Binary codes from graphs on triples and permutation decoding

Transcription

1 Biary codes from graphs o triples ad permutatio decodig J. D. Key Departmet of Mathematical Scieces Clemso Uiversity Clemso SC U.S.A. J. Moori ad B. G. Rodrigues School of Mathematics Statistics ad Iformatio Techology Uiversity of Natal-Pietermaritzburg Pietermaritzburg 3209 South Africa April Abstract We show that permutatio decodig ca be used ad give explicit PD-sets i the symmetric group for some of the biary codes obtaied from the adjacecy matrices of the graphs o ` 3 vertices for 7 with adjacecy defied by the vertices as 3-sets beig adjacet if they have zero oe or two elemets i commo respectively. 1 Itroductio MacWilliams [8] itroduced a algorithm for permutatio decodig employig it mostly for cyclic codes ad the Golay codes. It ivolves choosig appropriate iformatio sets for the code ad fidig a set of automorphisms that satisfies particular coditios (see below such a set beig called a PD-set. This work was supported by the DoD Multidiscipliary Uiversity Research Iitiative (MURI program admiistered by the Office of Naval Research uder Grat N ad NSF grat # Support of NRF ad the Uiversity of Natal (URF ackowledged. Post-graduate scholarship of DAAD (Germay ad the Miistry of Petroleum (Agola ackowledged. 1

2 2 BACKGROUND AND TERMINOLOGY 2 Appropriate iformatio sets ad PD-sets for ifiite classes of codes defied by some strogly regular graphs with the symmetric group as a automorphism group were foud i [6 7]. I [5] we examied the biary codes from a similar class of graphs ot strogly regular but with the symmetric group as automorphism group ad obtaied the dimesio the miimum weight ad some classes of miimum words for these codes: see Sectio 3 Result 1 for a statemet of the mai theorem obtaied. I that paper we aouced that we would iclude the establishmet of PD-sets for the iterestig members of this set of codes i a further paper ad this curret paper addresses that issue. Here we prove the followig: Theorem 1 Let Ω be a set of size where 7 ad is odd. Let P = Ω {3} the set of subsets of Ω of size 3 be the vertex set of the graph A 2 ( with adjacecy defied by two vertices (as 3-sets beig adjacet if the 3-sets meet i two elemets. Let C 2 ( deote the code formed from the row spa over F 2 of a adjacecy matrix for A 2 (. The dual C 2 ( is a [ ( ( ]2 code with I = {{i j } 1 i < j < } {{ 3 2 1}} \ {{ 2 1 }} as iformatio set. The C 2 ( has a PD-set i S give by the followig elemets of S i their atural actio o triples of elemets of Ω = { }: S = {( i( 1 j( 2 k 1 i 1 j 1 1 k 2} where (i i deotes the idetity elemet of S. Note that the size of the PD-set is of the order of 3 as is the legth of the code. The proof of the theorem is give i Sectio 3. I Sectio 4 we metio oe of the other dual codes ad give a PD-set. 2 Backgroud ad termiology Our otatio for desigs ad codes will be stadard ad as i [1]. A icidece structure D = (P B I with poit set P block set B ad icidece I is a t-(v k λ desig if P = v every block B B is icidet with precisely k poits ad every t distict poits are together icidet with precisely λ blocks. The desig is symmetric if it has the same umber of poits ad blocks. The code C F of the desig D over the fiite field F is the space spaed by the icidece vectors of the blocks over F. If the poit set of D is deoted by P ad the block set by B ad if Q is ay subset of P the

