ON THE RELATIVE GENERALIZED HAMMING WEIGHTS OF A 4-DIMENSIONAL LINEAR CODE AND A SUBCODE WITH DIMENSION ONE

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1 J Syst Sci Complex (2012) 25: ON THE RELATIVE ENERALIZED HAMMIN WEIHTS OF A 4-DIMENSIONAL LINEAR CODE AND A SUBCODE WITH DIMENSION ONE Zihui LIU Wende CHEN DOI: /s Received: 5 August 2010 / Revised: 2 January 2011 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2012 Abstract Finite projective geometry method is effectively used to study the relative generalized Hamming weights of 4-dimensional linear codes, which are divided into 9 classes in order to get much more information about the relative generalized Hamming weights, and part of the relative generalized Hamming weights of a 4-dimensional linear code with a 1-dimensional subcode are determined. Key words eneralized Hamming weight, relative difference sequence, relative generalized Hamming weight, support weight. 1 Introduction The relative generalized Hamming weights (RHWs) of a linear code were first introduced in [1] by improving the wire-tap channel of type II with the coset coding scheme invented by Ozarow and Wyner [2], and they generalized the well-known generalized Hamming weights (HWs) [3] to a two-code format. Like HWs, RHWs not only have usage in analyzing the coordinated multiparty wire-tap channel of type II [1], but also bring us much information about the structure of a linear code and its subcodes, so to determine HWs and RHWs is a basic and meaningful work. The HWs of linear codes with small dimensions (for example, dimension = 3, 4, 5) were extensively studied in [4 8]. In parallel with these research work, Ozarow and Wyner [2] determined all the RHWs of a 3-dimensional linear code with its subcodes. Part of the RHWs of a 4-dimensional linear code with its subcodes were determined in [10]. More concisely, in [10], almost all the RHWs of a 4-dimensional linear code with a subcode with dimension greater than one were determined, and for a 4-dimensional linear code with a subcode with dimension one, only the RHWs of the former four classes were studied according to the classification in this paper, and other classes remained unsolved. In this paper, we will study the properties of the RHWs of other classes and determine part of the RHWs. Zihui LIU Department of Mathematics, Beijing Institute of Technology, Beijing , China. lzhui@bit.edu.cn. Wende CHEN Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing , China. This research is supported by the National Natural Science Foundation of China under rant Nos and , and the Special Training Program of Beijing Institute of Technology. This paper was recommended for publication by Editor Xiao-Shan AO.

2 822 ZIHUI LIU WENDE CHEN 2 Finite Projective eometry Methods 2.1 Notations and Definitions Let J be a subset of I = {1, 2,,n}. The subcode C J of a linear [n, k] codec is defined as the set {(c 1,c 2,,c n ) C c t =0fort/ J}. For any subcode D of C, the support of D denoted by χ(d), is defined as the set of positions where not all the codewords of D have zero coordinates. The support weight of D denoted as w S (D) isdefinedasfollows:w S (D) = χ(d). Definition 1 [3] Let C be an [n, k] linear code. The HWs of C are defined as the parameters d r,1 r k, where d r (C) =min{w S (D) D is an [n, r] subcode of C}, 1 r k. In particular, d 1 is the minimum distance of C. Definition 2 [1] Let C be an [n, k] linear code and C 1 be a k 1 -dimensional subcode of C. Then the RHWs of C and C 1 are defined as the parameters M j,1 j k k 1,where Or equivalently, M j = M j (C, C 1 )=min{ J dim(c J ) dim(c 1 J) j}, 1 j k k 1. M j (C, C 1 )=min{ J dim(c J ) dim(c 1 J)=j}, 1 j k k 1. Obviously, if C 1 = {0}, then the HWs of C are retrieved. RHWs are therefore a generalization of HWs. In order to introduce finite geometry methods, we give another version of the definition of RHWs. Definition 3 [9] The RHWs of C and C 1 can also be described as the parameters M j, 1 j k k 1, where M j (C, C 1 )=min{w S (D) D is a subcode of C, dim D = j, D C 1 = {0}} =min{w S (D) D is a subcode of C, dim D dim D C 1 = j}. Definition 4 Assume the RHWs of an [n, k] codec andan[n, k 1 ] subcode C 1 are (M 1,,M k k1 ). Let M 0 = 0, then the relative difference sequence (RDS) of C and C 1 is the sequence (i 0, i 1,, i k k1 ), where i 0 = n M k k1, i j = M k k1 j+1 M k k1 j, 1 j k k 1. Obviously, (i 0, i 1,, i k k1 )and(m 1, M 2,, M k k1, n) can be determined from each other. 2.2 Projective Subspaces and Subcodes We describe projective geometry methods in what follows. Let u =(u 1,u 2,,u k ) F (q) k be a row vector (or the column vector (u 1,u 2,,u k ) T ), and L be a subset of the k coordinate positions of the vector u. Define P L (u) F (q) k such that the t-th component of P L (u) isu t if t L and 0 if t/ L. An example is as follows: Let L = {2, 4} and u =(u 1,u 2,u 3,u 4,u 5 ) F (q) 5 (or the column vector (u 1,u 2,u 3,u 4,u 5 ) T ), then P L (u) = (0,u 2, 0,u 4, 0) (or P L (u) = (0,u 2, 0,u 4, 0) T ). If L = {1, 2}, then P L (u) = (u 1,u 2, 0, 0, 0) (or P L (u) =(u 1,u 2, 0, 0, 0) T ). For a subset U F (q) k, define P L (U) ={P L (u) u U}. Obviously, if U is a subspace of F (q) k,soisp L (U).

