The Symmetries of the $pi$-metric

The Symmetries of the π -metric Marcelo Muniz Silv a Alv es ∗ Luciano P anek † Abstract Let V b e an n -dimensional vector space o ver a finite field F q . W e consider on V the π -metric d π recentl y introduced by K. F eng, L. Xu and F. J. Hick ernell. In this pap er w e give a complete description of the group of symmetries of the metric space ( V , d π ). Key wor ds : error-block cod e, π -metric, co d e automorphism, symmetry 1 In tro duction Let F n q be the finite field o f q elements. Let n b e a p ositive integer and let π be a partition of n , i.e., π is a non-empty sequence ( k 1 , k 2 , . . . , k m ) where each k i is a p ositive integer, n = k 1 + . . . + k m , m ≥ 1 and k 1 ≥ k 2 ≥ . . . ≥ k m ≥ 1. F eng, Xu and Hick er nell introduced in [1] a metric associa ted to the partition π , the π - metric , similar to the clas sic Hamming metric. The pa r tition π induces a direc t sum decomp osition F n q = F k 1 q ⊕ F k 2 q ⊕ . . . ⊕ F k m q where e very vector v of F n q is wr itten a s v = ( v 1 , v 2 , . . . , v m ) , with v i ∈ F k i q . The π -weight of v = ( v 1 , v 2 , . . . , v m ) ∈ F n q is ω π ( v ) = |{ i : 1 ≤ i ≤ m, v i 6 = 0 }| and the π - dis tance betw een u and v is given b y d π ( u, v ) = ω π ( u − v ). Note that when k i = 1 for all i (and m = n ) the π -weigh t is the Hamming weigh t ov er F n q . It is clear that d π ( u, v ) = |{ i : 1 ≤ i ≤ m, u i 6 = v i }| . A F q -linear c o de in  F n q , d π  is called a line ar err or-blo ck c o de . The linear sy mmetr ies (automorphisms) of F n q with r esp ect to a π -metr ic hav e already b een determined in [2]. In this pap er we describ e all s ymmetries of  F n q , d π  ; in o rder to do that, we de s crib e tw o subgroups M and S π of symmetries and prove that the full symmetry group is their s emi- direct pro duct. W e also reo btain the automo rphism gro up as a particular case. F rom this po int on we write simply d and ω instead of d π and ω π . ∗ Departamen to de Matem´ atica, Cen tro Polit´ ecnico, U FPR , Caixa Postal 019081, 81531-990, Curitiba, PR. E-m ail: marcelomsa@ufpr.br † Cen tro de Enge nharias e Ci ˆ encias, UNIOESTE, 85870 -650, F oz do Igua¸ cu, PR. E-mail: luc- panek@gmail.com 1 2 The Symmetry Group A s ymmet ry o f  F n q , d  is a bijection ϕ : F n q → F n q that pres e r ves distance, i.e., d ( ϕ ( u ) , ϕ ( v )) = d ( u, v ) for every u, v ∈ F n q . An automorphism is a linear symmetry . W e will denote the group of symmetries of  F n q , d  by S y mm  F n q , d  . Let M b e the gr o up of mappings T : F n q → F n q , where T (( v 1 , v 2 , . . . , v m )) = ( T 1 ( v 1 ) , . . . , T m ( v m )) and every “co o rdinate function” T i is a bijection from F k i q onto itself. Since each T i is a bijection, T i ( v i ) 6 = T i ( u i ) if a nd only if u i 6 = v i . It follows that d ( T ( u ) , T ( v ) ) = m X i =1 δ ( T i ( u i ) , T i ( v i )) = m X i =1 δ ( u i , v i ) = d ( u, v ) , so tha t every T ∈ M is a symmetry of  F n q , d  . Let B ( F k i q ) b e the gr oup o f bijections from F k i q onto itself. Clear ly , if M i =  ( id, . . . , id, T i , i d, . . . , i d ) ∈ M : T i ∈ B ( F k i q )  then M = M 1 × . . . × M m . Besides, M i is isomorphic to B ( F k i q ), whic h is in tur n isomo rphic to S q k i , the p er m utation gr oup on a set of q k i symbols; hence, w e have prov ed the following: Prop ositio n 1 L et M b e the gr oup of mappings T : F n q → F n q define d