Construction of a 3-Dimensional MDS code

Constrution of a 3 dimensional MDS-o de Dedia ted to the entenar y of the bir th of Feren Kár teszi (1907-1989) A. Aguglia and L. Giuzzi ∗ Abstrat In this pap er, w e desrib e a pro edure for onstruting q ary [ N , 3 , N − 2] MDS o des, of length N ≤ q + 1 (for q o dd) or N ≤ q + 2 (for q ev en), using a set of nondegenerate Hermitian forms in P G (2 , q 2 ) . 1 In tro dution The w ellkno wn Singleton b ound states that the ardinalit y M of a o de of length N with minim um distane d o v er a q ary alphab et alw a ys satises M ≤ q N − d +1 ; (1) see [7℄. Co des attaining the b ound are alled maximum distan e sep ar able  o des , or MDS o des for short. In teresting families of maxim um distane separable o des arise from geo- metri and om binatorial ob jets em b edded in a nite pro jetiv e spaes. In partiular linear [ N , k , N − k + 1] MDS o des, with k ≥ 3 , and N ars in P G ( k − 1 , q ) are equiv alen t ob jets; see [1℄. A general metho d for onstruting a q ary o de is to tak e N m ultiv ariate p olynomials f 1 , . . . , f N dened o v er a suitable subset W ⊆ GF( q ) n and onsider the set C giv en b y C = { ( f 1 ( x ) , . . . , f N ( x )) | x ∈ W } . In this pap er, w e deal with the ase |W | = q t and also assume that the evaluation funtion Θ : ( W 7→ C x 7→ ( f 1 ( x ) , f 2 ( x ) , . . . , f N ( x )) is injetiv e. ∗ Resear h supp orted b y the Italian Ministry MIUR, Strutture geometri he, om binatoria e loro appliazioni. 1 If C attains the Singleton b ound then the restritions of all the o dew ords to an y giv en t = N − d + 1 plaes m ust all b e dieren t, namely in an y t p ositions all p ossible v etors o ur exatly one. This means that a neessary ondition for C to b e MDS is that an y t of the v arieties V ( f m ) for m = 1 , . . . , N meet in exatly one p oin t in W . Here V ( f ) denotes the algebrai v ariet y asso iated to f . Applying the ab o v e pro edure to a set of nondegenerate Hermitian forms in P G (2 , q 2 ) w e onstrut some q ary [ N , 3 , N − 2] MDS o des, of length N ≤ q + 1 (for q o dd) or N ≤ q + 2 (for q ev en). The o des th us obtained an also b e represen ted b y sets of p oin ts in PG(3 , q ) ; this represen tation is used in Setion 4 in order to devise an algebrai deo ding pro edure, based up on p olynomial fatorisation; see [9 ℄. 2 Preliminaries Let A b e a set on taining q elemen ts. F or an y in teger N ≥ 1 , the funtion d H : A N × A N 7→ N giv en b y d H ( x , y ) = |{ i : x i 6 = y i }| , is a metri on A N . This funtion is alled the Hamming distan e on A N . A q ary ( N , M , d ) o de C o v er the alphab et A is just a olletion of M elemen ts of A N su h that an y t w o of them are either the same or at Hamming distane at least d ; see [4, 5℄. The elemen ts of C are alled  o dewor ds whereas the in tegers d and N are resp etiv ely the minimum distan e and the length of C . If A = GF ( q ) and C is a k dimensional v etor subspae of GF ( q ) N , then C is said to b e a line ar [ N , k , d ] o de. Under sev eral omm uniation mo dels, it is assumed that a reeiv