3 ON THE RELATIVE ENERALIZED HAMMIN WEIHTS 823 Let A be a generator matrix for a k-dimensional linear code C, thena can naturally determine a correspondence: m: F (q) k Z such that for any u F (q) k, m(u) denotes the number of occurrences of u as a column in A. m(u) is called the value of u. Define the value of a subset U of F (q) k as follows: m(u) = u U m(u). Lemma 1 [10] matrix of C is Let C be an [n, k] code, and C 1 an [n, k 1 ] subcode of C. Assume a generator A = ( Ak1 n A (k k1) n where A k1 n is a generator matrix for C 1. Then there is a one-one correspondence between the set of the j-dimensional subcodes D satisfying D C 1 = {0} and the set of the (k j)- dimensional subspaces U F (q) k satisfying dim(p L (U)) = k 1 such that if D corresponds to U, thenm(u) =n w S (D), where1 j k k 1,andL = {1, 2,,k 1 } represents the first k 1 coordinate positions of the vectors in F (q) k. Let C, C 1,D and A be as in Lemma 1. If y is a column of A, andx = αy for some nonzero α F (q), then we may replace y by x without changing the support weight of the subcode D. Therefore, as in [4] (also see [11]), we may look at the columns of A as the projective points in P(k 1,q). Then m(x) means the number of occurrences of x as a point in P(k 1,q)in the columns of A. Then the correspondence m : F (q) k Z defined before becomes a value function (also called a value assignment [4] ) ), m : P(k 1,q) Z( 0). For any point p P(k 1,q), we call m(p) thevalueofp. Correspondingly, m(u) in Lemma 1 is called the value of the projective subspace U whenever we view the vectors in U as projective points in P(k 1,q). A value function m( ) defines a generator matrix and a code (up to equivalence). Definition 3 tells us that M j (C, C 1 )=min{w S (D) D is a subcode of C, dim D = j, D C 1 = {0}}, 0 j k k 1. We can assume M j (C, C 1 )=w S (D )forsomed satisfying D C 1 = {0}. By Lemma 1, D corresponds to a (k j)-dimensional subspace U F (q) k satisfying dim(p L (U )) = k 1 such that m(u )=n w S (D ). Now, we consider U as a (k j 1)-dimensional projective subspace of P(k 1,q), and still denote it by U.Thendim(P L (U )) = k 1 1. So we can get max{m(u) U P(k 1,q), dim U = k j 1 and dim(p L (U)) = k 1 1} k k 1 j = m(u )=n w S (D )=n M j = i t, where 0 j k k 1. (1) t=0 Summarize finite projective geometry methods as follows: to construct a linear k-dimensional code C and a k 1 -dimensional subcode C 1 with the parameters (M 1, M 2,, M k k1, n), it is necessary to construct a value function m( ) satisfying (1).

