Johnson Type Bounds on Constant Dimension Codes
aa r X i v : . [ c s . I T ] S e p Johnson Type Bounds on Constant DimensionCodes ∗ Shu-Tao Xia † and Fang-Wei Fu ‡ Abstract
Very recently, an operator channel was defined by Koetter and Kschis-chang when they studied random network coding. They also introducedconstant dimension codes and demonstrated that these codes can be em-ployed to correct errors and/or erasures over the operator channel. Con-stant dimension codes are equivalent to the so-called linear authentica-tion codes introduced by Wang, Xing and Safavi-Naini when constructingdistributed authentication systems in 2003. In this paper, we study con-stant dimension codes. It is shown that Steiner structures are optimalconstant dimension codes achieving the Wang-Xing-Safavi-Naini bound.Furthermore, we show that constant dimension codes achieve the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures.Then, we derive two Johnson type upper bounds, say I and II, on con-stant dimension codes. The Johnson type bound II slightly improves onthe Wang-Xing-Safavi-Naini bound. Finally, we point out that a family ofknown Steiner structures is actually a family of optimal constant dimensioncodes achieving both the Johnson type bounds I and II. keywords:
Constant dimension codes, linear authentication codes, binaryconstant weight codes, Johnson bounds, Steiner structures, random network cod-ing. ∗ This research is supported in part by the NSFC-GDSF Joint Fund under Grant No.U0675001, and the open research fund of National Mobile Communications Research Labo-ratory, Southeast University. † S.-T. Xia is with the Graduate School at Shenzhen of Tsinghua University, Shenzhen,Guangdong 518055, P. R. China. He is also with the National Mobile Communications ResearchLaboratory, Southeast University, P.R. China. E-mail: [email protected] ‡ F.-W. Fu is with the Chern Institute of Mathematics, and The Key Laboratory of PureMathematics and Combinatorics, Nankai University, Tianjin 300071, P.R. China. Email:[email protected] Introduction
Throughout this paper, F q denotes the finite field with q elements, where q is aprime power. Let W be an n -dimensional vector space over F q and let P ( W )denote the set of all subspaces of W . For any A, B ∈ P ( W ), denote A + B = { a + b : a ∈ A, b ∈ B } , that is the smallest subspace containing both A and B . It is known [3] that the dimension distance between A and B defined by d ( A, B ) = dim( A + B ) − dim( A ∩ B ) (1)= dim( A ) + dim( B ) − A ∩ B ) (2)is a metric for the space P ( W ). A q -ary ( n, M, D ) or ( n, M, D ) q code C is simplya subset of P ( W ) with size M and minimum dimension distance D which isdefined by D = D ( C ) = min X = Y ∈C d ( X, Y ) . (3)For any positive integer l ≤ n , let P ( W, l ) denote the set of all l -dimensionalsubspaces of W . For integers 0 ≤ m ≤ n and q ≥
2, let h nm i q = m − Y i =0 q n − i − q m − i − q -binomial coefficient or Gaussian binomial coefficient [7, pp.443-444].It is well known that |P ( W, l ) | = (cid:2) nl (cid:3) q . A q -ary ( n, M, δ, l ) or ( n, M, δ, l ) q con-stant dimension code is simply a subset of P ( W, l ) with size M and minimumdimension distance 2 δ . Note that by (2) the dimension distance of any two code-words of a constant dimension code must be an even number and 1 ≤ δ ≤ l .An ( n, M, ≥ δ, l ) q constant dimension code is a subset of P ( W, l ) with size M and minimum dimension distance at least 2 δ . For fixed numbers n, l, δ, q , denote A q [ n, δ, l ] the maximum number M of codewords in an ( n, M, ≥ δ, l ) q constantdimension code. An ( n, M, ≥ δ, l ) q constant dimension code is said to be optimal if M = A q [ n, δ, l ]. One of the main research problems on constant dimensioncodes is to determine A q [ n, δ, l ] and find corresponding optimal constant dimen-sion codes. 