3 2 BACKGROUND AND TERMINOLOGY 3 we will deote the icidece vector of Q by v Q. Thus C F = v B B B ad is a subspace of V = F P the full vector space of fuctios from P to F. For ay vector w V the coordiate of w at the poit P P is deoted by w(p. All our codes here will be liear codes i.e. subspaces of the ambiet vector space. If a code C over a field of order q is of legth dimesio k ad miimum weight d the we write [ k d] q to show this iformatio. A geerator matrix for the code is a k matrix made up of a basis for C. The dual or orthogoal code C is the orthogoal uder the stadard ier product ( i.e. C = {v F (v c = 0 for all c C}. A check (or parity-check matrix for C is a geerator matrix H for C. If c is a codeword the the support of c is the set of o-zero coordiate positios of c. A costat vector is oe for which all the o-zero etries are equal to 1. The all-oe vector will be deoted by j ad is the costat vector of weight the legth of the code. Two liear codes of the same legth ad over the same field are equivalet if each ca be obtaied from the other by permutig the coordiate positios ad multiplyig each coordiate positio by a o-zero field elemet. They are isomorphic if they ca be obtaied from oe aother by permutig the coordiate positios. A automorphism of a code C is a isomorphism from C to C. The automorphism group will be deoted by Aut(C. Termiology for graphs is stadard: the graphs Γ = (V E with vertex set V ad edge set E are udirected ad the valecy of a vertex is the umber of edges cotaiig the vertex. A graph is regular if all the vertices have the same valecy. Ay code is isomorphic to a code with geerator matrix i so-called stadard form i.e. the form [I k A]; a check matrix the is give by [ A T I k ]. The first k coordiates are the iformatio symbols ad the last k coordiates are the check symbols. Permutatio decodig was first developed by MacWilliams [8]. It ivolves fidig a set of automorphisms of a code such that the set satisfies certai coditios that allow it to be used for decodig; such a set is called a PD-set. The method is described fully i MacWilliams ad Sloae [9 Chapter 15] ad Huffma [3 Sectio 8]. A PD-set for a t-error-correctig code C is a set S of automorphisms of C which is such that the every possible error vector of weight s t ca be moved by some member of S to aother vector where the s o-zero etries have bee moved out of the iformatio positios. I other words every t-set of coordiate positios is moved by at least oe member of S to a t-set cosistig oly of checkpositio coordiates. That such a set will fully use the error-correctio potetial of the code follows easily ad is proved i Huffma [3 Theorem 8.1]. Of course such a set might ot exist at all ad existece is ot ivariat uder equivalece or isomorphism of codes. Furthermore there

4 3 THE CODES AND PD-SETS 4 is a boud o the miimum size that the set S may have: see Gordo [2] or [3]. The algorithm for permutatio decodig the is as follows: we have a t-error-correctig [ k d] q code C with check matrix H i stadard form. Thus the geerator matrix G for C that is used for ecodig has I k as the first k colums ad hece the first k coordiate positios correspod to the iformatio symbols. So G = [I k A] ad H = [A T I k ] for some A ad ay vector v of legth k is ecoded as vg. Suppose x is set ad y is received ad at most t errors occur. Let S = {g 1... g s } be the PD-set. Compute the sydromes H(yg i T for i = 1... s util a i is foud such that the weight of this vector is t or less. Now look at the iformatio symbols i yg i ad obtai the codeword c that has these iformatio symbols. Decode y as cg 1 i. Note that this is valid sice permutatios of the coordiate positios correspod to liear trasformatios of F so that if y = x + e where x C the yg = xg + eg for ay g S ad if g Aut(C the xg C. 3 The codes ad PD-sets We describe first briefly how the codes are defied from graphs ad desigs. Let be ay iteger ad Ω a set of size ; to avoid degeerate cases we take 7. Takig the set Ω {3} to be the set of all 3-elemet subsets of Ω we defie three o-trivial udirected graphs with vertex set P = Ω {3} ad deote these graphs by A i ( where i = The edges of the graph A i ( are defied by the rule that two vertices are adjacet i A i ( if as 3-elemet subsets they have exactly i elemets of Ω i commo. For each i = we defie from A i ( a 1-desig D i ( o the poit set P by defiig for each poit P = {a b c} P a block {a b c} i by {a b c} i = {{x y z} {x y z} {a b c} = i }. Deote by B i ( the block set of D i ( so that each of these is a symmetric 1-desig o ( 3 poits with block size respectively: ( 3 3 for D0 (; 3 ( 3 2 for D1 (; 3( 3 for D 2 (.

5 3 THE CODES AND PD-SETS 5 The icidece vector of the block {a b c} i for i = respectively is the v {abc} 0 = v {xyz} ; (1 xyz Ω v {abc} 1 = v {axy} + v {bxy} + v {cxy} ; (2 xy Ω xy Ω xy Ω v {abc} 2 = v {abx} + v {acx} + v {bcx} (3 x Ω x Ω x Ω where Ω = Ω \ {a b c} ad as usual with the otatio from [1] the icidece vector of the subset X P is deoted by v X. Sice our poits here are actually triples of elemets from Ω we emphasize that we are usig the otatio v {abc} istead of the more cumbersome v {{abc}} as metioed i [1]. I [5] we examied the biary codes of these desigs i.e. deotig the block set of D i ( by B i ( we took C i ( = C 2 (D i ( = v b b B i ( where the spa is take over F 2. Alteratively this ca be regarded as the row spa over F 2 of a adjacecy matrix of the relevat graph. We obtaied the followig: Result 1 Let Ω be a set of size where 7. Let P = Ω {3} the set of subsets of Ω of size 3 be the vertex set of the three graphs A i ( for i = with adjacecy defied by two vertices (as 3-sets beig adjacet if the 3-sets meet i zero oe or two elemets respectively. Let C i ( deote the code formed from the row spa over F 2 of a adjacecy matrix for A i (. The 1. 0 (mod 4: (a C 2 ( = F P 2 ; (b C 0 ( = C 1 ( is [ ( ( 3 3 4]2 ad C 0 ( is [ ( ( ]2 ; 2. 2 (mod 4: C i ( = F P 2 for i = 0 1 2; 3. 1 (mod 4: (a C 0 ( = C 1 ( C 2 (; (b C 0 ( is [ ( ( 3 ( 3 2 8]2 ad C 0 ( is [ ( ( 3 2 2]2 ; C 1 (9 is [ ] 2 ad C 1 (9 is [ ] 2 ; C 1 ( is [ ( ( ]2 ad C 1 ( is [ ( 3 1 ( 2( 3] 2 for > 9; C 2 ( is [ ( ( ]2 ad C 2 ( is [ ( ( ]2 ;

