, if | B 1 ∩ B 2 | = 1 , then B 1 = { y , a, b, c } and B 2 = { y , d, e, f } , a nd we get at mo st 9 blocks containing x and intersecting B 1 and B 2 , on e for each pair with one elem ent taken fro m { a, b, c } and th e other from { d, e, f } . An upper bound for the num ber of blocks containing x but not y is then max { 9 , v / 2 + 3 } = v / 2 + 3 as v ≥ 16 . By the same argumen t there are at m ost v / 2 + 3 blocks that contain y but not x . Finally , at most ( v − 2) / 2 bloc ks contain b oth x and y , so | Q | ≤ ( v − 2) / 2 + 2( v / 2 + 3) = 3 v / 2 + 5 . 3) For every pair o f points there is a blo ck of Q th at do es not intersect the pair: (Note that in this c ase no p oint occur s in | Q | − 1 b locks). Any pair o f points m ay occur in at most 4 blocks of Q , since Q c ontains a block B that do es n ot intersect the pair , and each block th at c ontains the pair also contains a point of B . T ake any po int x . Th ere are at least two blo cks that do not con tain x . If these blocks inte rsect in two points, say B 1 = { a, b, c, d } and B 2 = { a, b, e, f } , we get tha t eac h block containing x must contain a (at most 4 blocks), b (at most 4), c an d e (at most 1) , c an d f (at mo st 1 ), d and e (at most 1), o r d an d f ( at most 1), gi ving a total of a t most 1 2 blocks. Similarly , for the situation with one p oint in the in tersection, B 1 = { a, b, c, d } , B 2 = { a, e, f , g } , we get a n up per bou nd of 4 + 3 2 = 13 blocks. T hus any po int occ urs in at mo st 1 3 b locks. If each po int occ urs in at most 8 block s, we ha ve | Q | ≤ 1 + 4(8 − 1) = 29 as a ny block m ust intersect a giv en b lock. Assuming that there is a point x o ccurring in at least 9 block s, a nd considerin g blo cks containing x and in tersecting a b lock B that does n ot contain x , we g et by the p igeonho le principle that some pair { x, y } with y ∈ B must occur it at least 3 b locks. Now consider a b lock { x, y , a, b } ∈ Q . There are at most 2 · 13 − 3 = 23 blo cks that intersect { x, y } . By considerin g blocks in tersecting three b locks { x, y , a, b } , { x, y , c, d } , and { x, y , e, f } , one obtains that a block th at does no t intersect { x, y } must contain one o f 2 3 = 8 sets, { a, c, e } , etc. Mo reover , since no two bloc ks may intersect in th ree p oints, the ir total num ber is at most 8. Summing up the numb er of blo cks that intersect { x, y } and those that do no t, we get tha t | Q | ≤ 23 + 8 = 31 . Combining the results above, we con clude that | Q | ≤ max(2 v − 3 , 3 v / 2 + 5 , 31) < ( v − 1)( v − 2) / 6 when v ≥ 16 , and the result follows. Lemma 3: For v ≥ 16 , an SQ S( v ) c an be reconstru cted (up to isom orphism) from its block grap h. Pr oo f: The blocks that co ntain a specified point form a clique of size ( v − 1 )( v − 2) / 6 , and the clique corr esponds to the b locks of a d eriv ed STS( v − 1) . By Lem ma 2, other types of cliques cannot be this large, so an SQS( v ) ca n be r econstructed from its blo ck g raph by fin ding max imum cliques and identify ing them with poin ts. W e h av e now m ade all p reparation s for the main result. Theor em 3: An extended 1 -perfe ct code can b e recon- structed (up to equiv alence) fro m its minim um distance gra ph. Pr oo f: For lengths n ≤ 8 the claim is trivial as these codes are unique , so we assume that n ≥ 16 . 3 Identify an ar bitrary vertex with the all-zero codew ord 0 . By L emma 1 we can construct th e blo ck grap h of the neighbo rhood SQS o f 0 , an d by Lemm a 3 the neighb orhood SQS itself. Now we h a ve reconstructe d all co dew ords with weight at most 4 . The codewords with weight 6 can be recogn ized b y Lemma 1 an d reconstructed as follows. A ssume that x is su ch a codeword. I f x i = 1 , then x has  5 2  = 10 neighb ors y with y i = 1 , wt( y ) = 4 ; if x i = 0 , then an upp er bound f or th e number o f suc h neighb ors is gi ven b y th e maximu m size of a co de of length 6, constant weig ht 3, an d minimu m distance 4, which is 4. W e proceed with indu ction on the weight of co dew ords. Assume that we have reconstructed all codewords with weight at most w , w ≥ 6 , and let x be a codew ord with weigh t w . For e ach coor dinate r there is a set { i, j, k } ⊂ supp( x ) such that { i , j, k , r } is n ot a block o f the neighb orhoo d SQS of x . Accord ingly , x has three d istinct neighb ors v , y , z such that { r , i, j } ⊂ supp( x − v ) , { r , i, k } ⊂ supp( x − y ) , and { r, j, k } ⊂ supp( x − z ) . Each of v , y , z h as weigh t at most w , and hence those codewords are kn own. Furth ermore, { r } = supp( x − v ) ∩ supp( x − y ) ∩ supp( x − z ) . (1) Consider th e block grap h o f the neighbo rhood SQS of x , and the maxim um cliques fro m Lemma 3. U sing (1) we can recognize th e clique co rrespond ing to th e coordinate r . Now we know which neig hbors of x differ from x in that coordin ate. By