ab ove. Then ( i ) M is a su b gr oup of  F n q , d  . ( ii ) M is isomorphic t o the dir e ct pr o duct Q m i =1 S q k i . Let S m be the p ermutation gr oup of { 1 , 2 , . . . , m } . W e will ca ll a p ermutation σ ∈ S m admissible if σ ( i ) = j implies that k i = k j . Clearly , the set S π of all a dmis s ible permutations is a s ubgroup of S m . Prop ositio n 2 [2] S π acts as a gr oup of line ar symmetries in V . Besides being linear , these sy mmetr ies sa tisfy an imp ortant prop er t y: if V i =  (0 , . . . , 0 , v i , 0 , . . . , 0) ∈ F n q : v i ∈ F k i q  , then σ ( V i ) = V σ ( i ) . As we see in the following, even non-linea r symmetr ie s satisfy a n analogo us prop erty . Lemma 1 L et F b e a symmetry of  F n q , d  and let v ∈ F n q . F or e ach i ∈ { 1 , 2 , . . . , m } , t her e exists j ∈ { 1 , 2 , . . . , m } su ch that k i = k j and F ( v + V i ) = F ( v ) + V j . 2 Pro of. Let v i ∈ V i , v i 6 = 0. Since d ( F ( v + v i ) , F ( v )) = d ( v + v i , v ) = 1, F ( v + v i ) − F ( v ) is a v ector of π -weigh t 1. But for any vector u in F m q , ω ( u ) = 1 iff u ∈ V j for some j . Hence there is a n index j such that F ( v + v i ) = F ( v ) + v j , with v j ∈ V j , v j 6 = 0. If v ′ i ∈ V i , v ′ i 6 = v i and v ′ i 6 = 0, then F ( v + v ′ i ) = F ( v ) + v k for some v k ∈ V k (with v k 6 = 0 ), but also d ( F ( v + v i ) , F ( v + v ′ i )) = d ( v + v i , v + v ′ i ) = 1. If k 6 = j then d ( F ( v ) + v j , F ( v ) + v k ) = 2; hence k = j a nd F ( v + v ′ i ) = v + v ′ j , with v ′ j 6 = v j . This prov es that F ( v + V i ) ⊆ F ( v ) + V j . Now apply the same reas oning to F − 1 : if v i 6 = 0 and F ( v + v i ) = F ( v ) + v j , then F − 1 ( F ( v ) + v j ) ∈ v + V i and therefore F − 1 ( F ( v ) + v j ) ⊆ v + V i . It f ollows that F ( v + V i ) = F ( v ) + V j . Finally , k i = dim ( V i ) = dim ( V j ) = k j bec ause F is bijective.  This result implies, in particular, that for each i there corresp onds a j s uch that F ( V i ) = F (0) + V j and that k i = dim ( V i ) = dim ( V j ) = k j . This sug g ests the following definition. Definition 1 L et F b e a symmetry of  F n q , d  . Th e admissible p ermu tation σ F induc e d by F is given by σ F ( i ) = j if and only if F ( V i ) = F (0 ) + V j . Theorem 1 Every symmet ry F is a pr o duct σ T , wher e σ ∈ S π and T ∈ M . This de c om- p osition is unique. Pro of. Supp ose that F (0) = 0; then F = ( σ F )  σ − 1 F F  , where σ F is the admissible p ermu- tation defined by F . Clearly ,  σ − 1 F  ( V i ) = V i for ea ch i , and ther efore this ma pping is in M . If F (0) = v 0 6 = 0, let S b e the translation S ( v ) = v − v 0 ; then S ∈ M , S F (0) = 0 a nd, therefore, S F = σ S F T , where T ∈ M . It follows that F can b e written as F = σ S F ( T S − 1 ). If fo llows tha t every symmetr y F is a pro duct σ T ; since S π ∩ M = { id } , this decomp o- sition is unique .  Theorem 2 ( i ) S y mm  F n q , d  ∼ = S π ⋉ M , wher e the semi-dir e ct st ructur e is induc e d by the action of S π on M by c onjugation. ( ii ) S y m m  F n q , d  ∼ = S π ⋉ Q m i =1 S q k i . Pro of. W e will show that the group M is normal in S y mm  F n q , d  . Since the last r esult shows that S y mm  F n q , d  = S π M , it is enough to check that if T = ( T 1 , T 2 , . . . , T m ) ∈ M and σ ∈ S π then σ − 1 T σ is also in M . Applying σ − 1 T σ to a vector we conclude that σ − 1 T σ =  T σ (1) , . . . , T σ ( m )  and hence that M is a no rmal subg roup o f S y mm  F n q , d  . Since S π ∩ M = { i d } , this shows that the symmetry group is isomorphic to the semi-dir ect pro duct S π ⋉ M , where m ultiplication is given by ( σ , T ) ( ϕ, S ) =  σ ϕ,  ϕ − 1 T ϕ  S  . The fact that M ∼ = Q m i =1 S q k i yields the se c o nd ass ertion.  