2enote X ⊥ the orthogonal complement of X ∈ P ( W ). For any two l -dimensional subspaces X, Y ∈ P ( W, l ), since X ⊥ ∩ Y ⊥ = ( X + Y ) ⊥ , we have d ( X ⊥ , Y ⊥ ) = dim( X ⊥ ) + dim( Y ⊥ ) − X ⊥ ∩ Y ⊥ )= n − dim( X ) + n − dim( Y ) − n − dim( X + Y ))= d ( X, Y ) . (4)Let C ⊆ P ( W, l ) be an ( n, M, δ, l ) q constant dimension code. Then by (4) weknow that ¯ C , { X ⊥ : X ∈ C} is an ( n, M, δ, n − l ) q constant dimension code.This implies that A q [ n, δ, l ] = A q [ n, δ, n − l ] . (5)Hence, we only need to determine A q [ n, δ, l ] for l ≤ n/ operator channel and found that an ( n, M, ≥ δ, l ) q constantdimension code C could be employed to correct errors and/or erasures over theoperator channel, i.e., the errors and/or erasures could be corrected by a minimumdimension distance decoder if the sum of errors and erasures is less than δ . Somebounds on A q [ n, δ, l ], e.g., the Hamming type upper bound, the Gilbert typelower bound, and the Singleton type upper bound, were derived in [3]. It isknown that the Hamming type bound is not very good [3] and there exist nonon-trivial perfect codes meeting the Hamming type bound [5, 6]. The Singletontype bound developed in [3] is the following: Proposition 1 [3, Th.3] (Singleton type bound) A q [ n, δ, l ] ≤ (cid:20) n − δ + 1 l − δ + 1 (cid:21) q . Moreover, Koetter and Kschischang [3] designed a class of Reed-Solomon likeconstant dimension codes and afforded decoding procedures. They showed thatthese codes were nearly Singleton-type-bound-achieving.In 2003, Wang, Xing and Safavi-Naini [4] introduced the so-called linear au-thentication codes when constructing distributed authentication systems. They[4, Th.4.1] showed that an ( n, M, ≥ δ, l ) q constant dimension code is exactly an[ n, M, t = n − l, d = δ ] linear authentication code over F q . Furthermore, theyestablished an upper bound [4, Th.5.2] on linear authentication codes, which isequivalent to the following bound on constant dimension codes:3 roposition 2 [4, Th.5.2] (Wang-Xing-Safavi-Naini Bound) A q [ n, δ, l ] ≤ (cid:2) nl − δ +1 (cid:3) q (cid:2) ll − δ +1 (cid:3) q . Moreover, Wang, Xing and Safavi-Naini [4] presented some constructions of lin-ear authentication codes (or corresponding constant dimension codes) that areasymptotically close to this bound.In this paper, we show that Steiner structures are optimal constant dimensioncodes achieving the Wang-Xing-Safavi-Naini bound in Proposition 2. Further-more, it is shown that constant dimension codes achieve the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures. Two Johnson typeupper bounds, say I and II, on constant dimension codes are derived. The John-son type bound II slightly improves on the Wang-Xing-Safavi-Naini bound. It isobserved that the Wang-Xing-Safavi-Naini bound is always better than the Sin-gleton type bound for nontrivial constant dimension codes. Finally, we point outthat a family of known Steiner structures is actually a family of optimal constantdimension codes achieving both the Johnson type bounds I and II.