repeating this for every r we can reconstruc t the codew ords corr esponding to the n eighbor s of x , and the result f ollows as each cod e word y tha t is n ot the all-one word has a neigh bor of weight wt( y ) − 2 . Cor ollary 1 : W eakly isom etric e xtended 1 -perfect codes are equiv alent. I V . R E C O N S T RU C T I N G 1 - P E R F E C T C O D E S W e will handle the prob lem of re constructing a 1 -perf ect code from its minimum d istance graph by reducing it to the problem of r econstructing an extend ed 1 -perfect code f rom its minimum distance graph . Lemma 4: The codew ords with Hammin g distance 4 can be recogn ized from the minimum distance graph of a 1 -perfect code. Pr oo f: If cod ew ords x , y have Hamming distance 4 , th en their neig hborho ods intersect in  4 2  = 6 vertices, since f or any two c oordinates of supp( x + y ) there is one neigh bor of x which differs fro m x in tho se coor dinates. If the codew ords are at distance 6 , size of the intersectio n of their neigh borhoo ds is at most 4 (attained by a Pasch configur ation), and for o ther distances the neig hborho ods d o not intersect. Theor em 4: A 1 -per fect code can be reconstructed (up to equiv alence) from its minim um distance gr aph. Pr oo f: Add new e dges b etween codew ords with Ham- ming distance 4 (Lemma 4). Th is giv es the minimu m distance graph for the extend ed co de (obtained by addin g a parity coordin ate). Using Th eorem 3 we can reconstruct the extend ed code. All codewords co nnected by n e w edges in the first step of the proof d iffer in th e par ity coo rdinate, wh ich can thereb y be detected . By pun cturing in the parity coo rdinate we get th e 1-perf ect c ode. Cor ollary 2 : W eakly isometric 1-perfect cod es a re equiv a- lent. V . A U T O M O R P H I S M G RO U P S The fact th at the auto morphism grou p of a 1- perfect code is isomorp hic to the autom orphism g roup of its m inimum distance graph (for length s n ≥ 15 ) fo llow implicitly fro m [2], [ 3], [4], and the an alogous result for extend ed 1-perf ect codes (for leng ths n ≥ 256 ) from [5] c ombined with Theorem 1. T he cu rrent study enables d irect and concise p roofs of these facts ( expanded to lengths n ≥ 16 for extended co des). Theor em 5: The automorp hism group of an extended 1 - perfect cod e o f length n ≥ 16 is isomorph ic to th at of its minimum distance graph . Pr oo f: The automo rphisms of the code c an be mapp ed to autom orphisms of the graph in th e obvious fashion. Using the con struction of Theor em 3 , this h omomor phism can be in verted; mo re specifically , we get an automorph ism of the code by check ing h ow α ∈ Aut( G ) maps the co dew ord 0 an d the c liques used in the con struction. The result for 1-per fect codes now fo llows e asily . Theor em 6: The automorph ism gro up of a 1 -perfe ct code of length n ≥ 15 is isomor phic to that o f its m inimum d istance graph. Pr oo f: W e use the co nstruction from Theorem 4. Assume that exten ding a 1 -p erfect code C with a p arity coo rdinate yields the code C ′ . Now Aut( C ) is the subgro up of Aut( C ′ ) that stabilizes the parity c oordinate. Similarly , if G is the minimum distanc e grap h of C an d G ′ the g raph con structed in Theorem 4, Aut( G ) is th e subg roup of Aut( G ′ ) that stabilizes the new edges setwise. By Theor em 5 these sub groups are isomorph ic, and hence Aut( C ) ∼ = Aut( G ) as well. V I . C O N C L U S I O N S The result that 1 - perfect and extended 1 -perf ect codes can be reconstruc ted from their minimum distance g raphs is not only of theoretical interest but also has practical implica tions. Sev eral methods have been used for deciding equ i valence of (extended) 1 -perf ect codes [1], [9], [10]—the mo st straightfo r- ward meth od o f representing the codes as gr aphs an d de ciding isomorph ism of these graph s is rath er inefficient [10]. The results obtained imp ly that this problem red uces to determining whether their minimum distance gr aphs are isomor phic. R E F E R E N C E S [1] K. T . Phelps and M. LeV an, “Switch ing equi v ale nce classes of perfect codes, ” Des. Codes Crypt ogr . , vol. 16, pp. 179–18 4, 1999. [2] S. V . A vgustino vich, “Perfect binary ( n, 3) codes: The structur e of graphs of minimum distances, ” Discr ete Appl. Math. , vol. 114, pp. 9–11, 2001. [3] S. V . 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Comp. , v ol. 73, pp. 2075–2092, 2004. [8] B. M. I. Rands, “ An extension of the Erd ˝ os, Ko, Rado theorem to t - designs, ” J . Combin. Theory Ser . A , vol. 32, pp. 391–395, 1982. [9] P . R. J. ¨ Osterg ˚ ard and O. Pottonen , “The perfec t binary one-error- correct ing codes of length 15 : Pa rt I—Cla ssificatio n, ” submitted for publica tion. Preprint at arXi v:0806.2513v1. [10] K. T . Phelps, “ An enu meration of 1-perfect bina ry codes, ” Austr alas. J. Combin. , vol . 21, pp. 287–298, 2000.