The automorphism group of a linear space with the Rosenbloom Tsfasman metric

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1 European Journal of Combinatorics 24 (2003) The automorphism group of a linear space with the Rosenbloom Tsfasman metric Kwankyu Lee Department of Mathematics, Sogang University, Seoul , Republic of Korea Received 3 April 2003; received in revised form 29 May 2003; accepted 30 May 2003 Abstract Let Mat s n (F q ) be the linear space of all s n matrices over the finite field F q,equipped with the new metric introduced by Rosenbloom and Tsfasman. We determine the full group of weightpreserving linear automorphisms of Mat s n (F q ) with the Rosenbloom Tsfasman metric Elsevier Ltd. All rights reserved. 1. Introduction Let Mat s n (F q ) be the linear space of all s n matrices over the finite field F q. Rosenbloom and Tsfasman [2] introduced a new metric on the space Mat s n (F q ), generalizing the usual Hamming metric on F n q.afterwards several authors studied codes in the space Mat s n (F q ) equipped with the Rosenbloom Tsfasman metric. In particular, Skriganov [1, 3] definedsome natural linear automorphisms of the space preserving the metric, and he asked if these linear automorphisms generate the full group of linear automorphisms preserving the metric. In this paper, we answer his question affirmatively, and determine the group explicitly. 2. The Rosenbloom Tsfasman metric We define weights for the matrices in Mat s n (F q ) as follows. Let v be a matrix in Mat s n (F q ).For1 j n, definethedepth of the jth column of v to be depth j (v) max{i v ij 0}, assuming max{} 0. Then the weight ρ(v) of the matrix v is defined by summing depths address: kwankyu@hotmail.com (K. Lee) /03/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi: /s (03)

2 608 K. Lee / European Journal of Combinatorics 24 (2003) of all columns, ρ(v) depth j (v). j1 It is easy to check that ρ indeed induces a metric on Mat s n (F q ) : ρ(v) 0with equality if and only if v 0andρ(v + w) ρ(v) + ρ(w) for every v, w Mat s n (F q ).Note that if s 1, then ρ induces just the usual Hamming metric on Mat 1 n (F q ) F n q.hence the induced metric from ρ, called the Rosenbloom Tsfasman metric, can be viewed as a generalization of the Hamming metric induced from the usual Hamming weight. Note that our definition of the Rosenbloom Tsfasman metric is slightly different from the usual definitions found in [1 3]. Our definition uses columns and counts weight from thetop while other definitions use rows or count from the bottom. Though obviously these definitions are equivalent, we found our definition more natural, at least for our present problem to determine the full group of linear automorphisms preserving the metric. 3. The automorphism group From now on, the weight of a matrix means its Rosenbloom Tsfasman weight. Note that a linear automorphism preserves the Rosenbloom Tsfasman metric if and only if it preserves weights of matrices. For convenience, let G Mat s n (F q ),andaut(g) denote the group of all weight-preserving linear automorphisms of G. We now determine the group Aut(G). Let T s be the group of all upper triangular matrices of size s over F q with nonzero diagonal elements. Then (T s ) n denotes the direct product of n copies of T s.thereisa natural action of (T s ) n on G. Letτ (τ 1,...,τ n ) (T s ) n.letc [c 1 c n ] G where c i denotes the ith column of c.thenthe action of τ on c is given by τ c [τ 1 c 1 τ n c n ]. Each τ i being an upper triangular matrix with nonzero diagonal elements guarantees that c and τ c have the same depths on each column (because being upper triangular gives that thedepth is not increased and having nonzero diagonal elements gives that the depth is not decreased), and havethe same weights. Clearly this action is faithful. Thus the permutation representation of this action of (T s ) n on G gives an injective group homomorphism α of (T s ) n into Aut(G). Let S n be the symmetric group of n letters. S n naturally acts on G. Letσ S n and c [c 1 c n ] G.Theaction of σ on c is given by σ c [c σ 1 (1) c σ 1 (n) ]. This action is also faithful. And hence the permutation representation of this action gives an injective group homomorphism β of S n into Aut(G). Now our answer to Skriganov s question whether (T s ) n and S n generate Aut(G) is Theorem 1. The group of all linear automorphisms of Mat s n (F q ) preserving the Rosenbloom Tsfasman metric is a semidirect product of (T s ) n and S n.