In this section we first introduce the combinatorial objectives Steiner structures.Then we show that constant dimension codes achieve the Wang-Xing-Safavi-Nainibound if and only if they are certain Steiner structures. This means that Steinerstructures are optimal constant dimension codes. Finally we describe the onlyknown family of nontrivial Steiner structures in combinatorics.Recall that W is the n -dimensional vector space over the finite field F q and P ( W, l ) denote the set of all l -dimensional subspaces of W . The following defini-tion and proposition on Steiner structures are from [5]. Definition 1 [5]
A subset
F ⊆ P ( W, l ) is called a Steiner structure S [ t, l, n ] q ifeach t -dimensional subspace of W is contained in exactly one l -dimensional sub-space from F . The l -dimensional subspaces in F are called blocks of the Steinerstructure S [ t, l, n ] q . Proposition 3 [5]
The total number of blocks in an S [ t, l, n ] q is (cid:2) nt (cid:3) q / (cid:2) lt (cid:3) q . Proposition 4
A Steiner structure S [ t, l, n ] q is an ( n, M, δ, l ) q constant dimen-sion code with M = (cid:2) nt (cid:3) q / (cid:2) lt (cid:3) q and δ = l − t + 1 .Proof: By Definition 1 and Proposition 3, we only need to show that δ = l − t + 1. For any two different blocks X, Y ∈ S [ t, l, n ] q , since every t -dimensional subspace is contained in exactly one block of S [ t, l, n ] q , we havedim( X ∩ Y ) ≤ t −
1. Thus, by (2), d ( X, Y ) = 2 l − X ∩ Y ) ≥ l − t + 1),which implies that δ ≥ l − t + 1. On the other hand, let V be a fixed ( t − W , choose two t -dimensional subspaces U and U of W such that V = U ∩ U . Let X and X be the unique blocks in S [ t, l, n ] q suchthat U ⊆ X and U ⊆ X , respectively. Then, V ⊆ X ∩ X , which implies thatdim( X ∩ X ) ≥ dim( V ) = t −
1. Hence, by (2), d ( X , X ) ≤ l − t + 1). Thus, δ ≤ l − t + 1 since 2 δ is the minimum dimension distance of S [ t, l, n ] q . Combiningthese assertions, δ = l − t + 1. This completes the proof.Next we give the necessary and sufficient condition for constant dimensioncodes to achieve the Wang-Xing-Safavi-Naini bound in Proposition 2. Theorem 1 An ( n, M, ≥ δ, l ) q constant dimension code C achieves the Wang-Xing-Safavi-Naini bound, i.e., M = [ nl − δ +1 ] q [ ll − δ +1 ] q , if and only if C is a Steiner structure S [ l − δ + 1 , l, n ] q .Proof: Since an ( n, M, δ, l ) q constant dimension code is an ( n, M, ≥ δ, l ) q constant dimension code, we know from Propositions 2 and 4 that a Steinerstructure S [ l − δ + 1 , l, n ] q is an ( n, M = [ nl − δ +1 ] q [ ll − δ +1 ] q , ≥ δ, l ) q constant dimensioncode achieving the Wang-Xing-Safavi-Naini bound.On the other hand, suppose there exists an ( n, M = [ nl − δ +1 ] q [ ll − δ +1 ] q , ≥ δ, l ) q con-stant dimension code C achieving the Wang-Xing-Safavi-Naini bound. Since thedimension distance between any two different codewords of C is not small than2 δ , it follows from (2) that each ( l − δ +1)-dimensional subspace could not be con-tained in two different codewords. Moreover, since each codeword of C contains (cid:2) ll − δ +1 (cid:3) q distinct ( l − δ + 1)-dimensional subspaces, all codewords of C contains to-tally M (cid:2) ll − δ +1 (cid:3) q = (cid:2) nl − δ +1 (cid:3) q pairwisely different ( l − δ + 1)-dimensional subspaces.Note that there are totally (cid:2) nl − δ +1 (cid:3) q distinct ( l − δ + 1)-dimensional subspaces5f W . Hence, each ( l − δ + 1)-dimensional subspace is contained in exactly onecodeword of C . Therefore, regarding the codewords of C as blocks, C forms aSteiner structure S [ l − δ + 1 , l, n ] q by its definition.Theorem 1 shows that Steiner structures are optimal constant dimensioncodes. The following corollary follows from Theorem 1 imediately. Corollary 1 A q [ n, δ, l ] = (cid:2) nl − δ +1 (cid:3) q (cid:2) ll − δ +1 (cid:3) q if and only if a Steiner structure S [ l − δ + 1 , l, n ] q exists. It is known [5 ,
