We discuss the class of projective systems whose supports are the complement of the union of two linear subspaces in general position. We express the weight enumerators of the codes generated by these projective systems using two simplex codes corresponding to given linear subspaces. We also prove these codes are uniquely determined upto equivalence by their weight enumerators

Homma Masaaki

Kim Seon Jeong

Yoo Mi Ja

本間 正明

Journal of the Korean Mathematical Society

We discuss the class of projective systems whose supports are the complement of the union of two linear subspaces in general position. We express the weight enumerators of the codes generated by these projective systems using two simplex codes corresponding to given linear subspaces. We also prove these codes are uniquely determined upto equivalence by their weight enumerators.Articlejournal articl

本間, 正明

5761

Homma, Masaaki

5762

Kim, Seon Jeong

5763

Yoo, Mi Ja

5764

KANAGAWA University Repository

Bull. Korean Math. Soc. 37 (2000), No. 3, pp. 493-505PROJECTIVE SYSTEMS SUPPORTED ON THECOMPLEMENT OF TWO LINEAR SUBSPACESMASAAKI HOMMA, SEON JEONG KIM, AND MI JA YooABSTRACT. We discuss the class of projective systems whose sup-ports are the complement of the union of two linear subspaces ingeneral position. We express the weight enumerators of the codesgenerated by these projective systems using two simplex codes cor-responding to given linear subspaces. We also prove these codes areuniquely determined upto equivalence by their weight enumerators.1. Introduction and preliminariesLet IFq be the finite field of order q and pm the m-dimensional pro-jective space over lFq. We denote the dual space of pm by pm* whichis the set of all hyperplanes in pm. A code C C JF; is said to be de-generate, if there is a position i (1 ::; i ::; n) such that Ci = 0 for anycodeword c = (Cl,··· ,en) E C; otherwise C is said to be nondegenerate.Throughout this paper, a code means a nondegenerate code.Let C be a nondegenerate rn, k]q-code with a generator matrixG = (al,··· ,an),where ~ is a column vector of length k. Since C is nondegenerate, we candefine a positive O-cycle I:~=l [~] on the projective space r-1, where [3i]Received October 4, 1999. Revised January 4,2000.2000 Mathematics Subject Classification: 94B05, 5IE20, 05B25.Key words and phrases: linear code, weight enumerator, simplex code, projectivespace, projective system.This work has been supported by the Japan Society for the Promotion of Scienceand the Korea Science and Engineering Foundation, Project No. 976-0100-001-2. Alsothe second author is partially supported by KOSEF-GARC and by the Korea ResearchFoundation made in the program year of 1998, Project No. 1998-015-D00006. Thethird author is supported by Research Intern Program, KOSEF.Masaaki Homma, Seon Jeong Kim, and Mi Ja Yoois the point of the projective space represented by 8i. Note that the sup-port of the o-cycle L:~l [8i] spans the whole space pk-l because the rankof G is k. If one chooses another generator matrix G' = (a'h ... ,a'n)of C, then the o-cyc1e L:~""l[a'i] is projectively equivalent to the previ-ous one over lFq. Therefore, for a given nondegenerate [n, k]q-code C,we can define a projective equivalence class of positive o-cycles on pit-Iwhose supports span the whole space, and denote by Xc one of the 0-cycles. We call Xc a projective system associated to C after [4]. It isobvious that a code C is equivalent to C' if and only if Xc is projectivelyequivalent to Xc,. Moreover, there is oneto one correspondence betweenthe equivalence classes of nondegenerate" [n, k]q-codes