Group code structures on affine-invariant codes

Group co de structures on affine-in v arian t co des Jos ´ e Joaqu ´ ın Bernal, ´ Angel del R ´ ıo and Juan Jacob o Sim´ on ∗ No v em b er 1, 2 018 Abstract A group code structure of a l inear co de i s a description of the code as one-sided or tw o-sided ideal of a group algebra of a finite gro up . In these realizations, the group algebra is identified with t he ambient space, and the group elements with th e co ordinates of the ambien t space. It is well known that every affine-inv ariant code of length p m , w ith p prime, can b e realized as an ideal of the group algebra F I , where I is th e underlying additive group of the field with p m elemen ts. In this pap er we d es crib e all the group co de stru ct ures of an affine- in v arian t cod e of length p m in terms of a family of maps from I to the group of automorphisms of I . Affine-in v ariant co des were firstly introduced by Kasa mi, Lin a nd Peterson [KLP2] as a g e ne r alization of Reed-Muller codes. This class of co des has received the atten tion of several autho r s bec ause of its go o d algebraic a nd decoding prop erties [D, BCh, ChL, Ho, Hu]. The length of an affine-inv ariant co de is a prime p o wer p m , wher e p is the characteristic of the finite field F which plays the role of alphab et. It is well known that every affine-inv ariant co de of leng th p m ov er the field F can b e r ealized as an ideal of the gro up alg ebra F I , where I is the under lying additive group of the field with p m elements, i.e. I is the elementary ab elian group of order p m . In this rea lization, the group elements are identified with the elements of the standard bas e o f the am bient space F p m . W e refer to these rea lizations o f co de s as one-s ide d or tw o-sided ide a ls in group algebras as gro up co de structures of the given co de. In this pap er we study all the p o ssible gro up co de structur es on an a ffine-in v ar ian t co de. Our main to o ls are an in trins ic a l characteriz a tion of group co des obtained in [BRS] and a descr iption of the group o f per mutation automor phisms of non- trivial affine-inv ariant co des given in [BCh]. The s e results are rev iew ed in Sec - tion 1 , where we also reca ll the definition a nd main prop erties of a ffine-in v ar ian t co des. In Section 2, we de s cribe all the gro up co de structures of an affine- inv ariant code C in terms of a family of maps I → G a,b where G a,b is a subgroup of the group of automorphism o f I dep ending on t wo in teger s a and b whic h ∗ Departamen to de Mate m´ aticas, Univ ersidad de Murcia, 30100 Murcia. Spain. email: josejoaquin.b ernal@alu.um.es, adelrio@um. e s, jsimon@um.es Pa rtial l y supported b y D.G.I. of Spain and F undaci´ on S´ en eca of Murcia. 1 are determined by the co de C . So me metho ds to calc ula te a a nd b were given in [D] and [BCh]. As an applica tio n we exhibit in Section 3 a family of gr oup co de str uctures on a n y affine-inv ariant co de C for which the in teger a is different from m and characterize the affine-inv aria n t code s C whic h hav e a non- abelian group co de structure. 