A Computer Search for N1L-Configurations

A Computer Searc h for N 1 L -Configuratio ns Martin Do wd MartDowd@ao l.com Abstract In an earlier pap er the author defin ed N 1 L configurations, and stated a conjecture concerning them whic h w ould lead to an impro vement b y a constant factor to th e sphere-p ac king b ound for linear double error correcting co des. Here a computer searc h is presented, in an effort to gather evidence on the conjecture. 1 In tro duction In [4], some configurations in binary linear co des, first co nsidered in [3], were further considered. In particular , a conjecture was made, o n the maximum num b er of rows o f a co nfiguration called an N 1 L configuratio n. If true, the conjecture w o uld yield an improv ement b y a consta n t factor to the cur rently kno wn b ound (spher e packing b ound) on the maximu m length n of a double error cor recting binary linear co de of redundancy r . As in [4], the following co n ven tio ns will b e use d. Let F q denote the finite field of or der q , for a prime power q . A binary co de of length n is a subse t of the vector s pace F n 2 ov er F 2 . Such a co de is linear if it is a subspace; if k is the dimension the redundancy r is defined to b e n − k . Co dewords are g e nerally denoted v , w , e tc. Positions a re gener ally deno ted i , j , etc., with 1 ≤ i ≤ n . As usual, v i denotes the element of F 2 in po sition i of v . A vector in F n 2 will b e called a bit vector, of length n . Generator matrices will b e considere d to b e k × n , and pa rit y chec k matr ices n × r . A vector v may b e ident ified with its “s uppor t” { i : v i = 1 } . The Hamming weigh t | v | of a vector is the c a rdinality of the s uppor t. An N 1 configuratio n is defined to b e a set S of weight 5 vectors, of minimum dis tance 6, where there is a weigh t 5 “a nc hor ” vector v and a p osition i ∈ v , such that for w ∈ S , i ∈ w and | v ∩ w | = 2. An N ′ 1 configuratio n is a configur ation of triples (weight 3 v ectors), whic h ma y be obtained fr o m a n N 1 configuratio n by deleting the p ositions of v . Again a s in [4] a par tial linear space is defined to b e a n incidenc e matrix (matrix over F 2 ), wher e tw o columns are incident to at most one r ow. Note that the requir emen t may equally b e stated as, t wo rows are incident to at most one co lumn; the requir emen t is that no “rectangle” of 1 ’s o ccur. The following is observed in [4]. Theorem 1 An N ′ 1 c onfigur ation is a p artial line ar sp ac e, of c onstant r ow weight 3, to gether with a p artition of the r ows into 4 or fewer p arts, such t hat in e ach p art the r ows ar e disjoint. Note that a n N 1 configuratio n from which an N ′ 1 configuratio n arises can b e determined from the partition. Clearly the ma xim um num b er o f rows in an N 1 configuratio n o f leng th n (equiv alently an N ′ 1 configuratio n of length n − 5) is 4 ⌊ ( n − 5) / 3 ⌋ . This b ound is achiev ed in a 3-( n ,5 ,1) design; these e x ist for n = 4 m + 1 where m ≥ 1 (see [1], theorem 6 .9 ). An N 1 configuratio n is said to b e an N 1 L configuratio n if the linear span of its rows and the a nc ho r vector has minimum weight 5, and an N ′ 1 L configuratio n is an N ′ 1 configuratio n whic h a rises from an N 1 L configuratio n. Let N 1 L ( n ) denote the ma xim um num b er of rows in an N 1 L configuratio n in a linear double error correcting c ode of leng th n . The following conjectur e was made in [4 ]. Conjecture 1 N 1 L ( n ) is ≤ c 1 ( n − 5 ) almost everywher e, for a c onstant c 1 smal ler t han 4 / 3 . It w as also shown that the co njecture yields an upp er b ound on the length of a linear double error correcting co de of redundancy r , b etter by a constant factor than the sphere packing b ound. A computer search show ed that for r ≤ 8, the co njecture holds with c 1 = 2 / 3. Howev er, an N 1 L ( n ) configuratio n with n − 5 = 60 and 44 rows was found in a cyclic co de. 1 In this pap er, a computer sea rch is car r ied out for all N ′ 1 L configuratio ns with n − 5 < = 18 and r < = 14, where r is the n umber of r ows. This ser v es t wo purp oses. First, the existence of N 1 L configuratio ns for small r, n pairs is determined. Second, data is provided for p ossibly infer ring rules for a n inductive pro of o f the conjecture . It thus repres en ts an a ttempt to achiev e a goa l s tated in [6], tha t “o ccasio nally a