and the projectiveequivalence classes of positive o-cycles of length n in pk-l whose supportsspan the whole space ([1], [2]).We use the notationqm+1_1N(m) := = qm + qm-l + ... + q2 + q + 1 for m ;:::: 1,q-lwhich is the number of all points on a projective space pm over lFq. Byconvention, we let N(O) = 1, N(-l) = O. Linear subspaces Li , i =1,2, ... ,r (r ;:::: 2) in pm are said to be in general position if, for any subset{ib •.. ,is} C {1,· .. ,r}, the dimension of the linear span < Uj""lL ij ) ofUj""ILij is equal to min{m,L:;=1 dimLij + s -I}.The code corresponding to the O-Cycle L:PEpk-l P is called a simplexcode, which is an [N(k - 1), k, qk-l]q-code, and its weight enumerator is1 + (qk - l)sqk-l.Let Ci" C lFi be an [nil ~]q-code for i = 1,2. Then we define theirdirect sum by Cl EB C2 = {(Cl,C2) I Cl E Cl, C2 E C2}· Then Cl $ C2is an [nl + n2, kl + k2]q-codeand its weight enumerator is WCl E&C2(S) =WCl(S)· WC2 (s).We use the following well-known facts.LEMMA 1.1([3], [4}).(Griesmer bound) For any rn, k, d]q-code, we havek-l dn;:::: L:fil,i""O qwhere fx1means the smallest integer greater than or equal to x.494Projective systems supported on the complement of two linear subspacesLEMMA 1.2([3], [4]).(MacWilliams identity) Let C be an rn, k]q-codeand Cl.. the dual code of O. Then1 1- sWC.L(s) = k(l + (q - l)s)n . Wc( 1 ( 1))·q + q- sLEMMA 1.3([3]). If H is a parity check matrix of a code of length n,then the minimum distance of the code is d if and only if every d - 1columns of H are linearly independent and some d columns are linearlydependent.2. Weight enumeratorsWe compute the weight enumerators of codes corresponding to theprojective systems supported on linear subspaces or their complements.For this we use the following lemmas.LEMMA 2.1. Let S be a subset ofplc-l, and assume that the linearspans of S and its complement SC are the whole space plc-I. Letand let C i be the corresponding code to Xi for i = 1,2. Then we haveWC2(S) = 1+ (WCI(~) -1)· Sri-I.sProof. Let ni be the length of the code C i for i = 1,2, Le., nl =# Sand nz =# SC. Since #(H n S) +# (H n SC) =# H = N(k - 2) for anyhyperplane H in pk-l, we havenz -#(H n SC) #Sc -#(H n SC)(N(k - 1) _# S) - (N(k - 2) -#(H n S))qk-l _ (#S -#(H n S))qk-l _ (nl -#(H n S)).Hence the sum of weights of codewords in Cl and Oz, corresponding tothe same hyperplane H, is qk-l, which implies the desired formula. 0495Masaaki Homma, Soon Joong!Gm, and Mi Ja YooIn the following lemmas, we deal with the case when the linear spanof B is not a whole space.LEMMA 2.2. Let B be a subset ofr-1 whose linear span is not thewhole space r-1• Then the linear span of BC is the whole space.Proof. Choose arbitrary points P and Q in B and (B)C, respectively.Then the line PQ intersects (B) at only one point P, otherwise Q shouldbe contained in (B). Thus the other q points on the line PQ is containedin (BY, and hence in BC. Thus the line PQ is contained in (BC), andhence P E (BC). 0LEMMA 2.3. Let B be a subset ofr-1 with dim (B) = [ - 1 < k - l.Letand let Ci be the corresponding nondegenerate code to Xi for i = 1,2.Then we haveProof. Note that Cl is an [-dimensional code. Let ni be the length ofthe code Ci for i = 1,2, Le., nl =# B and n2 =# BC. We may assume(B) = pi-I C r-1• Let H be any hyperplane in r-I.If (B) c H, thenn2 -#(H n BC) = (N(k -1) _#S) - (N(k - 2) _# B) = qk-I.If (B) et H, thenn2 -#(H n BC) (N(k - 1) _# B) - (N(k - 2) -#(H n B))qk-I - (#B -#(H(s) n B))- qk-I _ (nl -#(H(s) n B)),where H(s} = H n (B) is the hyperplane in (B) = pi-I.496Projective systems supported on the complement of two linear subspacesSince the number ofhyperplanes in r- l containing (S) is N(k-1-1),we haveWC2(S) = 1 + (N(k -1) - N(k - 1 -l))(WcI (~) - 1) . SI-I+(q - l)N(k - 1 _l)sQk-1k I 1 I-Il+(q-Wcl (-)-l)s ,Sas desired. 