4 824 ZIHUI LIU WENDE CHEN 3 On the RHWs of a 4-Dimensional Linear Code C and a 1-Dimensional Subcode C Classification Let E =(1, 0, 0, 0),F =(0, 1, 0, 0), =(0, 0, 1, 0),H =(0, 0, 0, 1) be the basis points of V = P(3,q) in this section (see Figure 1). The projective subspace spanned by two points, e.g., A and B, is called a line, and denoted AB. The projective subspace spanned by three points, e.g., A, B and C, is called a plane, and denoted ÂBC. By Lemma 1, the j-dimensional (1 j 3) subcodes D satisfying D C 1 = {0} are 1-1 corresponding to the (3 j)-dimensional subspaces U P(3,q) satisfying dim(p L (U)) = k 1 1=0,whereL = {1} {1, 2, 3, 4}. Itisafactthatdim(P L (U)) = 0 for L = {1} means U FH. So, the construction of a value function m( ) satisfying (1) is equivalent to that of m( ) such that max{m(p) p is a point of P(3,q)andp/ FH} = i 0, max{m(l) l is a line of P(3,q)andl FH} = i 0 + i 1, max{m( ) is a plane of P(3,q)and FH} = i 0 + i 1 + i 2, m(v )=m(p(3,q)) = i 0 + i 1 + i 2 + i 3. (2) Define MU 0 = {p m(p) =i 0,p is a point and p/ FH}, MU 1 = {l m(l) =i 0 + i 1,l is a line and l FH}, MU 2 = { m( ) =i 0 + i 1 + i 2, is a plane and FH}. We introduce some conditions which may or may not be true. Then according to these conditions we can divide the analysis of the RHWs of a 4-dimensional code and a 1-dimensional subcode into several classes. In such a way, we can obtain much information about the RHWs. (Con1) There exist p MU 0,l MU 1, and P MU 2 such that p l P. (Con2) There exist p MU 0 and l MU 1 such that p l. (Con3) There exist p MU 0 and P MU 2 such that p P. (Con4) There exist l MU 1 and P MU 2 such that l P. According to whether (Con1) (Con4) are true or false, we can divide codes (and value functions) into 9 classes: Class A: Con2, Con3, Con1, Con4 true. Then we call the RDS ARDS. Class B: Con2, Con3 true; Con1, Con4 false. Then we call the RDS BRDS. Class C: Con3, Con4 true; Con1, Con2 false. Then we call the RDS CRDS. Class D: Con2, Con3, Con4 true; Con1 false. Then we call the RDS DRDS. Class E: Con1, Con2, Con4 false, Con3 true. Then we call the RDS ERDS. Class F: Con1, Con3, Con4 false, Con2 true. Then we call the RDS FRDS. Class : Con1, Con3 false; Con2, Con4 true. Then we call the RDS RDS. Class H: Con1, Con2, Con3 false; Con4 true. Then we call the RDS HRDS. Class I: Con1, Con2, Con3, Con4 false. Then we call the RDS IRDS. We have described in [10] the properties of the RHWs of Class A Class D by a group of non-equalities satisfied by their RDS. Conversely, we have showed part of the sequences satisfying these non-equalities are the RHWs of Class A Class D, respectively.

5 ON THE RELATIVE ENERALIZED HAMMIN WEIHTS 825 In this paper, we will give the properties of the RHWs of all other 5 classes of codes and determine part of the RHWs. We also show that the sequences satisfying the upper bound of the non-equalities in Class B and Class D are the RHWs of these two classes, respectively. 3.2 Results about the RHWs Class A: See [10]. Class B: The result of Class B is Theorem 1 If a sequence (i 0,i 1,i 2,i 3 ) is a BRDS, then it must satisfy i 0 1, i 1 2, i 2 qi 1 (q +1), i 3 qi 2 (q +1),i 3 i 1. Conversely, almost all the sequences satisfying the upper bound are BRDSs. E F H Figure 1 The upper bound construction of Class B Proof We have showed a BRDS has the properties listed above in [10]. Conversely, to show almost all the sequences (i 0, i 1, i 2, i 3 ) satisfying the upper bound, that is, satisfying i 2 = qi 1 (q +1)andi 3 = qi 2 (q + 1), are BRDSs, it is necessary to construct a value function m( ) satisfying the conditions in Class B for almost all such sequences. In Figure 1, choose a point on each of the q + 1 lines passing through F in the plane FH and denote them by I 1, I 2,, I q+1, respectively. Let I t / H for 1 t q +1. Thenm( ) can be constructed as follows: i 1, p = F, i 1 1, p H, m(p) = i 1 3, p = I k, where k =1, 2,,q+1, i 1 1, other points in FH, 0, other points in P(3,q). Using this value function, we can check that m(e) =i 0, m(ef) =i 0 + i 1 and m(êh) = i 0 + i 1 + i 2. So E MU 0, EF MU 1 and ÊH MU 2. Using this value function, we can further check all the other conditions in Class B are satisfied. Note that such a value function is meaningful only if i 1 3. So, we can get all the sequences satisfying i 2 = qi 1 (q +1),i 3 = qi 2 (q +1), i 1 3, and i 0 1 are BRDS. Comparing with all the sequences in Theorem 1 satisfying the upper bound, we can see that almost all the sequences satisfying the upper bound are BRDS.