3 K. Lee / European Journal of Combinatorics 24 (2003) We now prove this theorem in several steps. First some notations. For each 1 l s, let H l {c G c ij 0, i > l, 1 j n}. Notethat H l is a linear subspace of G of dimension ln.for each 1 i s, 1 j n, lete ij denote the matrix whose (i, j)th component is 1 and the other components are all zero. Lemma 2. We have α((t s ) n ) β(s n ) {1 G } where 1 G denotes the identity automorphism on G. Proof. Let ϕ α((t s ) n ) β(s n ).Suppose ϕ β(σ) for some σ S n.ifσ is not the identity element of S n,thereisa j with 1 j n such that σ(j) j. Then ϕ(e 1 j ) σ e 1 j e 1σ(j).Notethat depth j (e 1 j ) 1butdepth j (e 1σ(j) ) 0. So ϕ changes the depths of columns of e 1 j.sincethe action of any τ (T s ) n does not change depths of columns, it is impossible that ϕ α(τ) for some τ (T s ) n.weconclude that σ is the identity element of S n,andhence ϕ β(σ) is the identity automorphism on G. Now let ϕ be a weight-preserving linear automorphism of G. Wewill show that ϕ can be written as the composition of α(τ) and β(σ) with some τ (T s ) n and some σ S n. See Lemmas 3 5. Let 1 j n. Notethat ρ(ϕ(e 1 j )) 1because ρ(e 1 j ) 1. So ϕ(e 1 j ) a j e 1σ(j) for some nonzero a j F q and σ(j) with 1 σ(j) n. Lemma 3. The map σ defined above is one-to-one, and hence a permutation in S n. Proof. Assume otherwise, and suppose σ(u) σ(v) for some 1 u,v n with u v. Then observe ϕ(a v e 1u a u e 1v ) a v ϕ(e 1u ) a u ϕ(e 1v ) a v a u e 1σ(u) a u a v e 1σ(v) 0, where 0 denotes the zero matrix. Since ρ(a v e 1u a u e 1v ) 2, this is a contradiction because ϕ preserves weights. This proves our assertion that σ is a permutation in S n. Lemma 4. For each 1 i s, 1 j n, we canwrite ϕ(e ij ) i (τ j ) ki e kσ(j) ( ) for some (τ j ) ki (1 k i) in F q with nonzero (τ j ) ii. Proof. We prove by induction on i. Leti 1. For every 1 j n, wehavealready shown ϕ(e 1 j ) a j e 1σ(j). Therefore if we take (τ j ) 11 a j for 1 j n,then( ) is satisfied. Let l > 1, and assume ( ) holds for 1 i < l. Wenow proceed to prove the claim ( ) for i l. Let 1 j n. Fromρ(e lj ) l, weseethat depth r (ϕ(e lj )) l for all 1 r n. Assume depth r (ϕ(e lj )) < l for all 1 r n. Wewill show that this assumption leads to