0REMARK 2.4.(1) If 1 = k and dim(SC) = k - 1, then the formula in Lemma 2.3 issame with that of Lemma 2.l.(2) If 1 = k and dim (SC) < k - 1, we may change the role of S andSC and use Lemma 2.3.Now we compute the weight enumerators of codes corresponding toO-cycle whose support is the complement of linear subspaces in generalposition in r- l .THEOREM 2.5. Let Li, i = 1,2"" ,r, be linear subspaces of di-mension Si in r- l , where Si'S satisfy 0 :s; SI :s; S2 :s; ... :s; Sr < k - 1and L:~=1 Si + r :s; k. Assume that Li's are in general position. LetT = r-l \ Ui=1 Li , X = L:PET P be a O-cycle, and let C be the codecorresponding to X. ThenWc(s) = 1 + (qk-l f( ~) - l)sl-l,Swhere 1- 1 = dim(Ui=ILi) = SI + S2 + + Sr + r - 1 andf(s) = (1 + (qSI+l - l)s£1'I) (1 + (qSr+l _ l)sQST).Proof. Since the nondegenerate code corresponding to the linear sub-space L i is a simplex code, its weight enumerator is 1 + (qs;+1 - l)sQS;.Hence the weight enumerator of the code corresponding to the union ofL/s is just f(s), since it is a direct sum of the codes corresponding to Li ,i = 1,2"" ,r. Now Lemma 2.3 completes the proof. 0497d -->Masaaki Homma, Soon Joong Kim, and Mi Ja YooWe prove a particular property of the codes in Theorem 2.5.THEOREM 2.6. Ifr < q, then the code in Theorem 2.5 achieves theGriesmer bound in Lemma 1.1.Proof The length of the code is n = N(k - 1) - :E~=l N(Si)' For theminimum distance d,we haven - max{#(H n T) I H is a hyperplane in r-1}n - (#H - min{#(H n (Ui=lLi» IH is a hyperplane in r-1})r r(N(k -1) - LN(Si») - (N(k - 2) - LN(Si -1»)i=l i=lrk-t- """ Si- q - L-q ,-i=land equality holds if and only if there exists a hyPerplane containingno Li , i = 1,2,," ,r. To prove the existence of such a hyperplane, wecompute#(all hyperplanes) - # (hyperplanes J Li for some i = 1, 2, ... ,r)r2 N(k -1) - LN(k -1- Si -1)i=l2 N (k-1)-rN(k-2)2 N(k -1) - qN(k - 2) = 1.Thus d = qk-l - :E~=l if; .Since r < q by assumption, we have'"" . """ 1 rL- qSi-J :::; L- - :::; - < 1,Si<j Si<j q qfor all j = 1,2, ... , k - 1, and henceIqk-l-j - L qS;-j - L qSi-jlSi?j Siljrk-l-j _ """ "si-jq L-'l'Si?j498Projective systems supported on the complement of two linear subspacesHence(the right hand side of Griesmer bound)r r r= (qk-1 _ LqSi) + rqk-2 - LqSi-11 + ... + r1- LqSi-(k-1)1i=1 i=1 i=1r= (qk-1 _ LqSi) + (qk-2 _ LqSi-1) + (1-3 _ LqSi-2)i=1 Si2=: 1 Si2=:2+ ... + (q - L qSi-(k-2)) + 1si2=:k-2r= N(k - 1) - LN(Si) = n.i=1Thus the proof is complete. oThe following example shows that the condition r < q is necessary inTheorem 2.6.EXAMPLE 2.1. In the projective space p3 over JF2, let 8 be the unionof two skew lines. LetX= L P,PEp3\Sand C be the code corresponding to o-cycle X. Then we can easily provethat C is [9,4, 4h-code but it does not satisfy the equality in Griesmerbound.3. Uniqueness for r = 2In this section we prove that any two codes with the same weightenumerator appeared in Theorem 2.5 are equivalent to each other.THEOREM 3.1. Let 8 be a subset ofIlJ>n with #8 = N (s )+N (t), s ~ t,and S + t = n - 1. Suppose that for any hyperplane H in Jll>n,#(H n 8) = N(s) + N(t -1), N(s -1) + N(t), or N(s -1) + N(t -1),499Masaa.ki Homma, Seon Jeong Kim, and Mi Ja Yooand{#{H E pm 1# (H n S) = N(s) +N(t -I)} = N(t),#{H E pm 1# (H n S) = N(s - 1) + N(t)} = N(s).)Then S is the union of two linear subspaces in general position ofdimen-sion s and t, respectively.Proof. We divide the proof into five claims.Claim 1. If