6 826 ZIHUI LIU WENDE CHEN Class C: See [10]. Class D: The result of Class D is Theorem 2 If a sequence (i 0,i 1,i 2,i 3 ) is a DRDS, then i 0 1, i 1 1, i 0 i 2 qi 1 (q +1), i 1 i 3 qi 2 (q +1). Conversely, almost all the sequences (i 0,i 1,i 2,i 3 ) satisfying the upper bound are DRDSs. E F I H Figure 2 The upper bound construction for Class D. Proof See [10] for the proof of the properties satisfied by DRDSs. It is still necessary to give the upper bound construction of m( ). Denote the points on the line H in Figure 2 by 0 =, 1,, γ,, q 1, q = H, and the points on the line HF by H 0 = H, H 1,, H γ,, H q 1, H q = F. Assume F i H i = O i for 1 i q 1. Choose a point on the line F and denote it by K. Suppose E i I = J i for 1 i q and i 0 +1=qα + β. Then we can get the upper bound construction m( ) whenever i 0 2q q 1 and i 1 3. i 1, p = F, i 1 1, p =, α +1, p = J i, 1 i β, α, other points on I\{}, i 1 1 α 1, p = i, 1 i β, m(p) = i 1 1 α, other points on H\{}, i 1 3, p = O i, 1 i q 1, i 1 3, p = H 1, i 1 3, p = K, i 1 1, other points in FH, 0, other points in P(3,q). We can show that m(eh) =i 0 + i 1 + i 2 and m(i) =i 0 + i 1, and other conditions stated in Class D can be checked one by one. So m( ) satisfies our demands. In the following upper bound constructions, we need Lemma 2 If there exist a point p MU 0 andalinel MU 1 such that p/ l then i 0 qi 1 ; If Con2 false, then i 0 qi 1 (q +1).

7 ON THE RELATIVE ENERALIZED HAMMIN WEIHTS 827 Proof Assume i 0 >qi 1,thenm(l) =i 0 + i 1 > (q +1)i 1,sothereexistsapointp l such that m(p ) >i 1,thenm(pp 0 ) >i 0 + i 1, a contradiction to (2). Similar arguments show that if Con2 false, then i 0 qi 1 (q +1). Class E: The result of Class E is Theorem 3 If a sequence (i 0,i 1,i 2,i 3 ) is an ERDS, then it must satisfy { i0 +1 i 2 qi 1 (q +1), i 0 +1 i 3 qi 2 (q +1). Conversely, almost all the sequences satisfying the upper bound are ERDSs. Proof If a sequence (i 0,i 1,i 2,i 3 ) is an ERDS, then there exist a point p MU 0 and a plane MU 2 such that p, but any line in passing through the point p has the value at most i 0 + i 1 1, so i 0 + i 1 + i 2 = m( ) (q +1)(i 0 + i 1 1) qi 0,thatis,i 2 qi 1 (q + 1). Further, any plane in P(3,q) passing through a fixed line l MU 1 has a value at most i 0 + i 1 + i 2 1, so i 0 + i 1 + i 2 + i 3 = m(p(3,q)) (q +1)(i 0 + i 1 + i 2 1) q(i 0 + i 1 ), that is, i 3 qi 2 (q +1). Assume p 0 = l,thenm(p 0 ) i 1 1sincem(pp 0 ) i 0 + i 1 1. Since m(p(3,q)\ ) = i 3 m(l\p 0 ), we get that i 0 +1 i 3. Further, the plane passing through the point p and the line l has a value at most i 0 + i 1 + i 2 1, so m(p)+m(l) =i 0 +(i 0 + i 1 ) i 0 + i 1 + i 2 1, that is, i 0 +1 i 2. Conversely (see Figure 3), assume i 0 +1=qα + β and ĜIJ FH = L, wherel FH and 0 β q 1. Denote the q points on the line I\{} by 1, 2,, q 1, q = I, respectively. Suppose E i H = O i for 1 i q, whereo q = H. Then the upper bound construction is given as follows: m(p) = i 1 3, p = F, i 1 1, other points on the line F 2, p = J α +1, p = i for 1 i β α, other points on the line I\{} i 1 2 α, p = O i for 1 i β i 1 1 α, other points on the line H\{, H} i 1 1, points on the plane FH i 1 5, p = L i 1 2 α, p = H = O q i 1 3, p FH\{F, H, L} 0, other points in P(3,q). We first note that i 1 α = i 1 i0+1 q > 0 by Lemma 2, so we can make m(p) 0 for all the points p P(3,q)onlyifi 1 is large enough. So the value function m( ) is meaningful. In addition, we can check that I MU 1, ÊF MU 2 and other details so that m( ) satisfies our demands.