1 Preliminaries In this section we recall the definition o f (left) gr oup co de and the intrinsical characterization given in [BRS]. W e a lso recall the definition of affine-inv ariant co de and the description of its gro up of p erm utation automo rphisms given in [BCh]. All througho ut p is a p ositiv e prime integer and all the fields in this pap er are finite of c har acteristic p . The field with p s elements is denoted by F p s . Two finite fields F and K = F p m (of characteristic p ) a re g oing to b e fixed througho ut the pap er. The roles of alphab et and length of the a ffine-in v ar ian t codes of this pap er a re going to b e represented b y F and p m , resp ectiv ely . W e denote the underlying additive g r oup of K with I . W e will abuse of the notation and some maps defined on I (resp ectiv ely , K ) will b e consider ed acting in K (respectively , I ) to o. W e use this double notation to e mpha size whether we are considering the additive or the field structure. The group algebra o f a gro up G with co efficients in the field F is going to b e denoted by F G . In particular F I denotes the group a lgebra of the underlying additive gro up of K with co efficien ts in F and F ( K ∗ ) is the gro up a lgebra of the group o f units of K with co efficien ts in F . T o avoid the a mbiguity of an expression o f the form P g ∈I a g g , as e ither a n element of F I or an element of a field containing b oth F and K , w e use the nota tion P g ∈I a g g to represent elements of F I . W e consider the gr oup algebra F ( K ∗ ) as a subspace of F I (not as a subalgebra ). All the group theoretica l notions used in this pap er can b e easily founded in [R]. Definition 1. If E = { e 1 , . . . , e n } is the standar d b asis of F n , C ⊆ F n is a line ar c o de and G is a gr oup (of or der n ) t hen we say that C is a left G -co de (r esp e ctively, a right G -code ; a G -co de ) if t he r e is a bije ction φ : { 1 , 2 , . . . , n } → G such that the line ar map φ : F n → F G , given by e i 7→ φ ( i ) , maps C to a left ide al (r esp e ctively, a right ide al; a two-side d ide al) of F G . A left group co de (r esp e ctively, gro up co de ) is a line ar c o de which is a left G -c o de (r esp e ctively, a G -c o de) for some gr oup G . A (left) cyclic gr ou p c o de (r esp e ctively, ab elian gr oup c o de, solvable gr oup c o de, et c.) is a line ar c o de which is ( le ft) G -c o de for some cyclic gr oup G (r esp e ctively, ab elian gr oup, solvable gr ou p, etc.). Let S n denote the g roup of permutations on n symbols. Ev er y σ ∈ S n defines an automor phis m of F n in the obvious w ay , i.e. σ ( x 1 , . . . , x n ) = ( x σ − 1 (1) , . . . , x σ − 1 ( n ) ). By definition, the g roup of p ermut atio n automor phisms o f a linea r co de C of 2 length n is P Aut( C ) = { σ ∈ S n : σ ( C ) = C } . (1) An intrinsical characterization o f (left) gro up co des C in terms o f P Aut( C ) has bee n obtained in [BRS]. Theorem 2. [BRS] L et C b e a line ar c o de of lengt h n over a field F and let G b e a finite gr oup of or der n . (a) C is a left G -c o de if and only if G is isomorphic to a tr ansitive su b gr oup of S n c ontaine d in P Aut( C ) . (b) C is a G -c o de if and only if G is isomorphi c to a tr ansitive su b gr oup H of S n such that H ∪ C S n ( H ) ⊆ P Aut( C ) , wher e C S n ( H ) denotes the c entr alizer of H in S n . Recall tha t I denotes the underlying additive group of K = F p m . Let S ( I ) denote the g roup of bijections I → I . Every element o f S ( I ) induces a unique F -linear bijectio n of the group algebr a F I . Affine-in v ariant