6] that trivial Steiner structures S [ t, n, n ] q and S [ t, t, n ] q existfor all t ≤ n . For nontrivial Steiner structures, by our knowledge, it is only known[5 ,
6] that S [1 , l, n ] q exists where l | n , and the blocks of S [1 , l, n ] q form a partitionof W (excluding the zero vector). For completeness, we review the construction[5 ,
6] of such an S [1 , l, n ] q where n = kl as follows. Let e = ( q kl − / ( q l −
1) andlet α be a primitive element of F q n . Define h α e i = { , α e , α e , . . . , α ( q l − e } . The cosets C i = α i h α e i = { α i , α i + e , α i +2 e , . . . , α i +( q l − e } , i = 0 , , . . . , e − cyclotomic classes of order e . Let E i = C i ∪ { } . Note that F q n is an n -dimensional vector space over F q . One can verify that E = F q l and the E i ’s,when viewed as subsets of F nq , are l -dimensional subspaces of W = F nq . Regardingall E i ’s as blocks, an S [1 , l, kl ] q is obtained.From Corollary 1, we have the following result. Corollary 2
For any positive integers k and l , we have A q [ kl, l, l ] = q kl − q l − . (6) In this section, we first review some basic definitions and the Johnson bound I forbinary constant weight codes in coding theory. It is shown that a corresponding6inary constant weight code can be obtained from a given constant dimensioncode. Then, using the Johnson bound I for this corresponding binary constantweight code, we obtain the Johnson type bound I for constant dimension codes.It is observed that this bound is tight in some cases.Let F n be the n -dimensional vector space over the binary field F . For anytwo vectors a , b ∈ F n , the Hamming distance d H ( a , b ) is the number of coordi-nates in which they differ, the Hamming weight w H ( a ) is the number of nonzerocoordinates in a . It is known that d H ( a , b ) = w H ( a ) + w H ( b ) − w H ( a ∗ b ) (7)where a ∗ b = ( a b , a b , . . . , a n b n ) . A binary code C of length n is a nonempty subset of F n . The minimum distanceof C is the minimum Hamming distance between any two distinct codewords in C . A binary constant weight code is a binary code such that every codeword hasa fixed Hamming weight. Denote A ( n, δ, w ) the maximum number of codewordsin a binary constant weight code with length n , weight w and minimum distanceat least 2 δ . We state the Johnson bound I for binary constant weight codes inthe following proposition. Proposition 5 (Johnson bound I) [8] If w > n ( w − δ ) , then A ( n, δ, w ) ≤ (cid:22) nδw − n ( w − δ ) (cid:23) , where ⌊·⌋ denotes the floor function. Below we show that a corresponding binary constant weight code can beobtained from a given constant dimension code.Recall that W is the n -dimensional vector space over the finite field F q and P ( W, l ) denote the set of all l -dimensional subspaces of W . Let denote theall-zero vector in W and W ∗ = W \ { } . Denote N = q n −
1. Suppose all thevectors in W ∗ are ordered from 1 to N . Define the incidence vector of a subset X ⊆ W by v X = ( v , v , . . . , v N ) ∈ F N v i = 1 if the i -th vector of W ∗ is contained in X , and v i = 0 otherwise.For any two l -dimensional subspaces X, Y ∈ P ( W, l ), by (7) it is easy to see that w H ( v X ) = w H ( v Y ) = q l − , (8) w H ( v X ∗ v Y ) = q dim( X ∩ Y ) − , (9) d H ( v X , v Y ) = 2( q l − q dim( X ∩ Y ) ) . (10)Let C be an ( n, M, δ, l ) q constant dimension code. By (8), the incidence vectorsof the codewords in C form a binary constant weight code C , which is called the derived binary constant weight code of C . From (8), (10) and the definition ofconstant dimension codes, we have the following result. Proposition 6