3 Corollary 1 L et π = ( k 1 , k 2 , . . . , k m ) b e a p artition of n . If k 1 = . . . = k m 1 = l 1 , . . . , k m 1 + ... + m l − 1 +1 = . . . = k n = l r with l 1 > l 2 > . . . > l r , then   S y mm  F n q , d    =   l X j =1 m j !   · m Y i =1  q k i  ! ! . Pro of. In fact, since S y mm  F n q , d  ∼ = S π ⋉ Q m i =1 S q k i and S π ∼ = Q m j =1 S l j ,   S y mm  F n q , d    = | S π | · m Y i =1   S q k i   =   l X j =1 m j !   · m Y i =1  q k i  ! ! .  When m = n , k 1 = k 2 = . . . = k n = 1, the π -weight is the Ha mming weight on F n q . In this ca se each M i is isomor phic to S q and every per m utation in S n is admissible; th us we reobtain the symmetry groups o f Ha mming spaces from our previo us calculations. Corollary 2 L et d H b e the Hamming metric over F n q . The symmetry gro up of  F n q , d H  is isomorphi c to S n ⋉ S n q . Another pr o of of this can b e found in [3]. 3 Automorphisms The group of automor phis ms of  F n q , d  is easily deduced from the results a bove. Let F = σ T be a symmetry . Since σ is linear, the linear it y of F is a matter of whether T is linear or not. Now, if T = ( T 1 , T 2 , . . . , T m ) is linear, then each comp onent T i m ust also b e linear ; since each T i is bijective, T i is in the gro up Aut ( V i ) of linear automor phisms of V i . Ther efore T ∈ Q m i =1 Aut ( V i ). On the other hand, any element of this group is a linear symmetry; hence: Theorem 3 The automorphism gr oup Aut  F n q , d  of  F n q , d  is isomorph ic to S π ⋉ Q m i =1 Aut ( V i ) . Corollary 3 L et π = ( k 1 , k 2 , . . . , k m ) b e a p artition of n . If k 1 = . . . = k m 1 = l 1 , . . . , k m 1 + ... + m l − 1 +1 = . . . = k n = l r with l 1 > l 2 > . . . > l r , then   Aut  F n q , d    =   l X j =1 m j !   · m Y i =1  q k i − 1   q k i − q  . . .  q k i − q k i − 1  ! . Pro of. Note initially that there is a bijection from Aut ( V i ) and the family of all or- dered bases of V i : let ( e 1 , e 2 , . . . , e k i ) b e an order ed basis o f V i ; if T ∈ Aut ( V i ), then ( T ( e 1 ) , T ( e 2 ) , . . . , T ( e k i )) is an ordered basis of V i ; if ( v 1 , v 2 , . . . , v k i ) is an ordered basis 4 of V i then there exist a unique auto mo rphism T with T ( e j ) = v j for all j ∈ { 1 , 2 , . . . , k i } . Since the n umber o f o rdered bas is of V i equal  q k i − 1   q k i − q  . . .  q k i − q k i − 1  follows that | Aut ( V i ) | =  q k i − 1   q k i − q  . . .  q k i − q k i − 1  . F rom ab ove theo rem   Aut  F n q , d    = | S π | · m Y i =1 | Aut ( V i ) | . Since | S π | = P l j =1 m j ! the c orollar y follows.  Restricting to the Hamming case again, Aut ( V i ) = Aut ( F q ) = F ∗ q , and S π = S n . Hence: Corollary 4 The automorphism gr oup of  F n q , d H  is S n ⋉  F ∗ q  n . References [1] K. F eng, L. Xu and F.J. Hick er nell, Linear E rror -Blo ck Co des , Finite Fields and Their Applic ations 12 638-6 52 (2006). [2] M.M.S. Alves, L. Panek and M. Firer , Err or-Blo ck Co des a nd Poset Metrics, Ad vanc es in Mathematics of Communic ations , 2 9 5 -111 (200 8). [3] I. Consta n tinescu and W. Heise, On the co ncept of code-is omorphy , Journal of Ge ometry 57 63-6 9 (1996). 5