co des of length p m are defined as subspaces of the gr oup algebra F I , ra ther than subspaces of F p m , in terms o f its group o f p e rm utation a utomorphisms consider ed as elements of S ( I ). W e explain now the transfer from P Aut as a subgroup of S p m to the group of p ermut atio n automor phisms as elements of S ( I ). F or an F -subspace C of F I , let P Aut( C ) = { σ ∈ S ( I ) : σ ( C ) = C } . (2) Observe that if φ : { 1 , . . . , p m } → I is a bijection and D is a linear c o de in F p m then φ induces an isomorphism P Aut( D ) ≃ P Aut( φ ( D )), wher e the left side P Aut uses (1) a nd the second one uses (2). Therefore, if C is a subspac e of F I and G is a gr oup of order p m then C is a left G -code if and only if P Aut( C ) contains a tra nsitiv e s ubgroup H of S ( I ) iso morphic to G a nd it is a G -co de if H can b e se lected such that C S ( I ) ( H ) ⊆ P Aut( C ). Definition 3. An affine-invariant c o de is an F -su bsp ac e C of F I such that P Aut( C ) c ontains the maps of the form x ∈ I 7→ αx + β , with α ∈ K ∗ and β ∈ I and every element P g ∈I a g g of C satisfies P g ∈I a g = 0 . Affine-in v ariant co des can b e seen as extended cyclic co des as follows. If J is a n ideal of F ( K ∗ ) then the parity chec k extension of J is    X g ∈I a g g : X g ∈ K ∗ a g g ∈ J a nd X g ∈ K ∗ a g = 0    . If C ⊆ F I is a n a ffine-in v ar ian t co de then C ∗ = { P g ∈ K ∗ a g g : P g ∈I a g g ∈ C } is a n ideal of F ( K ∗ ) and C is the par it y check extension of C ∗ . W e reca ll a characteriz a tion o f Kasami, Lin and P eterso n of the parit y c heck extensions of ide a ls o f F ( K ∗ ) which a re affine-inv aria n t in terms of the p -adic 3 expansion of its defining s et [KLP1]. Let C ⊆ F I b e the pa r it y ch eck extension of a n ideal of F ( K ∗ ). The defining set of C is D C =  i : 0 ≤ i < p m and P g ∈I a g g i = 0 , for every P g ∈I a g g ∈ C  , where, by conv ention, 0 0 = 1 . If q is the c a rdinalit y of F then D C \ { p m − 1 } is a union of q -cycloto mic cla sses mo dulo p m − 1. Conv ersely , if D is a subset of { 0 , 1 , . . . , p m − 1 } , such that D \ { p m − 1 } is a unio n of q -cyclotomic clas s es mo dulo p m − 1, then ther e is a unique idea l J of F ( K ∗ ) s uc h that D is the defining s e t of the parity chec k extension of J (see e.g. [Ch]). The p -adic expansion of a non-neg ativ e integer x is the list of in teger s ( x 0 , x 1 , . . . ), uniquely defined by 0 ≤ x i < p and x = P i ≥ 0 x i p i . The p - adic expansion yields a partial ordering in the set o f p ositive integers b y setting x  y if x i ≤ y i , for every i , where ( x i ) and ( y i ) ar e the p - adic expans ions of x and y , re s pectively . Prop osition 4. [KLP1][Hu, Cor ol lary 3.5] L et C ⊆ F I b e t he p arity che ck extension of an ide al F ( K ∗ ) . The n C is affine-invariant if and only if D C satisfies the fol lowi ng c ondition for every 1 ≤ s, t ≤ p m − 1 : s  t and t ∈ D C ⇒ s ∈ D C . (3) Three obvious affine-inv ariant co des are the zero co de, the rep etition co de and its dual, i.e. { 0 } , the ideal of F I generated by P g ∈I g , and the augmen- tation idea l { P g ∈I a g g : P g ∈I a g = 0 } , resp ectiv ely . Their defining sets are { 0 , 1 , . . . , p m − 1 } , { 0 , 1 , . . . , p m − 2 } and { 0 } , res pectively . These three co des are known a s the trivial affine-inv ariant co des [BCh, Hu]. F