A Note on Matrix Rigidity

Size: px

Start display at page:

Download "A Note on Matrix Rigidity"

Liliana Carson
5 years ago
Views:

1 A Note o Matrix Rigidity Joel Friedma Departmet of Computer Sciece Priceto Uiversity Priceto, NJ Jue 25, 1990 Revised October 25, 1991 Abstract I this paper we give a explicit costructio of matrices over fiite fields which are somewhat rigid, i that if we chage at most k etries i each row, its rak remais at least C(log q k)/k, where q is the size of the field ad C is a absolute costat. Our matrices satify a somewhat stroger property, we which explai ad call strog rigidity. We itroduce ad briefly discuss strog rigidity, because it is i a sese a simpler property ad may be easier to use i givig explicit costructios. Recetly there has bee iterest i givig explicit costructios of matrices which are rigid, i the sese that their rak is high ad remais high whe a few of their coefficiets are chaged (see [Val77], [Gri76], [Raz], ad [PSR] 1 ). It is easy to costruct matrices over ifiite fields, F, such that whe o more that k of the etries of each row are altered, the rak remais at least /k; oecatakeavader Mode matrix, for example. I this ote we give a explicit costructio of a matrix which is slightly more rigid tha such a costructio, for fiite fields, F, ad k larger tha some costat (depedig o the size of the field). These matrices are actually strogly rigid, i a sese that we will discuss later. This paper was writte while o leave from Priceto, at the Hebrew Uiversity. The author wishes to ackowledge the Natioal Sciece Foudatio for supportig this research i part uder PYI grat CCR , ad a grat from the program of Medium ad Log Term Research at Foreig Ceters of Excellece. 1 Pudlak ad Savitzky have show that over the real umbers, a Hadamard matrix of dimesio remais of rak r if o more tha 2 / ( r 4 log 2 r ) of its etries are chaged. Razborov has improved this to 2 / ( r 3 log r ). 1

2 Theorem 1 For ay costat C 1 > 0 there is a costat C 2 > 0 such that the followig holds. Let F be a fiite field of q elemets. Let A be a matrix such that the first /2 rows are the basis of a liear error-correctig code i F of miimum distace C 1.IfB is ay matrix over F with at most k o-zero etries i each row, where k /C 2, the we have rak(a + B) ( logq k +log C 2 k q (q 1) ). I the above theorem it is the log q k as opposed to the log q (q 1) which is of iterest. We have icluded the log q (q 1) toremarkthatwheq with fixed ad k, the costructio does ot completely degeerate. The above matrices satisfy a stroger property, which we call strog rigidity. After provig this theorem we defie ad discuss strog rigidity, because it is somewhat easier to work with ad may be a useful poit of view i costructig other explicit examples. We recall that there are may types of explicitly specified codes, which for ay value of q give a sequece of s ad a code for each such of dimesio /2 ad miimum distace C 1 with C 1 idepedet of q ad (oe ca take Justese codes or Goppa codes, see respectively [vl82] ad [vdgvl88]). Thus the above theorem, for ay value of q ad may of, gives matrices rigid i the above sese. Proof The first /2 rowsofa represet vectors v 1,...,v /2 which are a basis of liear code (subspace) C F of miimum distace C 1.Letb 1,,b /2 be the first /2 rows of B. If the rak of the matrix cosistig of the first /2 rowsofa + B is t, the this is just to say that E w =(w 1,...,w /2 ) F /2 (here 0 is the origi i F )satisfies dim E = 2 t. /2 i=1 w i (v i + b i )=0 Sice E is a (/2) t dimesioal subspace of F /2, we ca fid a o-zero elemet w E of weight r for ay r satisfyig Hammig ball of radius r/2 if /2 q t, or more crudely, ( ) /2 (q 1) r/2 q t. (1) r/2 2

3 But for such a w we have /2 i=1 w i b i C {0}, ad so such a r must satisfy rk C 1. So takig r 0 = C 1 /k 1, equatio 1 caot hold for r = r 0,thatistosaythatt is bouded below by [( ) ] ( ) /2 t log q (q 1) r 0/2 /2 log r 0 /2 q + r 0 r 0 /2 2 log q(q 1) ( logq k+log C 2 k q (q 1) ), for large eough C 2, also assumig k /C 2, where we have estimated ( ) [ ( /2 log q log r 0 /2 q 2 r ) r0 /2 / 0 (r 0 /2) 0/2] r 2 C 2 k log q k. Defiitio 2 C F is (k, t)-strogly rigid if every subspace B spaed by c =dimc vectors b 1,...,b c, each of weight k, has dim(b C) dim C t. (2) Glacig at the proof of theorem 1 shows that matrix A is actually (k, t)-strogly rigid with t = log q k/(c 2 k). Strog rigidity is a simpler property to check, i the sese that we do ot care about the precise relatios betwee a basis for C ad of oe for B. Therefore, as i theorem 1, we hope that it may be easier to work with strog rigidity to give explicit costructios. To compare strog rigidity to the usual otio of rigidity, we ll say that a matrix A is (k, t)-rigid if wheever o more tha k etries i each row of A are altered, the A s rak remais at least t. It is easy to see (ad well-kow) that if A is a radom matrix i a ifiite or sufficietly large field, F, thewithhigh probability 2 A will be (k, t)-rigid with ay t (k) 1/2. While we caot assert so large a value of t for the existece of strogly rigid matrices, we ca obtai existece for a value of t which is iterestig for some of the iteded applicatios of rigid matrices. Theorem 3 Let k, be itegers with k /2 ad (for simplicity) eve. The over ay fiite field, F, there exists a (k, t)-strogly rigid code C F of dimesio /2, for ay t with t k log q k k 4 log q, 4 where q is the size of F. The same holds for ay ifiite field, where i the above equatio we substitute 0 for the two occurreces of log q (), ad where we require strict iequality. If we substitute /10 forthe /8 i the above equatio for t, the ay matrix over F is (k, t)-strogly rigid with high probability. 2 If F is ifiite, the etries of A should be chose from a distributio which has each field elemet weighted sufficietly small. 3