ed w ord r should b e deo ded as the o dew ord c ∈ C whi h is nearest to r aording to the Hamming distane; this is the soalled maxim um lik eliho o d deo ding. Under these assumptions the follo wing theorem, see [4, 5℄, pro vides a basi b ound on the guaran teed error orretion apabilit y of a o de. Theorem 1. If C is a  o de of minimum distan e d , then C  an always either dete t up to d − 1 err ors or  orr e t e = ⌊ ( d − 1) / 2 ⌋ err ors. Observ e that the theorem do es not state that it is not p ossible to deo de a w ord more than e errors happ ened, but just that in this ase it is p ossible that the orretion fails. Managing to reo v er from more than e errors for some giv en reeiv ed o dew ords is alled orreting b ey ond the b ound. The weight of an elemen t x ∈ GF ( q ) N is the n um b er of nonzero omp onen ts x i of x . F or a linear o de the minim um distane d equals the minim um w eigh t of the nonzero o dew ords. The parameters of a o de are not indep enden t; in general it is diult to determine the maxim um n um b er of w ords a o de of presrib ed length N and minim um distane d ma y on tain. 2 Observ e that, for an y arbitrary linear [ N , k , d ] o de, ondition ( 1) ma y b e rewritten as d ≤ N − k + 1 ; (2) th us C is a linear MDS o de if and only if equalit y holds in ( 2). In Setion 3 w e shall mak e extensiv e use of some nondegenerate Hermitian forms in P G (2 , q 2 ) . Consider the pro jetiv e spae P G ( d, q 2 ) and let V b e the underlying v etor spae of dimension d + 1 . A sesquiline ar Hermitian form is a map h : V × V − → GF ( q 2 ) additiv e in b oth omp onen ts and satisfying h ( k v , l w ) = k l q h ( v , w ) for all v , w ∈ V and k , l ∈ GF ( q 2 ) . The form is de gener ate if and only if the subspae { v | h ( v , w ) = 0 ∀ w ∈ V } , the r adi al of h , is dieren t from { 0 } . Giv en a sesquilinear Hermitian form h , the asso iated Hermitian v ariet y H is the set of all p oin ts of P G ( d, q 2 ) su h that { < v > | 0 6 = v ∈ V , h ( v , v ) = 0 } . The v ariet y H is de gener ate if h is degenerate; nondegenerate otherwise. If h is a sesquilinear Hermitian form in P G ( d, q 2 ) then the map F : V − → GF ( q ) dened b y F ( v ) = h ( v , v ) , is alled the Hermitian form on V asso iate d to h . The Hermitian form F is nonde gener ate if and only if h is nondegenerate. Complete in tro dutions to Hermitian forms o v er nite elds ma y b e found in [2, 6℄. 3 Constrution Let S b e a transv ersal in GF( q 2 ) of the additiv e subgroup T 0 = { y ∈ GF ( q 2 ) : T ( y ) = 0 } , where T : y ∈ GF( q 2 ) 7→ y q + y ∈ GF( q ) is the trae funtion. Denote b y Λ the subset of GF ( q 2 ) satisfying  α − β γ − β  q − 1 6 = 1 (3) for an y α, β , γ ∈ Λ . Cho ose a basis B = { 1 , ε } of GF ( q 2 ) , regarded as a 2  dimensional v etor spae o v er GF ( q ) ; hene, it is p ossible to write ea h elemen t α ∈ GF ( q 2 ) in omp onen ts α 1 , α 2 ∈ GF( q ) with resp et to B . W e ma y th us iden tify the elemen ts of GF ( q 2 ) with the p oin ts of AG (2 , q ) , b y the bijietion ( x, y ) ∈ AG (2 , q ) 7→ x + εy ∈ GF ( q 2 ) . Condition (3 ) orresp onds to require that Λ , regarded as p oin tset in AG (2 , q ) , is an ar. Th us, setting N = | Λ | , w e ha v e N ≤ ( q + 1 for q o dd q + 2 for q ev en. (4) 3 No w, onsider the nondegenerate Hermitian forms F λ ( X, Y , Z ) on GF ( q 2 ) 3 F λ ( X, Y , Z ) = X q +1 + Y q Z + Y Z q + λ q X q Z + λX Z q , as λ v aries in Λ . Lab el the elemen ts of Λ as λ 1 , . . . , λ N and let Ω = GF ( q 2 ) × S . Theorem 2. The set C = { ( F λ 1 ( x, y , 1) , F λ 2 ( x, y , 1) , . . . , F λ N ( x, y , 1)) | ( x, y ) ∈ Ω } is a q -ary line ar [ N , 3 , N − 2] MDS  o de. Pr o of. W e rst sho w that C onsists of q 3 tuples from GF ( q ) . Let ( x 0 , y 0 ) , ( x 1 , y 1 ) ∈ Ω and supp ose that for an y λ ∈ Λ , F λ ( x 0 , y 0 , 1) = F λ ( x 1 , y 1 , 1) . Then, T ( λ ( x 1 − x 0 )) = x q +1 0 − x q +1 1 + T ( y 0 − y 1 ) . (5) In partiular, T ( λ ( x 1 − x 0 )) = T ( α ( x 1 − x 0 )) = T ( γ ( x 1 − x 0 )) (6) for an y α, λ, γ ∈ Λ . If it w ere x 1 6 = x 0 , then (6) w ould imply  α − β γ − β  q − 1 = 1 , on traditing the assumption made on Λ . Therefore, x 1 = x 0 and from ( 5) w e get T ( y 0 − y 1 ) = 0 . Hene, y 0 and y 1 are in the same oset of T 0 ; b y denition of S , it follo ws that y 0 = y 1 , th us C has as man y tuples as | Ω | . W e are no w going to sho w that C is a v etor subspae of GF( q ) N . T ak e ( x 0 , y 0 ) , ( x 1 , y 1 ) ∈ Ω . F or an y λ ∈ Λ , F λ ( x 0 , y 0 , 1) + F λ ( x 1 , y 1 , 1) = F λ ( x 2 , y 2 , 1) , (7) where x 2 = x 0 + x 1 and y 2 = y 0 + y 1 − x q 0 x 1 − x q 1 x 0 . Lik ewise, for an y κ ∈ GF( q ) , κ F λ ( x 0 , y 0 , 1) = F λ ( x, y , 1) , (8) where x = κx 0 and y is a ro ot of y 2 + y = ( κ − κ 2 ) x q +1 0 + κ ( y q 0 + y 0 ) . Therefore, C is a v etor subspae of GF( q ) N ; as it onsists of q 3 tuples, C is indeed a 3 dimensional v etor spae. Finally w e pro v e that the minim um distane d of C is N − 2 . Sine C is a v etor subspae of GF( q ) N , its minim um distane is N − z , where z = max c ∈C c 6 = 0 |{ i : c i = 0 } | . 4 First observ e that z ≥ 2 b eause of Singleton b ound ( 2). In order to sho w that z = 2 w e study the follo wing system      F α ( x, y , 1) = 0 F β ( x, y , 1) = 0 F γ ( x, y , 1) = 0 (9) for α, β , γ distint elemen ts of Λ . Set U = x q +1 + y q + y , V = x q and W = x ; then, (9) b eomes      U + α q V + αW = 0 U + β q V + β W = 0 U + γ q V + γ W = 0 (10) Sine  α − β γ − β  6 = 1 , the only solution of (10 ) is U = V = W = 0 , that is x = 0 and y + y q = 0 . In partiular, there is just one solution to (9 ) in Ω , that is x = (0 , 0) . This implies that a o dew ord whi h has at least three zero omp onen ts is the zero v etor, hene z = 2 and th us the minim um distane of C is N − 2 . Example. When q is o dd, a transv ersal S for T 0 is alw a ys giv en b y the subeld GF( q ) em b edded in GF( q 2 ) . In this ase it is then extremely simple to onstrut the o de. F or q = 5 , a omputation using GAP [ 3 ℄, sho ws that in order for Λ to satisfy prop ert y (3), w e ma y tak e Λ = { ε 3 , ε 4 , ε 8 , ε 15 , ε 16 , ε 20 } , where ε is a ro ot of the p olynomial X 2 − X + 2 , irreduible o v er GF (5) . The orresp onding Hermitian forms are X q +1 + Y q Z + y Z q + ε 15 X q Z + ε 3 X Z q X q +1 + Y q Z + Y Z q + ε 20 X q Z + ε 4 X Z q X q +1 + Y q Z + Y Z q + ε 16 X q Z + ε 8 X Z q X q +1 + Y q Z + Y Z q + ε 3 X q Z + ε 15 X Z q X q +1 + Y q Z + Y Z q + ε 8 X q Z + ε 16 X Z q X q +1 + Y q Z + Y Z q + ε 4 X q Z + ε 20 X Z q A generator matrix for the [6 , 3 , 4] MDS o de obtained applying Theorem 2 to these Hermitian forms is, G =   1 1 1 1 1 1 0 1 0 2 1 2 0 0 1 2 2 1   5 R emark 1 . In P G (2 , q 2 ) , tak e the line ℓ ∞ : Z = 0 as the line at innit y . Then, in the ane spae AG (2 , q 2 ) = P G (2 , q 2 ) \ ℓ ∞ , an y t w o Hermitian urv es V ( F λ ) ha v e q 2 ane p oin ts in ommon, q of whi h in Ω ⊂ AG ( 2 , q 2 ) . Lik ewise, the full in tersetion \ λ ∈ Λ V ( F λ ) onsists of the q ane p oin ts { (0 , y ) | y q + y = 0 } , orresp onding to just a single p oin t in Ω . R emark 2 . Denote b y A i the n um b er of w ords in C of w eigh t i . Sine C is an MDS o de, w e ha v e A i =  N i  ( q − 1) i − N +2 X j =0 ( − 1) j  i − 1 j  q i − j − N +2 ; see [8 ℄. Th us, A N − 2 = 1 2 ( N 2 − N )( q − 1) A N − 1 = N q 2 − ( N 2 − N ) q + N 2 − 2 N A N = q 3 − N q 2 + 1 2  ( N 2 − N ) q − N 2 + 3 N  . 4 Deo ding In this setion it will b e sho wn ho w the o de C w e onstruted ma y b e deo ded b y geometri means. Our approa h is based up on t w o remarks: 1. An y reeiv ed w ord r = ( r 1 , . . . , r N ) an b e uniquely represen ted b y a set e r of N p oin ts of PG(3 , q ) e r = { ( λ 1 i , λ 2 i , r i , 1) : λ = λ 1 i + ελ 2 i ∈ Λ } . These p oin ts all lie on the one Ψ of basis Ξ = { ( λ 1 i , λ 2 i , 0 , 1) : λ = λ 1 i + ελ 2 i ∈ Λ } and v ertex Z ∞ = (0 , 0 , 1 , 0) . 2. The funtion φ ( a,b ) ( x, y , z , t ) =  a q +1 + T ( b )  t + T (( x + εy ) a ) is a homogeneous linear form dened o v er GF( q ) 4 for an y a, b ∈ GF( q 2 ) . Reall that the o dew ord c orresp onding to a giv en ( a, b ) ∈ Ω is c =  φ ( a,b ) ( λ 1 1 , λ 2 1 , 0 , 1) , φ ( a,b ) ( λ 1 2 , λ 2 2 , 0 , 1) , . . . , φ ( a,b ) ( λ 1 N , λ 2 N , 0 , 1)  ; 6 th us, e c , the set on taining the p oin ts ( λ 1 i , λ 2 i , c i , 1) , is the full in tersetion of the plane π a,b : z = φ ( a,b ) ( x, y , z , t ) with the one Ψ . It is lear that kno wledge of the plane π ( a,b ) is enough to reonstrut the o dew ord c . In the presene of errors, w e are lo oking for the nearest o dew ord c to a v etor r ; this is the same as to determine the plane π ( a,b ) on taining most of the p oin ts of e r . In order to obtain su h a plane, w e adopt the follo wing approa h. Assume ℓ to b e a line of the plane π 0 , 0 : z = 0 external to Ξ and denote b y π ∞ the plane at innit y of equation t = 0 . F or an y P ∈ ℓ , let e r P b e the pro jetion from P of the set e r on π ∞ . W rite L P r for a urv e of π ∞ of minim um degree on taining e r P . Observ e that deg L P r ≤ q + 1 and deg L P r = 1 if, and only if, all the p oin ts of e r lie on a same plane through P , that is e r orresp onds to a o dew ord asso iated with that plane passing through P . W e no w an apply the follo wing algorithm using, for example, [3 ℄. 