4 610 K. Lee / European Journal of Combinatorics 24 (2003) acontradiction. Our induction hypothesis implies that {ϕ(e ij ) 1 i < l, 1 j n} forms a basis of H l 1.Soϕ,restricted to the subspace H l 1,isanautomorphism of H l 1. Therefore there is a c H l 1 such that ϕ(c) ϕ(e lj ).Thenϕ(c e lj ) ϕ(c) ϕ(e lj ) 0. This is a contradiction because ρ(c e lj ) l. Itfollows that depth r (ϕ(e lj )) l for some 1 r n.sinceρ(e lj ) l,thisalsoimplies that depth s (ϕ(e lj )) 0fors r.therefore we can write l ϕ(e lj ) (τ j ) kl e kr, for some (τ j ) kl F q (1 k l) with nonzero (τ j ) ll.itremains to show that actually r σ(j).ifnot, then the weight of ϕ(e lj + e 1 j ) ϕ(e lj ) + ϕ(e 1 j ) l (τ j ) kl e kr + (τ j ) 11 e 1σ(j) is l + 1buttheweight of e lj + e 1 j is l.this is a contradiction because ϕ preserves weights. Thus we showed l ϕ(e lj ) (τ j ) kl e kσ(j) with nonzero (τ j ) ll.byinduction our claim ( ) is proved. Let 1 j n. Fork, l with 1 k l s, seethat ( ) defines (τ j ) kl.for k > l, set(τ j ) kl 0. Then note that each τ j is an upper triangular matrix over F q. Let τ (τ σ 1 (1),...,τ σ 1 (n) ) (T s) n. Lemma 5. We have ϕ α(τ) β(σ). Proof. Let us observe that for c G, ϕ(c) ϕ c ij e ij c ij ϕ(e ij ) i, j i, j i, j c ij j1 i1 j1 j1 i (τ j ) ki e kσ(j) c ij (τ j ) ki e kσ(j) ( ) (τ j ) ki c ij e kσ(j) i1 (τ j c j ) k e kσ(j),

5 K. Lee / European Journal of Combinatorics 24 (2003) where in the last expression, (τ j c j ) k denotes the kth component of the column vector τ j c j. So we have α(τ) β(σ)(c) τ (σ c) [τ σ 1 (1) c σ 1 (1) τ σ 1 (n) c σ 1 (n) ] (τ σ 1 ( j) c σ 1 ( j) ) ke kj j1 j1 ϕ(c) (τ j c j ) k e kσ(j) for every c G. Hence α(τ) β(σ) ϕ. We now showed that every weight-preserving linear automorphism ϕ of G can be written as ϕ α(τ) β(σ) for some τ (T s ) n, σ S n.lastly we check the normality of thesubgroup α((t s ) n ) in Aut(G). Lemma 6. α((t s ) n ) is a normal subgroup of Aut(G). Proof. For σ S n, τ (τ 1,...,τ n ) (T s ) n, observe that for c G, (β(σ ) α(τ) β(σ) 1 )(c) (β(σ ) α(τ) β(σ 1 ))(c) (β(σ ) α(τ))([c σ(1) c σ(n) ]) β(σ)([τ 1 c σ(1) τ n c σ(n) ]) [τ σ 1 (1) c 1 τ σ 1 (n) c n] α((τ σ 1 (1),...,τ σ 1 (n) ))(c). So we have β(σ) α(τ) β(σ) 1 α((τ σ 1 (1),...,τ σ 1 (n) )) for every σ S n, τ (τ 1,...,τ n ) (T s ) n.thisobservation implies that α((t s ) n ) is a normal subgroup of Aut(G). Proof of Theorem 1. With all the above lemmas, we conclude that we have an isomorphism (T s ) n ψ S n Aut(G) given by (τ, σ ) α(τ) β(σ),with ψ : S n Aut((T s ) n ) defined by ψ(σ)(τ 1,...,τ n ) (τ σ 1 (1),...,τ σ 1 (n) ) for σ S n, (τ 1,...,τ n ) (T s ) n. Hence we proved that the group of all linear automorphisms of Mat s n (F q ) preserving therosenbloom Tsfasman metric is isomorphic to the semidirect product (T s ) n ψ S n.

6 612 K. Lee / European Journal of Combinatorics 24 (2003) Acknowledgements Ithank my supervisor Prof. Kim for his useful suggestions and first introduction to the paper [1]. I also thank M. M. Skriganov for kindly providing me with his papers and for good advice. Finally I thank the referee for his comments which improved the presentation of this paper. The author was supported by the Seoam Scholarship Foundation and by grant No.R (2003) from the Basic Research Program of the Korea Science and Engineering Foundation. References [1] S.T. Dougherty, M.M. Skriganov, MacWilliams duality and the Rosenbloom Tsfasman metric, Mosc. Math. J. 2 (2002) [2] M.Yu. Rosenbloom, M.A. Tsfasman, Codes for the m-metric, Probl. Inf. Transm. 33 (1997) [3] M.M. Skriganov, Coding theory and uniform distributions, St. Petersburg Math. J. 13 (2002)