6 3 THE CODES AND PD-SETS (mod 4: (a C 1 ( = v P + j P P is [ ( ( ]2 ; (b C 0 ( = C 2 ( is [ ( ( ]2 ad C 2 ( is [ ( ( ] 2 ; For all 7i = C i ( C i ( = {0} ad the automorphism groups of these codes are S or S ( 3. Notice that the code C 2 ( for odd is always [ ( ( ]2 ad furthermore iformatio sets are give i [5]. We ow prove the theorem: Proof of Theorem 1: That the iformatio symbols ca be take as give above follows from Propositio 1 of [5] where we have replaced the poit { 2 1 } i the iformatio set by { 3 2 1} i order to be able to costruct a PD-set. Let I deote the iformatio positios ad C the check positios. Thus I = {{i j } 1 i < j < } {{ 3 2 1}} \ {{ 2 1 }}. Let P = { 3 2 1} I ad Q = { 2 1 } C. Notice that the code will correct t = 3 2 errors. Take a set T of t poits of P ad let T = {abc} T {a b c}. We eed to exhibit a elemet σ S such that T σ C. For this we eed to cosider the differet types of compositio of T so the proof goes through a umber of cases. Notice that if T C the the idetity automorphism which is i S will do. Thus suppose T C. Let z i deote the umber of elemets i Ω that occur i times i T. The t i=0 z i = ad t i=0 iz i = 3t. Thus t (2i 3z i + 9 = 3z 0 + z 1. (4 i=2 Case (I: T I. The at least t 1 members of T cotai. (i P T : the T 2t + 1 = 2 so there are at least two distict elemets a ad b i Ω ot equal to that are ot i T. If a 3 the σ = ( a will satisfy T σ C. If {a b} = { 2 1} the agai ( a will do. (ii P T : the T 2(t = 1 so there is at least oe elemet a Ω such that a T. Clearly a 4. If T = 1 the appear oly i P ad i o other elemet of T. Thus ( 3 will do. If T < 1 the there are at least two elemets a b T with a b 4. The T ( a( 1 b C.

7 3 THE CODES AND PD-SETS 7 Case (II: suppose T meets both I ad C o-trivially. (i Suppose first that z 0 0. The if a Ω a ad a T the traspositio τ = ( a will trasform T ito T τ which will ot cotai so that if we fid a elemet σ to take T τ ito C where σ S but fixes the τσ S will map T ito C. So assume that is abset from T. The P T ad the umber of poits of the form {i 2 3} with i 1 is 4 = 2t 1 so there is a elemet b 4 such that {b 2 3} T. The T ( a( 1 b C. (ii Now suppose that T = Ω ad thus z 0 = 0 ad z 1 9 from Equatio (4. For a Ω let x(a deote the umber of times a appears i poits i T. So 1 x(a t for each a Ω ad 3t = i=1 x(i = t i=1 iz i. For ay a Ω x(a = 3t b a x(b 3t ( 1 = t 2 ad so z i = 0 for i t 1. We will ow show that we ca fid a poit {a b c} T such that x(a = x(b = 1 a b 4 ad c. Suppose x( = m ad that 1 < m t 2. The 3t = z 1 + m + x(i z 1 + m + 2( 1 z 1 x(i 2i which simplifies to z 1 t+4+m. If m = 1 the this iequality still applies if we take z 1 to be the umber of elemets with x(a = 1 excludig. Suppose that as may pairs with x(a = 1 as possible occur as part of a triple with i T. This uses 2m elemets leavig z 1 2m t + 4 m elemets with x(a = 1 (always excludig from the cout. We wat to exclude which still leaves at least z 1 2m 3 t m+1 elemets. The umber of poits of T available is t m so we must have at least oe poit X = {a b c} with two elemets a ad b with x(a = x(b = 1 a b 4 ad c. Fially we show how this poit X = {a b c} ca be used to defie group elemets that will map T ito C. We eed to look at the various possibilities for c. 1. c 4: the σ = ( a( 1 b( 2 c will satisfy T σ C sice {d e }σ C {a b c}σ = { 1 2} C ad { 3 2 1}σ 1 = {b c 3} T sice x(b = 1 ad a c = 1: the σ = ( a(b 2 will work as above otig that { 3 2 1}σ 1 = {b 1 3} T sice x(b = 1 ad a c = 2: the σ = ( a(b 1 will work as i the previous case.