or future use w e describ e the a ffine-in v ar ian t co des of length 4. Example 5 (Affine-in v ariant co des of length 4) . Let D b e the defining set of an affine-in v ariant co de of leng th 4 ov er F 2 r . Thus D satisfies condition (3) and D \ { 3 } is a unio n of 2 r classes mo dulo 3. If r is even then the 2 r -cyclotomic classes mo dulo 3 are { 0 } a nd { 1 , 2 } . This implies that if r is even then D = { 0 } , { 0 , 1 , 2 } or { 0 , 1 , 2 , 3 } , i.e. C is trivial as affine-inv aria nt co de. How e ver, if r is o dd then the cyclotomic classes mo dulo 3 are { 0 } , { 1 } and { 2 } . So, in this ca s e there are tw o a dditional possibilities f or D , namely { 0 , 1 } and { 0 , 2 } . Resuming, if r is ev en then ther e are not 2 r -adic non-trivial affine-in v ariant co des of length 4 a nd if r is o dd then there are tw o 2 r -adic non-trivial affine in v ariant co des of length 4 . If C is a trivial affine-inv aria n t co de then P Aut( C ) = S n , and therefore C is G -co de for ev ery group G of order p m . So to av oid trivialities, in the remainder of the pape r all the a ffine-in v ar ian t co des are supp ose to be non-trivia l. The group of p erm utations o f a (non-trivial) affine-inv a rian t code has been described by Berger and Cha rpin [BCh]. W e identif y every elemen t y ∈ I with the translatio n x 7→ x + y , so that the group I can b e identified with the gro up o f translations of K . If L/ E is a 4 field extension then Gal( L/E ) denotes the Ga lo is gro up of L over E , i.e. the group of field automorphisms of L whic h fix the elemen ts of E . T o refer to L as a vector space ov er E we write L E . Accor dingly GL( L E ) denotes the gro ups o f linear transfor ma tions of L as vector space ov er E . Theorem 6. [BCh , Cor ol lary 2] L et C b e a non-trivial affine-invariant c o de of length p m over F = F p r and let K = F p m . L et a = a ( C ) = min n d | m : GL( K F p d ) ⊆ P Aut( C ) o and b = b ( C ) = min    d ≥ 1 : D C \ { p m − 1 } is a u nion of cyclotomic p d − classes mo dulo p m − 1    . Then b | r , b | a | m and P Aut( C ) = hI , GL( K F p a ) , Gal( K / F p b ) i . A metho d to co mput e a ( C ) and b ( C ) was fir stly o btained by Delsarte [D]. Later, Berger and Charpin gav e tw o alternative metho ds which a re so metimes computationally s impler [BCh]. Now w e present an alternative description of P Aut( C ) = hI , GL ( K F p a ) , Gal( K / F p b ) i , for C a n affine-inv aria n t co de as in Theor em 6. W e use the notatio n N ⋊ G to represent a semidirect pro duct o f N by G via s ome action of G o n N . That is , N a nd G are tw o g roups a nd there is a group homomorphism σ : G → Aut ( N ). The ma p σ is referr ed to as the actio n of the semidire c t pro duct a nd in mo s t examples it will b e clea r fro m the context. W e use σ g = σ ( g ), to minimize the nu mber of parenthesis. The underlying set o f N ⋊ G is the direct product N × G and the pr oduct is given by ( n 1 , g 1 )( n 2 , g 2 ) = ( n 1 σ g 1 ( n 2 ) , g 1 g 2 ). Given tw o divisors a a nd b of m with b | a , let G a,b =  f ∈ GL( K F p b ) : f is τ − s emilinear for some τ ∈ Gal  F p a / F p b   . W e claim that G a,b = h GL( K F p a ) , Gal( K / F p b ) i . Indeed, if f is τ -semilinear with τ ∈ Gal( F p a / F p b ) then τ is the r estriction of σ for so me σ ∈ Gal( K / F p b ). Then f σ − 1 ∈ GL( K F p a ). This prov es one inclusion; the other one is obvious. Using that I (iden tified with the gro up of tra nslations of K ), is normalized by GL( K F p ) we deduce the following from Theorem 6. Corollary 7 . If C is an affine-invariant c o de as in The or em 6 then P Aut( C ) = I ⋊ G a,b . Remark 8. The map T : G a,b → Gal( F p a / F p b ) which asso ciates f to τ , when f is τ -semilinear, is a sur jective g r oup homomorphis m with kernel GL( K F p a ). 