practica l algorithm a nd there by a classifica tio n result is obtained.” W e note here that there is a “ replication” ar gument which shows that if a ratio r / ( n − 5 ) is a c hieved then it is achiev ed infinitely often. Theorem 2 L et M b e an incidenc e matrix whose r ows ar e divide d int o classes M i , 0 ≤ i ≤ 3 . F or m ≥ 1 let M m i have m c opies of M i down t he diagonal. L et M m b e the matrix whose classes ar e the M m i . a. if M is an N ′ 1 c onfigur ation then M m is; and b. if M is an N ′ 1 L c onfigur ation then M m is. Pro of: Part a follows readily using theorem 1 and is left to the r eader. F o r pa rt b, theor e m 5b of [4] w ill be used. By a “section” of M m we mean the rows or c o lumns of one of the replications. Consider a sum of rows of M m . If the weight o f the sum is zero in a column section then the cor resp onding rows can b e deleted from the sum. If there is more than o ne column se ction where the sum is nonze r o then the total weigh t o f the sum is alrea dy at least 6 . Other wise, the requirements of theorem 5b ar e satisfied in the no nz e ro section. 2 Outline of sea rch pro cedure A search for N ′ 1 L configuratio ns can b e car ried out inductively , computing the isomorphism class es with r rows fro m those with r − 1 rows. A b ound can be impos ed on the num b er of columns c ( n − 5 in the preceding section), since o nly co nfig urations with larg e r /c are o f interest. In this paper , the b ound on c is 18, and the maximum v alue of r co nsidered is 14 (although se e section 4). General pro cedur es for isomorphism tes ting of incidence matrices exist, including Brenda n McKay’s “Nauty” [11], a nd [8]. F or one use of Nauty in co ding theory , see [5]. Discussions of using iso mo rphism detection pro cedure s in se a rching for combinatorial configura tio ns can be found in the literature, for example [2], [7], [10], and [12]. One common metho d is to co n vert each configura tion a s it is gener ated to a “ca no nical representative” of its equiv alence class under isomorphism. In this pa p er, sp ecialized metho ds ar e used to r educe the matrix to one of several “pa rtial canonical- izations”. E ach such is partitioned int o 4 row class e s and 15 column cla s ses. Sta nda rd metho ds are then used to canonicaliz e the pa rtitioned pa r tial canonica lizations, and the lexico graphically highest such is used. Source co de may b e requested by email from the a uthor, and the description her e omits v arious details. An N ′ 1 configuratio n is assumed to b e given as an incidence matrix M , and a partition into 4 or fewer parts of the rows. It may b e ass umed that the rows of a part a re consecutive; in some co n texts missing par ts are considered to b e empt y parts. The par ts are also ca lled “row classes ”. Num b ering the pa r ts from 0 to 3, a column may b e given a type, namely the function f ma pping { 0 , 1 , 2 , 3 } to { 0 , 1 } , where f ( i ) = 1 iff the column has a one in part i . The type may be considered a s a bit string of length 4, and denoted b y a hexadecimal dig it 0- F (bit 0 being the low o rder bit). Only co nfigurations with no columns of type 0 ne e d b e co ns idered. W riting the nonzer o t yp es in the order F7BDE359 6 A C12 48, a configura tion has a “signature” , the 1 5-tuple o f natura l num b ers which in each po sition gives the num b er o f co lumns of the c orresp onding type. The sy mmetric gr oup S 4 acts on the row classe s, a p ermutation α ∈ S 4 being co nsidered as“ moving” the class in p osition i to po sition α ( i ). This induces an action o n the column types, namely T 7→ α [ T ] wher e T is the supp ort (note that, considering T as a characteristic function, T ′ ( p ( i )) = T ( i ) wher e T ′ is the image). Considering the signa ture to be a function σ from { T } to the na tural num b ers, α acts o n the signatures b y mapping σ to σ α , where σ α ( α [ T ]) = σ ( T ). Given a signature σ , the “canonica lized sig nature” σ c is defined to be the lexicog raphically g reatest among the σ α . As will b e see n, it is useful to determine this, and also the r ight coset Gr of elements of α for which σ α = σ c . Thes e can r eadily b e determined by tr y ing all 2 4 po ssiblities. Indeed, since sig nature canonicaliza tion is of s econdary cost, an efficient implmentation of this metho d would undoubtedly suffice. As will b e seen, one refinement was made. F or c a nonicalizing the signature it is us e ful to have a libr ary of ro utines for co mputing with p ermutations in S 4 . The pe rm utatio ns can b e order e d (a recursive order where the first 6 elements ar e S 3 was used), and tables co ded which apply a permutation given as an index in the order, b y an array r eference. 