1. T ak e P ∈ ℓ ; 2. Determine the pro jetion r P and ompute the urv e L P r ; 3. F ator L P r in to irreduible fators, sa y L 1 , L 2 , . . . , L v ; 4. Coun t the n um b er of p oin ts in e r P ∩ V ( L i ) for an y fator L i of L P r with deg L i = 1 . 5. If for some i w e ha v e n i > N +1 2 , then return the plane spa wned b y P and t w o p oin ts of L i ; else, as long as not all the p oin ts of ℓ ha v e b een onsidered, return to p oin t 1. 6. If no urv e with the required prop ert y has b een found, return failure. R emark 3 . The ondition on n i in p oin t 5  he ks if the plane on tains more than half of the p oin ts orresp onding to the reeiv ed w ord r ; when this is the ase, a putativ e o dew ord c is onstruted, with d ( c , r ) ≤ N − 3 2 ; th us, when c ∈ C , then it is indeed the unique w ord of C at minim um distane from r . Ho w ev er, the aforemen tioned algorithm ma y b e altered in sev eral w a ys, in order to b e able to try to orret errors b ey ond the b ound; p ossible approa hes are 1. iterate the pro edure for all the p oin ts on ℓ and return the planes on tain- ing most of the p oin ts orresp onding to the reeiv ed v etor; 2. use some further prop erties of the one Ψ ; in partiular, when Ξ is a oni it seems p ossible to impro v e the deo ding b y onsidering also the quadrati omp onen ts of the urv e L P r . R emark 4 . The  hoie of P on a line ℓ is due to the fat that an y line of π 0 , 0 meets all the planes of PG(3 , q ) . In general, w e migh t ha v e  hosen ℓ to b e just a blo  king set disjoin t from Ξ . Observ e that when q is o dd and | Λ | = q + 1 , then the line ℓ is just an external line to a oni of π 0 , 0 . 7 Referenes [1℄ E.F. Assm us Jr and J.D. Key . Designs and their  o des , Cam bridge Uni- v ersit y Press (1992). [2℄ S.N. Bose and I.M. Chakra v arti, Hermitian varieties in a nite pr oje tive sp a e PG( N , q 2 ) , Canad J. Math. 18 (1966), 11611182. [3℄ The GAP Group, GAP  Gr oups, A lgo- rithms, and Pr o gr amming, V ersion 4.4 (2006) (\protet\vrule width0pt\protet \h ref {h tt p:/ /w ww. ga p- sys te m. org }{ ht tp: // ww w.g ap -sy st em .or g} ) . [4℄ L. Giuzzi, Co dii  orr ettori , SpringerV erlag (2006). [5℄ R. MEliee, The ory of Information and  o ding , Cam bridge Univ ersit y Press (2002). [6℄ B. Segre, F orme e ge ometrie hermitiane,  on p arti olar e riguar do al  aso nito , Ann. Mat. Pura Appl. (4) 70 (1965), 1201. [7℄ R. C. Singleton, Maximum distan e q nary  o des , IEEE T rans. Inf. Theory 10 issue 2 (1964), 116118. [8℄ L. Staiger, On the weight distribution of line ar  o des having dual distan e d ′ > k , IEEE T rans. Inf. Theory 35 issue 1 (1989), 186188. [9℄ M. Sudan, De  o ding R e e d Solomon  o des b eyond the err or orr e tion b ound , Journal of Complexit y , 13 issue 1 (1997), 180193 Angela A guglia Dipartimen to di Matematia P olitenio di Bari Via G. Amendola 126/B 70126 Bari Italy a.agugliapolib a.i t Lua Giuzzi Dipartimen to di Matematia P olitenio di Bari Via G. Amendola 126/B 70126 Bari Italy l.giuzzipoliba.i t 8