a line contains at least 3 points of S, then it is containedin S.Proof of Claim 1. Let{P, a point not contained in SM, a line through P~, the set of all hyperplanes containing PM, the set of all hyperplanes containing M.Then #(P \ M) = N(n -1) - N(n - 2) = qn-l. Let #(M n S) = u. Byassumption, #(H n S) ~ N(s -1) +N(t -1) for all hyperplane H, so wehaveL #(H n S) > #(P\M)· (N(s -1) + N(t-l»HEP\M_ ~-l. (N(s - 1) + N(t - 1)).On the other hand, for any point Q E S \ M,#(all hyperplanes in P \ M containing Q) - #((p n Q) \ (M n Q))- N(n - 2) - N(n - 3)_ ~-2.Hence we haveL #(HnS) - #(S\M)·#((PnQ)\(MnQ))HEP\M= (N(s) + N(t) - u) . qn-2.Comparing above two formulas, we have(N(s) + N(t) - u) . qn-2 ~ qn-l . (N(s - 1) + N(t -1»,which is equivalent to u ~ 2. Thus Claim 1 is proved.500Projective systems supported on the complement of two linear subspacesClaim 2. Let Ho be a hyperplane such that #(Hon S) = N(s -1) +N(t). Then every line containing two points in S \ Ho meets Ho at apoint in S.Proof of Claim 2. Suppose not. Then there exists a line M such that{#(Mn(S\Ho)) 2 2MnHo ft S.Let P = M n Ho. By Claim 1, #(M n S) = 2. In the computationin the proof of Claim 1, we have equality, which means #(H n S) =N (s - 1) + N (t - 1) for any hyperplanes H E P \ M. This contradictsthe choice of Ho.Claim 3. S contains a linear subspace of dimension s. More precisely,S contains the linear span Lo= (Ho n S) of Ho n S and dim Lo = s.Proof of Claim 3. By Claim 1 and 2, we conclude that Lo is containedin S. Thus Ho n Lo = Ho n S. Let dimLo = r. ThenN(r) =# (Lo) - #(H8 n Lo) +# (H9n Lo)= #(Hg n S) +#(Hon Lo)(N(s) - N(s - 1)) + N(r - 1),since Ho n Lo = Ho n Sand Lo is not contained in Ho.Thus we have r = s and we have proved Claim 3.Claim 4. Let S' = S \ Lo. Then #(S') = N(t) and#(H n S') = N(t) or N(t - 1)for any hyperplane H in JPTI. Moreover, there exist H l and H2 such that{#(Hl n S') = N(t)#(H2 n S') = N(t - 1).Proof of Claim 4. For any hyperplane H, note that{#(H n S) =# (H n Lo) +# (H n S')#(H n Lo) = N(s) or N(s -1),and by assumption of the theorem#(H n S) = N(s) + N(t - 1), N(s - 1) + N(t), or N(s - 1) + N(t - 1).501MasaakiHomma, Soon Joong Kim, and Mi Ja YooIf s > t, then N(s) is bigger than N(s - 1) + N(t) and N{s - 1) +N(t - 1). Hence, for a hyperplane H with #(H n Lo) = N(s), we have#(H n S) = N(s) +N(t -1). Indeed,#{H E pn* 1# (H n Lo) = N(s)} - #{H E pn* ILo C H}- N(n - s - 1) = N(t),which is equal to#{H E pn* \# (H n S) = N(s) + N(t -I)}.Thus #(H n Sf) = N(t) or N(t - 1) for any H E p~ and{#{H E pn* 1# (H n Sf) = N(t)} = N(s)#{H E r 1# (H n Sf) = N(t - I)} = N(n) - N(s).If s = t, then N(s) >N{s -'- 1) + N(t-..:- 1) and N(s) + N(t - 1) =N(s - 1) +N(t). Counting the number of hyperplanes again, we have{#{H Er /# (H n Sf) = N(t)} = N(s)#{H E pn* /#(H n Sf) = N(t -I)} = N(n) - N(s).Thus we have proved the Claim 4.Claim 5. S' is a t-dimensionallinear subspace of pn.Proof of Claim 5. We modify the proof of Claim 1 to show that anyline through two points in Sf is contained in Sf.Let{P, a point not contained in SfM, a line through PP, the set of all hyperplanes containing PM, the set of all hyperplanes containing M.Then #(P \ M) = N(n -1) - N(n - 2) = qn-l. Let #(M n Sf) = u. Byassumption, #(H n Sf) 2: N(t -1) for all hyperplane H, so we haveL #(HnSf ) > #(P\M)·N(t-l)HeP\M_ qn-l. N(t - 1).502Projective systems supported on the complement of two linear subspacesOn the other hand, for any point Q E 8' \ M,#(all hyperplanes in P \ M containing Q) - #((p n Q) \ (M n Q))- N(n - 2) - N(n - 3)_ qn-2.Hence we have:E #(H n S') = #(S' \ M) .