8 828 ZIHUI LIU WENDE CHEN E J F I H L Figure 3 The upper bound construction for Class E Class F: The result of Class F is Theorem 4 If a sequence (i 0,i 1,i 2,i 3 ) is an FRDS, then it must satisfy { i2 qi (I) 1 +1 i 0 i 3 qi 2 (q +1) or (II) qi 1 +1 i 2 i 0 i 3 qi 2 (q +1) i 2 + i 3 (q 2 + q)i 1, conversely, almost all the sequences satisfying the upper bound in (I) are FRDSs. Proof The proof of i 0 i 3 qi 2 (q + 1) is similar to Class E, and the details are omitted. Since any plane passing through a fixed point p MU 0 has the value at most i 0 + i 1 + i 2 1 or i 0 +(q +1)i 1,wegetthati 2 qi 1 +1ifi 0 + i 1 + i 2 1 i 0 +(q +1)i 1 or i 2 qi 1 +1if i 0 +(q +1)i 1 i 0 + i 1 + i 2 1. There are altogether (q 2 + q + 1) lines passing through the point p in P(3,q), so i 0 + i 1 + i 2 + i 3 = m(p(3,q)) i 0 +(q 2 + q +1)i 1,soi 2 + i 3 (q 2 + q)i 1. Note that i 2 + i 3 (q 2 + q)i 1 is not necessary in (I) since we can get it from (I) itself. E F I H Figure 4 the upper bound construction for Class F (I) Conversely (see Figure 4), assume i 0 +1 = q 2 ζ +(q +1)η +σ, where0 η q 1, 0 σ q and σ =0ifη = q 1. Denote the points on the line Fby F 0 = F, F 1,, F q =, and denote the q 1 points on the line IF i \{I,F i } by ξ i1, ξ i2,, ξ iq 1 for 0 i q. LetQ represent an arbitrary point in FH\F, and suppose EQ FI = E Q. Then we can construct a desired

9 ON THE RELATIVE ENERALIZED HAMMIN WEIHTS 829 upper bound value function as follows. m(p) = i 1, p F, ζ +1, p = ξ i1,ξ i2,ξ iη+1 for 0 i, σ 1, ζ +1, p = ξ i1,ξ i2,ξ iη for σ i q, ζ, the other points in FI\F, i 1 m(e Q ), p = Q FH\F, 0, the other points in P(3,q). Using this value function, we can get all the sequences satisfying the upper bound in (I) are FRDSs only if i 0 >q+1. Class : The result of Class is Theorem 5 If a sequence (i 0,i 1,i 2,i 3 ) is a RDS, then it must satisfy (I) { i0 +1 i 2 qi 1 +1, i 0 i 3 qi 2 (q +1), or (II) { max{i0 +1,qi 1 +1} i 2, i 2 + i 3 (q 2 + q)i 1, conversely, almost all the sequences satisfying the upper bound of (I) are RDSs. E J F I H Figure 5 The upper bound construction for Class (I) Proof Similarly as before, we only give an upper bound value function, the details are omitted (see Figure 5). Assume i 0 = qα + β and choose a point J IF\{I,F}. Suppose EJ FH = K. If a sequence is a RDS, then there exist a point p MU 0, two lines l 1 MU 1, l 2 MU 1 and a plane MU 2 such that p l 1, l 2 but p / l 2. So i 1 i0 q = α by Lemma 2. Denote the q points on I\{} by I 0 = I, I 1,, I q 1. Suppose EI k H = H k,where0 k q 1andH 0 = H. Then we can give an upper bound value function as follows: 1, p = J, α +1, p = I k for 0 k β 1, α, the other points of I\{}, m(p) = i 1 α 1, p = H k for 0 k β 1, i 1 α, the other points of H\{}, i 1 1, p = K, i 1, the other points in FH, 0, the other points in P(3,q).