Let C be an ( n, M, δ, l ) q constant dimension code. Then itsderived binary constant weight code C has the following parameters: length N = q n − , size M , minimum distance q l − q l − δ ) , and weight q l − . Although every constant dimension code corresponds to a binary constantweight code, the reverse proposition may not hold. Given a binary ( q n − , M, q l − q l − δ ) , q l −
1) constant weight code, since its codewords may not be the incidencevectors of any subspaces, the code may not correspond to any ( n, M, δ, l ) q con-stant dimension code. From Propositions 5 and 6 we obtain the Johnson typebound I for constant dimension codes. Theorem 2 (Johnson type bound I for constant dimension codes) If ( q l − > ( q n − q l − δ − , then A q [ n, δ, l ] ≤ (cid:22) ( q l − q l − δ )( q n − q l − − ( q n − q l − δ − (cid:23) . The Johnson type bound I for constant dimension codes is tight in somecases. By Proposition 4 and Corollary 2, the Steiner structure S [1 , l, kl ] q is a( kl, q kl − q l − , l, l ) q constant dimension code achieving the Johnson type bound I forconstant dimension codes. Remark 1
There is another method to obtain a binary constant weight codefrom a constant dimension code. Let ˜ W be the set of all 1-dimensional subspacesof W . Denote ˜ N = q n − q − . We can regard ˜ W as P G ( n − , q ), the ( n − F q with ˜ N points [7, Appendix B], where8ach X ∈ P ( W, l ) corresponds to an ( l − P G ( n − , q ), say ˜ X . Supposeall points in P G ( n − , q ) are ordered from 1 to ˜ N . Define the punctured incidencevector of X ∈ P ( W, l ) as the incidence vector of ˜ X ∈ P G ( n − , q ). By puttingtogether the punctured incidence vectors of all codewords of an ( n, M, δ, l ) q constant dimension code C , we obtain a corresponding binary constant weightcode ˜ C which has length ˜ N = q n − q − , size M , minimum distance q l − q l − δ ) q − , andweight q l − q − . Note that the derived binary constant weight code C can be obtainedby concatenating ( q −
1) times of ˜ C . Hence, we have A q [ n, δ, l ] ≤ A (cid:18) q n − q − , q l − q l − δ ) q − , q l − q − (cid:19) . (11)However, by employing the Johnson bound I for binary constant weight codes,(11) implies the same results with Theorem 2. In this section, we derive an upper bound for constant dimension codes. We callthis upper bound the Johnson type bound II for constant dimension codes sinceit is similar to the Johnson bound II for binary constant weight codes [8]. TheJohnson type bound II for constant dimension codes slightly improves on theWang-Xing-Safavi-Naini bound.Let V , V ∈ P ( W ) and V ⊆ V . Define( V | V ) ⊥ = { a ∈ V : ∀ b ∈ V , ab T = 0 } , i.e., ( V | V ) ⊥ is the orthogonal complement of V in V . For any S ⊆ W , denote h S i the minimum subspace containing S . Theorem 3 A q [ n, δ, l ] ≤ (cid:22) q n − q l − A q [ n − , δ, l − (cid:23) . Proof:
Suppose C is an optimal ( n, M, ≥ δ, l ) q constant dimension codewith M = A q [ n, δ, l ]. Consider the binary M × ( q n −
1) matrix, say P , whoserows consist of all the codewords of C , where C is the derived binary constantweight code of C in Proposition 6. Denote ∆ the total number of 1’s in the matrix P . Since each codeword of C has weight q l − M ( q l −