8 4 CONCLUSION 8 4. c = 3: the {a b c} = {a b 3} ad take σ = ( a(b 1( 2 3. The {a b c}σ = { 1 2} C ad { 3 2 1}σ 1 = {b 2 3} T sice x(b = 1 ad a 2. We have show that every t-tuple ad hece every s-tuple for s t ca be moved by a elemet of S ito the error positios. Thus S is a PD-set for C 2 (. 4 Coclusio Note: 1. We expect that a similar sort of costructio ca be made for PD-sets for the other codes from these graphs but oe requires suitable iformatio sets. I particular takig the code C 0 ( for 1 (mod 4 ad 9 we have a [ ( ( 3 2 2]2 code that will correct 3 2 errors. From Lemma 10 of [5] we ca take as the ( 2 iformatio positios the poits {i j } for 1 i j 1 {i 2 1} for 1 i 3 ad two extra poits: { 4 3 1} ad { 4 3 2}. To costruct a PD-set we eed to switch oe of these poits with a poit from the check positios ad it is easy to verify that the followig will form a iformatio set: if I 1 = {{i j } 1 i < j 1} \ {{ 2 1 }}; I 2 = {{i 2 1} 1 i 3}; I 3 = {{ 4 3 2} { 4 3 1} { 5 4 3}} the I = I 1 I 2 I 3 is a iformatio set for C 0 ( for 1 (mod 4 ad 9. The set {( i( 1 j( 2 k( 3 l 1 i 1 j 1 1 k 2 1 l 3} where (i i deotes the idetity elemet of S ca be show to be a PD-set for the code the proof followig the lies of the proof for the code we have give but havig more cases to be dealt with separately. 2. The existece of a PD-set for a code is ot ivariat uder equivalece ad part of the problem of fidig PD-sets is to fid suitable iformatio sets: for example i the theorem the poits {1 2 } {1 3 }... { 2 1 } ca be take as iformatio symbols but if they are the code will ot have a PD-set to correct all the allowed errors sice for = 9 the [ ] 2 code corrects t = 3 errors but there is o elemet i S 9 that move the three poits {1 2 9} {3 4 5} {6 7 8}

9 REFERENCES 9 ito the check positios. However if {7 8 9} is placed i the check positios ad {6 7 8} i the iformatio positios the it ca be doe: i this example (6 9 will do it. 3. A boud foud i Gordo [2] (ad quoted ad derived i Huffma [3] for the smallest size a PD-set ca be whe compared with the order of the group of the code shows that for some well-kow codes from geometries PD-sets will ot exist beyod a certai code legth: see [4] for details. Refereces [1] E. F. Assmus Jr. ad J. D. Key. Desigs ad their Codes. Cambridge: Cambridge Uiversity Press Cambridge Tracts i Mathematics Vol. 103 (Secod pritig with correctios [2] D. M. Gordo. Miimal permutatio sets for decodig the biary Golay codes. IEEE Tras. Iform. Theory 28: [3] W. Cary Huffma. Codes ad groups. I V. S. Pless ad W. C. Huffma editors Hadbook of Codig Theory pages Amsterdam: Elsevier Volume 2 Part 2 Chapter 17. [4] J. D. Key T. P. McDoough ad V. C. Mavro. Partial permutatio decodig of codes from fiite plaes. To appear: Europea J. Combi. [5] J. D. Key J. Moori ad B. G. Rodrigues. Biary codes from graphs o triples. Discrete Math. To appear. [6] J. D. Key J. Moori ad B. G. Rodrigues. Permutatio decodig for biary codes from triagular graphs. Europea J. Combi. 25: [7] J. D. Key ad P. Seevirate. Permutatio decodig of biary codes from lattice graphs. Discrete Math. To appear. [8] F. J. MacWilliams. Permutatio decodig of systematic codes. Bell System Tech. J. 43: [9] F. J. MacWilliams ad N. J. A. Sloae. The Theory of Error-Correctig Codes. Amsterdam: North-Hollad 1983.