10 830 ZIHUI LIU WENDE CHEN Class H: The result of Class H is Theorem 6 If a sequence (i 0,i 1,i 2,i 3 ) is an HRDS, then it must satisfy i 0 +1 i 2 qi 1 q, max{i 0 +1,qi 1 q} i 2, (I) i 2 + i 3 (q 2 + q)i 1 (q 2 + q +1), or (II) i 2 + i 3 (q 2 + q)i 1 (q 2 + q +1), i 0 i 3, i 0 i 3, conversely, almost all the sequences satisfying the upper bound of (I) are HRDSs (see Figure 6). E F I H Figure 6 The upper bound construction for Class H (I) Proof Still denote the q points on I\{} by I 0 = I, I 1,, I q 1. Suppose EI k H = H k,where0 k q 1andH 0 = H. Assume i 0 +1 = qα + β, then we can get i 1 > i0+1 q = α again from Lemma 2. So an upper bound value function below is meaningful: m(p) = α +1, p = I k for 0 k β 1, α, p = I k for β k q 1, i 1 2 α, p = H k for 0 k β 1, i 1 1 α, p = H k for β k q 1, i 1 1, the other points in FH, 0, the other points in P(3,q). It can be checked similarly that this value function satisfies our demands only if i 0 is large enough, that is, i 0 >f(q), where f(q) is a function of q. Class I: The result of Class I is Theorem 7 If a sequence (i 0,i 1,i 2,i 3 ) is an IRDS, then it must satisfy { i0 +1 i (I) 2 qi 1 q, 2i 0 +1 i 3 qi 2 (q +1), or (II) max{i 0 +1,qi 1 q} i 2, 2i 0 +1 i 3 qi 2 (q +1), i 2 + i 3 (q 2 + q)i 1 (q 2 + q +1), conversely, almost all the sequences satisfying the upper bound of (I) are HRDSs (see Figure 7). Proof Assume i 0 +1 = qα+β and α =(q 1)ξ +γ, where0 β q 1 and0 γ q 2, then i 0 +2=q(q 1)ξ + qγ + β + 1. Choose the points I,J EH\{E,H}. Denotetheq +1 points on the line FH by F 1 = F, F 2,, F q+1 = H. Denote the q points on H\{} by H 1 = H, H 2,, H q. Assume ĜJF 2 EF = K and EH k J = J k for 1 k q. LetQ be an arbitrary point in ĜIF \{I F }, andeq ĜF H = E Q.ThenE Q ĜHF \{H F }.

11 ON THE RELATIVE ENERALIZED HAMMIN WEIHTS 831 E K F I H J Figure 7 The upper bound construction for Class I (I) We first assign to qγ + β + 1 points among ĜIF \{I F } each the value ξ + 1, and to the other points in ĜIF \{I F } each the value ξ. Then we can give m(p) = 1, p = K, i 1 2, p = F, i 1 1, p F\F, α +1, p = J k for 1 k β, α, p = J k for β k q, i 1 2 α, p = H k for 1 k β, i 1 1 α, p = H k for β k q, i 1 3 m(q), p = E Q = F 2, i 1 2 m(q), p = E Q = F k for 3 k q, i 1 1 m(q), p = E Q, ĜHF \{H F FH}, 0, the other points in P(3,q). Using this value function, we can similarly show that all the upper bound sequences are IRDSs only if i 0 is large enough, that is, i 0 h(q), where h(q) is a function of q. 4 Conclusion Finite geometry methods are effective tools to study RHWs. In this paper, we classify the codes with dimension 4 into 9 classes, and then use finite geometry methods to study the properties of the RHWs of each class and determine part of the RHWs of each class. We have shown almost all the sequences satisfying the upper bound of each class are the corresponding RHWs. Acknowledgements The authors wish to thank the editor and the referees for their comments and suggestions that helped to improve this paper. References [1] Y. Luo, C. Mitrpant, A. J. Han Vinck, and K. F. Chen, Some new characters on the wire-tap channel of type II, IEEE Trans. Inform. Theory, 2005, 51(3):

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