2 S 4 has 3 0 subgroups [9]. Not a ll of them ca n o ccur as a sta bilizer of a s ignature, but it is simplest to co de tables for all of them, in particular a table of ele ments p er subg r oup index. There are 234 cos ets. A coset may b e re pr esented as a bit vector o f length 24. As noted ab ov e, the right coset for a c anonicalized signature is r eadily computed alo ng with it. A hash table can b e used to obtain the subgr oup index and a coset r epresentativ e fro m the bit vector. T o sp eed up the pro ces s of cano nic a lizing the s ignature, the columns may b e gro uped acco rding to weight, and for each weight , the ca nonicalized signature and right co set determined succe s sively . F o r a given weigh t, only p ermutations in the stabilizer of the higher w eig h t columns need b e considered. F or the weight 3 c o lumns, the weights may b e sorted. Ther e a re 8 po ssibilities S 1 RS 2 RS 3 RS 4 where R is < or = among the sorted siz es; each yields a partition of { 1 , 2 , 3 , 4 } , a nd thereby a subgro up of S 4 , consisting of the pro duct of the symmetric gr oups ac ting on the par ts. F or the rema ining weights, all p ossibilities within the stabilizer so far are tried. It might b e p ossible to acheiv e a sp eed up in the c ase o f w eig h t 2 vectors with the full gr o up acting, and this is certainly true in the case of weight 1 vectors; but this was o mitted. The columns o f type f for some f will b e ca lled a column class. In a ddition to req uiring an incidence matrix to hav e the rows of each row class contiguous, the columns of ea ch column class will be required to be also . F urther, the column cla sses are re q uired to b e in the order g iven ab ov e. A matrix M with a such a r ow partition consists of 60 = 4 × 15 blo cks, one for ea c h row cla ss and column cla ss (some of blo cks may be empty , i.e., have 0 rows o r 0 columns). Suppo sing a metho d is specified for sp ecifying the ca nonicalization M cf of an N ′ 1 configuratio n M when the row classes ar e fixed, the canonica lization of M c of M may be defined as the lexicogr aphically highest of the M cf α , where M α are those matr ices o btained from M by p ermuting the row classe s , to yield the canonicalized s ignature (i.e, where α ∈ Gr where Gr is as a bove. When a pplying a p ermutation o f the row classes , the columns may be p ermuted in any manner to ob ey the column restr ic tio n. In obtaining M cf , only p ermu ta tions which preserve the blo cks need be cons ide r ed. A t first, the a utho r intended to wr ite a canonicaliza tio n pr o c e dure from scratch, under the b elief that this would b e faster a nd thus more likey to c o mplete. How ever, Nauty ha s a reputation for being fast; it per mits sp ecifying an initial par tition, in this ca se into blo c ks as ab ov e; and a preliminary version using it would p ermit debugging the other c o de and provide a chec k. A version using Nauty was thus coded. The gener ation of the configura tions pr o ceeds in stages, for r increasing up to some maximum v alue, where the n umber of co lumns is limited to some maximum v alue. At the b eginning o f the stage for r , the r − 1 row configur ations are pack ed in an array; the rest of memory is used for a hash table for the r row configuratio ns. At the end of a stage, the ha sh table is pack ed down to the b eginning of memory . Each r − 1 row configura tion is unpack ed, and the span gener a ted. F or each part of its row partition, a row is a dded in every po ssible wa y . F o r each r e sulting configuration, a check is made whether it is N 1 L . If so its signature is computed and canonicalized, and M c is obta ined as desc ribed ab ov e. M c is added to the hash ta ble if it is not alrea dy in it. 3 Results of searc h Initially the prog ram was run with maximum v alues of 10 and 15 for r and c . This ran in 21 sec onds, so the limits w ere raised to 1 2 and 1 8. This run found tha t co nfigurations with r = 12 and c = 16 exist. The limit on r was increase d to 1 4. The time for the run