2 Group co de stru c tures on affine-in v arian t co des In this sec tion w e pr esen t the ma in result of the pap er, namely a descriptio n of all the group co de structures of a non-trivia l affine-inv ariant co de C with a = a ( C ) and b = b ( C ) in ter ms of some maps α : I → G a,b . 5 Given a map α : I → G a,b let I α = { ( x , α ( x ) − 1 ) : x ∈ I } . The pr o of of the following lemma is straightforward. Lemma 9. I α is a sub gr oup of I ⋊ G a,b if and only if α ( x + y ) = α ( α ( y )( x )) α ( y ) (4) for every x, y ∈ I . In p articular, if α satisfies (4) then { α ( x ) : x ∈ I } is a p -sub gr oup of G a,b . W e need one more lemma which is an easy consequence of Sylow’s Theorem [R, 1.6.16 ]. Lemma 10. If P is a p -sub gr oup of GL( K F p a ) then P ρ ∈ P Im( ρ − 1) 6 = K . Pr o of. Select a basis b 1 , . . . , b n of K F p a , with n = m/a , and let U b e the set of endomor phisms f of K F p a such that for every i = 1 , 2 , . . . , n , f ( b i ) − b i be- longs to the F p a -subspace of K gener ated by the b j ’s with 1 ≤ j < i . That is, U is the gro up of automorphis ms o f K F p a having upper unitriangular as- so ciated matrix in the given basis. An ea sy coun ting ar gumen t shows that | U | = p a n ( n − 1) 2 and | GL( K F p a ) | = ( p a − 1 )( p na − p a )( p na − p 2 a ) . . . ( p na − p ( n − 1) a ) = | U | ( p a − 1)( p ( n − 1) a − 1)( p ( n − 2) a − 1) . . . ( p a − 1). Then U is a Sy- low’s p -subgroup of GL( K F p a ). By Sylow Theorem, there is g ∈ GL( K F p a ) such that P ⊆ g U g − 1 . Then P ρ ∈ P Im( ρ − 1 ) ⊆ P u ∈ U Im( g ug − 1 − 1) ⊆ g ( P u ∈ u Im( u − 1 )) = g ( h b 1 , . . . , b n − 1 i ) 6 = K . W e ar e ready to present our main result. Theorem 1 1. Le t F and K b e finite fields of char acteristic p and let I b e the underlying additive gr oup of K . L et C ⊆ F I b e a n o n- t riv ial affine-invariant c o de and let a = a ( C ) and b = b ( C ) . Then the fol lowing assertions hold for every fin i te gr oup G : (a) C is a left G - c o de if and only if G is isomorphic to I α for some map α : I → G a,b satisfying c ondition (4). (b) C is a G - c o de if and only if G is isomorphic t o I α for some map α : I → GL( K F p a ) satisfying c ondition (4) and such that t h e map β : K × K → K given by β ( x, y ) = α ( x ) − 1 ( y ) − y is F p a -biline ar. Pr o of. (a) First of all note that for every 0 6 = y ∈ I there exist ( y , α ( y ) − 1 ) ∈ I α ⊆ S ( I ) such that ( y , α ( y ) − 1 )(0) = y + α ( y ) − 1 (0) = y . Hence, if α satisfies condition (4 ) then I α is a transitive subgr oup o f S ( I ). Since |I α | = |I | = p m , the sufficiency follows from Theorem 2. Conv ersely , assume that C is a left G -co de for some g roup G , neces sarily of order p m . By Theo rem 2 and Corolla ry 7, we may a ssume without loss of 6 generality that G is a tra nsitiv e subgro up of S ( I ) con tained in I ⋊ G a,b . Thus, if ( x, g ) and ( y, h ) are tw o differen t