Pavel Rytí. November 22, 2011 Discrete Math Seminar - Simon Fraser University

Transcription

1 Geometric representtions of liner codes Pvel Rytí Deprtment of Applied Mthemtics Chrles University in Prgue Advisor: Mrtin Loebl November, 011 Discrete Mth Seminr - Simon Frser University

2 Bckground Liner code Liner code C of length n nd dimension d over eld F Liner subspce of dimension d of vector spce F n W C (x) := c C x w(c), w(c) number of non-zero entries Puncturing C long S S {1,..., n}, C/S = {(c i i / S) n i=1 c C} The puncturing C long S mens deleting the entries indexed by S from C. C/{1} = {(c, c 3,..., c n ) (c 1, c,..., c n ) C}

3 Bckground Motivtion Incidence mtrix A = (A ij ) of grph G A ij := { 1 if vertex v i belongs to edge e j, 0 otherwise. e j v i 1 The cycle spce C of grph G is the kernel of A over GF (). Grph G embedded s one dimensionl simplicil complex in R 3 my be considered s geometric representtion of C. It is useful: For grph G of xed genus, there exists polynomil lgorithm for computtion of W C (x) by Glluccio nd Loebl. This lgorithm uses geometric properties of G nmely embedding on closed Riemnn surfces.

4 Bckground D simplicil complexes Are there geometric representtion of liner codes tht re not cycle spces of grphs?

5 Bckground D simplicil complexes Are there geometric representtion of liner codes tht re not cycle spces of grphs? My representtions will be two dimensionl simplicil complexes. D simplicil complex = {vertices, edges, tringles} Every fce of simplex from belongs to Intersection of every two simplices of is fce of both

6 Bckground D simplicil complexes Incidence mtrix A = (A ij ) of A ij := { 1 if edge e i belongs to tringle t j, 0 otherwise. t j e i 1 Cycle spce ker of over F ker = {x A x = 0}

7 Bckground Liner code C is tringulr representble if: There exists tringulr congurtion s. t. C = ker /S for some set S There is bijection between C nd ker

8 Bckground Liner code C is tringulr representble if: There exists tringulr congurtion s. t. C = ker /S for some set S There is bijection between C nd ker Do we need two dimensionl simplicil complexes? Lets try C is grphic representble if: There exists grph G s. t. C = ker G/S for some set S The clss of liner codes tht re cycle spces of grphs is closed under opertion of puncturing. If C is not cycle spce of grph, there is no such grph G

9 Geometric representtions My results Theorem Let C be liner code over rtionls or over GF (p), where p is prime. Then C is tringulr representble. Theorem If C is over GF (p), where p is prime, then there exists tringulr representtion such tht: if m i=0 i x i is the weight enumertor of ker then m W C (x) = i x (i mod e), i=0 where e = (number of punctured coordintes)/ dim C.

10 Geometric representtions My results Theorem Let F be eld dierent from rtionls nd GF (p), where p is prime. Then there exists liner code over F tht is not tringulr representble.

11 Geometric representtions Work in progress My work immeditely rises the following questions: Which binry codes cn be represented by D simplicil complex embeddble into R 3? (every D complex cn be embedded into R 5 ) Reltion with permnents nd determinnts of 3D mtrices (tensors). Appliction of the geometric representtions to the Ising problem.

12 A trivil one dimensionl code The most trivil cse is code generted by vector tht contins only entries,. C = spn({(,,,..., )}). This code is represented by the following complex: This is tringultion of -dimensionl sphere by tringles such tht there is n ssignment of + nd to tringles such tht every edge is incident with + nd tringle. For every k there exists such tringultion with l tringles, l > k.