1) = A q [ n, δ, l ]( q l − . (12)9n the other hand, we will show that the number of 1’s in each column of P is not greater than A q [ n − , δ, l − W . Without loss of generality,suppose α ∈ W is the non-zero vector which indexes the first column of P . Let C = { X ∈ C : the first component of v X is 1 } . Hence, the weight of the column indexed by α equals |C | . Noting that α ∈ X for any X ∈ C , let C ′ = { ( h α i| X ) ⊥ : X ∈ C } and W = ( h α i| W ) ⊥ . Clearly, W is an ( n − F q and each element of C ′ is an( l − W . Hence, C ′ ⊆ P ( W , l −
1) is a q -ary constantdimension code with length n −
1, size |C | , and dimension l −
1. Moreover, for anytwo different codewords of C ′ , e.g., ( h α i| X ) ⊥ and ( h α i| Y ) ⊥ , where X = Y ∈ C , d (( h α i| X ) ⊥ , ( h α i| Y ) ⊥ )= 2( l − − h α i| X ) ⊥ ∩ ( h α i| Y ) ⊥ ]= 2( l − − h α i| X ∩ Y ) ⊥ )= 2 l − X ∩ Y )= d ( X, Y ) ≥ δ. Hence, C ′ is an ( n − , |C | , ≥ δ, l − q constant dimension code, which impliesthat |C | ≤ A q [ n − , δ, l − . The weight of the column indexed by α is not greater than A q [ n − , δ, l − P , we have that∆ ≤ ( q n − A q [ n − , δ, l − . Combining this with (12) and noting that A q [ n, δ, l ] is an integer, we obtain therequired conclusion.Using Theorem 3 recursively, we obtain the Johnson type bound II for con-stant dimension codes. Corollary 3 (Johnson type bound II for constant dimension codes) A q [ n, δ, l ] ≤ (cid:22) q n − q l − (cid:22) q n − − q l − − (cid:22) · · · (cid:22) q n − l + δ − q δ − (cid:23) · · · (cid:23)(cid:23)(cid:23) . B S , B W XS , B J denote respectively the Singleton type bound in Propo-sition 1, the Wang-Xing-Safavi-Naini bound in Proposition 2, and the Johnsontype bound II in Corollary 3. For example, letting q = 2, n = 6, δ = 2 and l = 3, we have B S = 155, B W XS = 93 and B J = 90. Below we show that theWang-Xing-Safavi-Naini bound is always better than the Singleton type boundfor δ > n > l . Since for i = 0 , , . . . , l − δ ,( q n − l + i +1 − q i +1 − ≥ ( q n − l + δ + i − q δ + i − ⇐⇒ ( q n − l + i +1 − q i +1 )( q δ − − ≥ , we have that B S = ( q n − δ +1 − q n − δ − · · · ( q n − l +1 − q l − δ +1 − q l − δ − · · · ( q − ≥ ( q n − q n − − · · · ( q n − l + δ − q l − q l − − · · · ( q δ −
1) = B W XS (13)and the equality holds if and only if δ = 1 or n = l . Furthermore, by [3, Lemma5], 1 < q − l ( n − l ) (cid:2) nl (cid:3) q < < l < n . Using the similar arguments in the proofof [3, Lemma 5], we obtain1 < q − m ( u − v ) ( q u − q u − − · · · ( q u − m +1 − q v − q v − − · · · ( q v − m +1 − < ≤ m ≤ v < u. (14)Hence, by (13) and (14), it is easy to see that for δ > < l < nB W XS < B S < q ( l − δ +1)( n − l ) < B W XS . (15)For example, letting n = 100, l/n = 0 . δ/n = 0 .
2, it is computed with the
Mathematica software that B S B W XS ≈ . , . , . , . , for q = 2 , , , , respectively. In this paper we show that Steiner structures, e.g., S [1 , l, kl ] q , are optimal con-stant dimension codes or linear authentication codes, and could be applied inrandom network coding or distributed authentication systems. Furthermore, it isshown that constant dimension codes achieve the Wang-Xing-Safavi-Naini bound11f and only if they are certain Steiner structures. We derive two Johnson typebounds for constant dimension codes. It would be interesting to construct moreconstant dimension codes which achieve Johnson type bounds I or II. It is a hardproblem to determine A q [ n, δ, l ] in general. However, one can first make effortsto determine A q [ n, , l ], A q [ n, , l ] and A q [ n, l − , l ] in the following steps. Acknowledgment
The authors would like to thank Professor Cunsheng Ding for reading this paperand giving valuable comments that helped to improve the paper.
References