with these v alues w a s 228 min utes. F or this pa per, further increases to the limit were omitted, as this would hav e required a dditional work. F or example, the input graph to Na ut y ha s 14 +18=3 2 no des. If the g raph has more than 32 no des the rows of the adjacency matrix no longer fit in a word, and Nauty’s execution time w o uld increase. Aga in, though, see s e c tion 4. F or this pap er, with the limits of 14 and 18 , the Naut y version is the final version. T able 1 shows the n umber of iso mprhism clas s es of N 1 L configuratio ns with r rows and c co lumns, for 2 ≤ r ≤ 14 and 5 ≤ c ≤ 18. F rom this, the v alue of c 1 is la rger than 2 /3. Indeed, wr iting c min for the smallest c for which configuratio ns exist, for even r with 8 ≤ r ≤ 14, c min increases by 2 as r do es. This suggests that c 1 is at leas t 1. In [4] the following observ ations ar e made. • An N ′ 1 L configuratio n w ith 2 flags in each column is a cubic graph. • Such a cubic graph must b e triangle free. • Two o f the 6 cubic g raphs on 8 vertices a re tria ngle free. 3 r 5 6 7 8 9 10 11 12 13 14 15 16 17 18 2 1 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 3 2 3 0 0 0 0 0 0 0 0 0 4 0 0 0 2 10 11 5 5 0 0 0 0 0 0 5 0 0 0 0 0 12 42 38 24 8 6 0 0 0 6 0 0 0 0 0 0 23 15 3 257 2 13 108 48 14 9 7 0 0 0 0 0 0 0 30 583 1635 1 927 126 2 607 223 8 0 0 0 0 0 0 0 5 13 2442 118 13 18982 16 261 9187 9 0 0 0 0 0 0 0 0 1 30 9153 87725 20069 0 219285 10 0 0 0 0 0 0 0 0 0 1 17 0 26957 6529 26 222066 5 11 0 0 0 0 0 0 0 0 0 0 0 840 486 2 4 4677339 12 0 0 0 0 0 0 0 0 0 0 0 6 2 513 858 36 13 0 0 0 0 0 0 0 0 0 0 0 0 24 3372 14 0 0 0 0 0 0 0 0 0 0 0 0 0 100 T able 1: Iso mo rphism class counts • Among the N ′ 1 L configuratio ns with r = 8 and c = 12 , bo th tria ngle-free cubic gra phs o ccur. F rom the table, there are 5 configura tions with r = 8 and c = 12. It is r eadily verified that 3 of these ar e the cube , and 2 are the other po ssible cubic gra ph. F or all 5 config urations the list o f par tition sizes is 2 ,2,2,2. 4 A second searc h Since the “gcc” compiler for the “x8 6” pro cessor supp o rts a 64 bit “ long lo ng” type, a version of Nauty can be used wher e the rows of an a djacency matrix fit in a “ long long” . A pa rtial sear c h was conducted using this feature . Star ting from the configur ations with r = 13, and c = 17 or c = 18, only ex tensions by up to t wo columns were co nsidered, and o nly the minimum t wo c v alues fo r each r used as input to the next stage. This search yields an upp er b ound o n c min . The r esults are given in the following table. After r = 19 the program a bor ted due to insufficient memo ry . r 15 16 17 18 19 bo und 19 19 21 22 22 The res ults of this search suggest tha t the status of conjecture 2 is unclear. The ratio r/c increa s es, but so slowly that the co mputer searches done here don’t g iv e any clear indication of its limiting v alue. F or example, it is still op en whether it can exceed 1. It sho uld als o b e noted that the b est known linear do uble error correc ting co des do no t contain N ′ 1 L configuratio ns with v alues of r/c as high as those of configuratio ns found here (see [4]). T opics for further resear ch clearly include the following. • An N 1 L configuratio n yields a linear double error correcting co de containing it, with n = c + 5 and k the rank of the config uration, augmented with v . This co de may not b e very go o d. A “go o dness” measure o f interest is  1 + n +  n 2  / 2 ( n − k ). • Go o d co des for a given configur ation, and configur a tions for a given go o d co de, s ho uld be mo re exten- sively inv es tigated. • Metho ds for obtaining N config urations from N 1 configuratio ns (for exa mple using classical groups) might b e of in ter est. • Although omitted here, determination of a ll configuratio ns up to r = 16 and c = 19 can b e pr obably be achieved by the methods pre sen ted here. 4 References [1] T. Be th, D. Jungnick el, and H. Lenz. Design The ory . Ca mbridge Universit y Press , 1993. [2] L. Chen. Gra ph isomor phism and identification matr ices: Parallel algor ithms. IEEE T r ans. Par al lel Distrib. Syst. , 7:308 –319, 1996. [3] M. Dowd. Questions r elated to the E rdos-Tura n co njecture. SIA M Journal in D iscr ete Mathematics , 1:142– 150, 1 988. [4] M. Dowd. Co nfigurations in binar y linear co des. 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