4 Proof This is, as usual, a easy coutig argumet (i the case of ifiite fields oe couts dimesios). For ay subspaces B, C of F with dim B =dimc, dim(b C) dim C t, (3) it is easy to see that there exist v 1,...,v t F ad l 1,...,l t (F ) (i.e. the dual of F as a F-vector space) such that C = { v sp B,v 1,...,v t li (v) =0 i }, (4) where sp deotes the liear spa. To see this, otice that equatio 3 also implies hece we ca choose v i so that dim(b + C) dim C + t =dimb + t ; B + C =sp B,v 1,...,v t adthechoosel i so that equatio 4 holds. Equatio 4 shows that to every code C of dimesio /2 which is ot (k, t)-strogly rigid there correspods (at least oe collectio of) a B geerated by /2 vectors of weight k ad vectors v i ad (dual) vectors l i as above. I the case of F fiite, the umber of vectors of weight k is ( ) (q 1) k k (assumig k /2), ad so the total umber of collectios (B, {v i }, {l i }) is bouded by [ ( /2 (q 1) k)] k q 2t. (5) O the other had, the total umber of subspaces of F of dimesio /2 is (q 1)(q q) ( q q (/2)+1) ( q ) /2 1 ) > ( q (q /2 1)(q /2 q) ( /2) /2. q /2 q (/2) 1 q /2 1 It follows that a (k, t)-strogly rigid C of dimesio /2 exists provided that [ ( /2 q 2 /4 (q 1) k)] k q 2t, i particular, usig ( ) k ( k) k /k k, t k log q k k 4 log q. 4 4

5 Also, if we replace the /8 i the above by /10, the a radomly chose C of dimesio /2 has probability /40 of ot beig (k, t)-rigid. If the field is ifiite, the the dimesio of the set of tuples (B, {v i }, {l i })asaf variety 3 is k 2 +2t. This, of course, is the q limit of log q of the expressio i equatio 5. The same goes through for the set of all C, i.e. subspaces of dimesio /2 if, meaig that oe couts its dimesio as a F variety (ad gets the q limit of log q of the previously derived expressio for fiite fields). Therefore we get the aforemetioed results, except that we eed strict iequality i the iequality for t, ad that whe we use /10 istead of /8, high probability meas occurs as a subvariety of codimesio /40. J. Håstad has poited out to me that the costructio give i theorem 1 will, i geeral, be o better that (k, t)-rigid for t proportioal to log q k/k; thatis,by startig with /2 radom vectors {b i } of small weights, choosig a radom subspace of F of dimesio t, ad choosig /2 radom vectors i this subspace, {u i },oe gets (may) matrices A as i the theorem which are ot (k, t)-rigid for appropriate t of order log q k/k (i.e. where A s first /2 rowsare{b i + u i }). The author wishes to thak A. Wigderso, J. Håstad, ad A. Razborov for helpful discussios ad commets. Refereces [Gri76] [PSR] [Raz] [Val77] D.Y. Grigorjev. Notes of the Leigrad Brach of the Steklov Mathematical Istitute of the Academy of Sciece of the USSR, 60:38 48, Pudlak, Savitzky, ad A. Razborov. Observatios o rigidity of Hadamard matrices. Persoal Commuicatio. A. Razborov. O rigid matrices. Problems of Pure ad Applied Mathematics (Literal Traslatio from Russia). toappear. L.G. Valiat. Graph-theoretic argumets i low-level complexity. Techical Report, Uiversity of Ediburgh, Computer Sciece Report Also i Proc. 6th Symp. o Mathematical Foudatios of Computer Sciece, Tratraska Lomica, Czechoslovakia Throughout this paragraph we will, by a slight abuse of termiology, use F variety to mea algebraic set over F. We do ot require these sets to be irreducible. 5

6 [vdgvl88] G. va der Geer ad J. va Lit. Itroductio to Codig Theory ad Algebraic Geometry. Birkhäuser Verlag, Bosto, [vl82] J. va Lit. Itroductio to Codig Theory. Spriger-Verlag, New York,

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special