13 An exmple of tringulr representtion of C = spn({(,, )}) I ssign to + tringles vlue nd to tringles vlue. Eqution given by the row of the incidence mtrix indexed by ny edge e hs form = C = ker /{ non-green tringles } dim C = dim ker = 1

14 An exmple of tringulr representtion of C = spn({(,, )}) Let p be the eld chrcteristic. The weight enumertor of ker equls W (x) = 1 + (p 1)x k, k is the number of tringles of W C (x) = 1 + (p 1)x (k mod (k 3)) = 1 + (p 1)x 3

15 Representtion of code C over primeeld generted by vector of form ( 1,, 1,,... ) Here I need tht the eld is primeeld. I use tht the dditive group of every primeeld is cyclic. C is generted by vector tht contins only four dierent elements 1,, 1,. 1 = n 1 g nd = n g for some genertor g of the cyclic group. Such code cn be represented by two tringulr spheres interconnected by tunnels t S 1 n n 1-1 S

16 Tringulr tunnel b + + c b

17 Representtion of C = spn({( 1,, 1,,... )}) 1 = n 1 g, = n g, g genertor of the dditive group t S 1 n n 1-1 S

18 Representtion of C = spn({( 1,, 1,,... )}) t S 1 n n 1-1 S The eqution indexed by the edges dierent from the middle empty tringle re 1 1 = 0 or 1 1 = 0. The eqution indexed by the edges of the middle empty tringle re n 1 n 1 = n (n 1 g) n 1 (n g) = 0. So the generting vector belongs to ker

19 Representtion of C = spn({( 1,, 1,,... )}) t S 1 n n 1-1 S The equtions 1 = x nd = x hve obviously unique solutions 1 nd, respectively. The eqution n 1 = n 1 x hs unique solution, since the dditive group hs prime or n innite order. Therefore dim ker = dim C = 1.

20 Representtion of code C over primeeld generted by vector of form ( 1,,..., k, 1,... ) t t t (k-1)k... S S S 1 k This code cn be represented by k tringulr spheres interconnected by tunnels nlogously s in the previous cse. I supposed tht ll i 0. If the genertor of the code contins zeros, I dd to the representtion one isolted tringle for ech zero entry. I cn represent ll one dimensionl codes over primeelds.

21 1 3 (k-1)k 1 k Geometric representtions of liner codes More dimensionl codes Let C be code over primeeld nd let B = {b 1,..., b d } be bsis of C. For every b i I construct representtion bi tht represents the code spn({b i }), s in the previous steps. Let B n = {B1 n,..., B n n } be the tringles of bi tht correspond to the entries of b i. spn({b i }) = ker bi /(non-b n tringles). I deform every bi so tht the tringles B n re in this position. B n 1 B n n n-1... B B n n t t t S S S

22 More dimensionl codes The representtion of C with respect to B is C B = d i=1 b i. B n 1 B n n Bn-1... B n n... b b b 1 d The solutions of equtions indexd by edges of B n tringles re ll liner combintions of solutions of ech prt bi, i = 1,..., d. Theorem ker C B /(non-b n tringles) = C dim ker C B = dim C

23 Weight enumertor, blnced representtions I cn mke the representtion such tht bi w(b i ) = e for ll i = 1,..., d nd e is greter thn the length of C. Such representtion is clled blnced. B n 1 B n n Bn-1... B n n... b b b 1 d

24 Blnced representtion exists I cn pply the following subdivisions, the rst increse the number of tringles by 6 nd the second by

25 Weight enumertor, blnced representtions B n 1 B n n Bn-1... B n n... b b b 1 d Let C be code nd C B be its blnced representtion with respect to bsis B Let c = α b B bb. I dene mpping f : C ker C s B f (c) := α b B b b Combintion degree of c is the number of non-zero α b 's (deg(c)) Let b B, then w(f (b)) = w(b) + e Let c C, then w(f (c)) = w(c) + deg(c)e w(f (c)) mod e = (w(c) + deg(c)e) mod e = w(c) Note tht, w(c) < e for every c

26 Weight enumertor, blnced representtions B n 1 B n n Bn-1... B n n... b b b 1 d if m i=0 i x i is the weight enumertor of C B then W C (x) = m i x (i mod e), i=0 where e = (number of non-b n tringles)/ dim C

27 Tringulr non-representbility My results Theorem Let F be eld dierent from rtionls nd GF (p), where p is prime. Then there exists liner code over F tht is not tringulr representble.

28 Tringulr non-representbility Non-representble code Let GF (4) = {0, 1, x, 1 + x}. The liner code over GF (4) generted by vector (1, x) is not tringulr representble. By n lgebric rgument there is no 0, 1 mtrix with the dimension of kernel equls one nd hving vector of form (1, x,,,..., ) in the kernel.

29 Tringulr non-representbility Thnk you for your ttention