#((p n Q) \ (M n Q))Hef>\M= (N(t) - u) . qn-2.Comparing above two formulas, we have(N(t) - u) . qn-2 ~ qn-l . N(t - 1)which is equivalent to u ~ 1. That means that any line not containedin S' has at most one point of S', or equivalently, any line containingtwo points in 8' is contained in S'. Hence S' is a linear subspace, anddim 8' = t since #(S') = N(t).Combining above 5 claims, we conclude that 8 is the union of twolinear subspaces of dimension s and t, and they are in general positionsince there is no hyperplane containing both of them and s+t = n-1. 0Finally we prove a uniqueness theorem using the following lemma.LEMMA 3.2. Let C and C' be nondegenerate codes with the sameweight enumerator. If every coefIi.cient in the o-cycle Xc is ~ 1, i.e.,Xc = :EpeSupp(.:!<:) P, then Xc' = :EpeSuPp(xcl) P. In this case, we canidentify O-cycle with its support.IProof. The assumption .:\::C = :EpeSupp(.:!<:) P means that any twocolumns of a generator matrix of C are linearly independent, which isequivalent to the minimum distance of Cl. being ~ 3, by Lemma 1.3.Since the weight enumerators of Cl. and c'l. are same by MacWilliamsidentity, we conclude that the minimum distance of c'l. is ~ 3, and hencethe lemma is proved. 0THEOREM 3.3. If a code C' over lFq has the same weight enumeratorwith the code in Theorem 2.5 with r = 2, then C' is equivalent to C.503Masaaki Homma, Seon Jeo~ Kim, and Mi Ja YooProof By Lemma 3.2, we have Xc' = LpeSuPp(Xet) P, and we identifythe o-cycle Xc' with its support. Let T = Supp(Xc') and let S = re inpk-l. Then, by Lemma 2.3, we can check that S satisfies the conditionsin Theorem 3.1 where n = s + t + 1. Hence S is the union of two linearsubspaces in general position in JPl'l. Therefore, we can find a projectiveautomorphism on pk-l which maps S onto Ll UL2 in Theorem 2.5. Thusthe code C' is equivalent to C. 0The following example shows that a code appeared in Theorem 2.5 isnot determined by its parameters [n, k, d] alone.EXAMPLE 3.1. In the projective space p3 over lF2, let S, X, C be sameas Example 2.1. Let 8 1 be a punctured plane. LetXl = L P,Pep3\Sland let Cl be a code corresponding to o-cycle Xl. Then we can proveeasily that Cl is [9,4, 4h-code, same as C. However their weight. enumer-ators are{Woes) = 1 + 9s4 + 6s6WC1(s) = 1 + 6s4 + 8s5 + s8.Note that the weight enumerator WcCs) is the polynomial appeared inTheorem 2.5.References[1] J. W. P. Hirschfeld, Projective Geometry over Finite Fields, Oxford UniversityPress, Oxford, 1998.[2] M. Homma, S. J. Kim, S. H. Park, and S. Y. Yoon, A family of linear rodesbetween a first order Reed-Muller rode and a si'f.nplex rode, preprint, 1998. '[3] F. J. MacWilliams and N. J. A. Sloane, The theory of Error-Correcting Codes,Elsevier Science Publishers, Amsterdam, 1977.[4] M. A. Tsfasman and S. G. VIadu\, Algebmic-Geometric Codes, Kluwe~ AcademicPublishers, Dordrecht, 1991.MASAAKI HOMMA, DEPARTMENT OF MATHEMATICS, KANAGAWA UNIVERSITY,ROKKAKUBASHI KANAGAWA-KU YOKOHAMA 221, JAPANE-mail: homma@cc.kanagawa-u.ac.jp504Projective systems supported on the complement of two linear subspacesSEON JEONG KIM, DEPARTMENT OF MATHEMATICS AND RESEARCH INSTITUTEOF NATURAL SCIENCE, GYEONGSANG NATIONAL UNIVERSITY, CHINJU 660-701,KOREAE-mail: skim.@nongae.gsnu.ac.krMY JA Yoo, DEPARTMENT OF MATHEMATICS AND RESEARCH INSTITUTE OF NAT-URAL SCIENCE, GYEONGSANG NATIONAL UNIVERSITY, CHINJU 660-701, KOREA505

Projective systems supported on the complement of two linear subspaces

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Projective systems supported on the complement of two linear subspaces

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