elemen ts of G with x, y ∈ I and g , h ∈ G a,b then x = ( x, g )(0) 6 = ( y , h )(0) = y , and this shows that the pr o jection of G ont o I is bijectiv e. If λ : I → G is the inv ers e of this bijection then G = { ( x, λ ( x )) | x ∈ I } . Define α ( x ) := λ ( x ) − 1 , for any x ∈ I . Then G = I α and α s atisfies condition (4), by Lemma 9 . (b) Le t α : I → GL( K F p a ) satisfy the conditio ns of (b). By pa rt (a) and Theorem 2 and Coro llary 7 to prov e that C is a n I α -co de it is enough to show that C S ( I ) ( I α ) ⊆ I ⋊ G a,b . In fact w e are going to show that C S ( I ) ( I α ) ⊆ I ⋊ GL( K F p a ). F or every x ∈ I we set λ ( x ) = α ( x ) − 1 . Since I α is a tra nsitiv e subgro up of S ( I ) of order |I | = p m , the centralizer of I α in S ( I ) is C S ( I ) ( I α ) = { f ( x ) : x ∈ I } , where f ( x )( y ) = ( y , λ ( y ))( x ) = y + λ ( y )( x ) . (See [BRS, Le mma 1.1], sp ecialized to i 0 = 0 .) F or every x, y ∈ I set λ ′ ( x )( y ) = y + λ ( y )( x ) − x = y + β ( y , x ) . W e claim that λ ′ ( x ) ∈ GL( K F p a ), for ev er y x ∈ I . Indeed, o n the one hand λ ′ ( x ) is the comp osition of f ( x ) and the translation y 7→ y − x . This sho ws that λ ′ ( x ) is bijective. On the other hand, since β is F p a -linear, if y , y 1 , y 2 ∈ I and γ ∈ F p a then λ ′ ( x )( y 1 + y 2 ) = y 1 + y 2 + β ( y 1 + y 2 , x ) = y 1 + β ( y 1 , x ) + y 2 + β ( y 2 , x ) = λ ′ ( x )( y 1 ) + λ ′ ( x )( y 2 ) and λ ′ ( x )( γ y ) = γ y + β ( x, γ y ) = γ ( y + β ( x, y )) = γ λ ′ ( x )( y ). So C S ( I ) ( I α ) = { ( x, λ ′ ( x )) : x ∈ I } ⊆ I ⋊ GL( K F p a ) as wan ted. Conv ersely , a ssume that C is a G -co de. By Theorem 2 and the first part, G ≃ I α for a map α : I → G a,b satisfying condition (4) and such that C S ( I ) ( I α ) ⊆ P Aut( C ) = I ⋊ G a,b . W e set λ ( x ) = α ( x ) − 1 and β ( x, y ) = λ ( x )( y ) − y . W e have to show that β is F p a -bilinear. As in the pr e vious par agraph the centralizer of I α in S ( I ) is formed by the ma ps f ( x ) : y ∈ I 7→ y + λ ( y )( x ), with x ∈ I . So f ( x ) ∈ I ⋊ G a,b , for every x ∈ I . Since f ( x )(0) = x , we hav e f ( x ) = ( x, λ ′ ( x )), with λ ′ ( x )( y ) = y + λ ( y )( x ) − x . In other words, C S ( I ) ( I α ) = I α ′ for α ′ ( x ) = λ ′ ( x ) − 1 . By Lemma 9, α ′ satisfies condition (4). Then I α and I α ′ are the centralizer of each o ther in S ( I ) a nd their ro les and the roles o f λ and λ ′ can be in terchanged. Since β ( x, y ) = λ ( x )( y ) − y = λ ′ ( y )( x ) − x , to prove that β is F p a -bilinear it is enough to show that λ ( x ) , λ ′ ( x ) ∈ GL( K F p a ) for every x . By symmetry , we only prove that λ ′ ( x ) ∈ GL( K F p a ), for every x ∈ I . Using that λ ′ ( y ) is additive for every y ∈ I , we have the following equality for every x 1 , x 2 ∈ I : x 1 + x 2 + λ ( x 1 + x 2 )( y ) − y = λ ′ ( y )( x 1 + x 2 ) = λ ′ ( y )( x 1 ) + λ ′ ( y )( x 2 ) = x 1 + λ ( x 1 )( y ) − y + x 2 + λ ( x 2 )( y ) − y . Thu s λ ( x 1 + x 2 ) + 1 = λ ( x 1 ) + λ ( x 2 ) . (5) Let Q b e the subgro up gener a ted b y the λ ( x )’s. Since Q is a p -subgroup of G a,b , by Remar k 8, T ( Q ) is a p -subgr oup of Gal( F p a / F p b ). Let P = Q ∩ 7 GL( K F p a ). W e fix a tr a nsv ersa l T of P in Q containing 1 , and for every x ∈ I we put δ ( x ) = λ ( x ) − t x , where t x is the o nly element of T with λ ( x ) t x − 1 ∈ P . Define J = P x ∈I Im( δ ( x ) ). Then J = X x ∈I Im  λ ( x ) t − 1 x − 1  t x  ⊆ X ρ ∈ P Im( ρ − 1) 6 = K by Lemma 1 0. F or ev er y x ∈ I , let τ x = T ( λ ( x )) and τ ′ x = T ( λ ′ ( x )), i.e. λ ( x ) is τ x - semilinear and λ ′ ( x ) is τ ′ x -semilinear. Observe that the condition λ ( x ) t − 1 x ∈ P is equiv a len t to λ ( x ) t x − 1 ∈ GL( K F p a ) and hence t x is τ x -semilinear. Having in mind that λ ′ ( x )( y ) = y + λ ( y )( x ) − x a nd λ ( x ) = δ ( x ) + t x we hav e λ ′ ( x )( γ y ) = γ y + δ ( γ y )( x ) + t γ y ( x ) − x and λ ′ ( x )( γ y ) = τ ′ x ( γ ) λ ′ ( x )( y ) = τ ′ x ( γ ) ( y + δ ( y )( x ) + t y ( x ) − x ) for any γ ∈ F p a , and x, y ∈ I . Ther efore ( γ − τ ′ x ( γ )) y = x − t γ y ( x ) + τ ′ x ( γ )( t y ( x ) − x )+ δ ( y )( τ y − 1 τ ′ x ( γ ) x ) − δ ( γ y )( x ) . (6) Recall tha t the go a l is pr oving that λ ′ ( x ) ∈ GL( K F p a ), or equiv alently that τ ′ x 6 = 1, for every x ∈ I . By means of contradiction as sume that this is not the case and fix an elemen t x in I s uc h that τ ′ x 6 = 1. W e also fix γ ∈ F p a with τ ′ x ( γ ) 6 = γ . Observe that the or der of τ ′ x is a p o wer of p b ecause so is the order of G . Therefor e p divides a/b . F or every y ∈ I let z y = x − t γ y ( x ) + τ ′ x ( γ )( t y ( x ) − x ) and s e t Z = Z x,γ = { z y : y ∈ I } . Using (6 ), we hav e I = { ( γ − τ ′ x ( γ )) y : y ∈ I } ⊆ { z + j : ( z , j ) ∈ Z × J } . Since the t y ’s takes a t most a/b different v alues and x and γ are fixed, | Z | ≤ ( a/b ) 2 . On the other hand J is a prop e r s ubspace of K F p a . Th us p m ≤ | Z × J | ≤ a 2 p m − a b 2 and so p a ≤ ( a/b ) 2 ≤ a 2 . This implies that p = 2, b = 1 a nd a is either 2 or 4 (recall that p divides a ). Then, the index o f J in I , as an additiv e subgroup, is | Z | = a 2 = 2 a and the elements of Z for m a set of repre sen tatives of K modulo J . This implies that T ◦ λ ′ is sur jectiv e a nd hence we may ch o ose x so that τ ′ x is the F rob enius automorphism that maps s ∈ F 2 a to s 2 . W e a lso may choose γ ∈ F 4 \ F 2 . Hence γ − τ ′ x ( γ ) = 1, so tha t y ∈ z y + J , b y (6). Therefore y − y ′ ∈ J if a nd only if z y = z y ′ if and only if t y = t y ′ and t γ y = t γ y ′ , for every 8 y , y ′ ∈ I . Given u 1 , u 2 ∈ T , let K u 1 ,u 2 = x − u 1 ( x ) + τ ′ x ( γ )( u 2 ( x ) − x ) + J . Then I /J = { K u 1 ,u 2 : u 1 , u 2 ∈ T } and y ∈ K u 1 ,u 2 if and only if t y = u 2 and t γ y = u 1 . Th us λ − 1 ( P ) = { y ∈ I : t y = 1 } = ∪ u ∈T K u, 1 and, by (5), λ − 1 ( P ) is a s ubgroup of I . Hence [ I : λ − 1 ( P )] = [ λ − 1 ( P ) : J ] = a . W e claim tha t if u + v ∈ λ − 1 ( P ) and u 6∈ λ − 1 ( P ) then λ ( u ) = λ ( v ). By means of contradiction assume that u + v ∈ λ − 1 ( P ), u 6∈ λ − 1 ( P ) and λ ( u ) 6 = λ ( v ). Then λ ( u + v ) is F p a -linear, so that λ ( u ) + λ ( v ) is also F p a -linear, b y (5). Thus for every ζ ∈ F p a and z ∈ I we hav e ( λ ( u ) + λ ( v ))( ζ z ) = ζ λ ( u )( z ) + ζ λ ( v )( z ) and ( λ ( u ) + λ ( v ))( ζ z ) = τ u ( ζ ) λ ( u )( z ) + τ v ( ζ ) λ ( v )( z ) . Therefore ( γ + τ u ( γ )) λ ( u )( z ) = ( γ + τ v ( γ )) λ ( v )( z ) . (7) F rom this equality and the ass ump tion λ ( u ) 6 = λ ( v ) one deduces that τ u 6 = τ v and the fixed fields of τ u and τ v coincides. Therefor e a = 4 and τ u and τ v are the tw o generators o f Gal( F 16 / F 2 ). So one may assume that τ u ( ζ ) = ζ 2 and τ v ( ζ ) = ζ 8 , for every ζ ∈ F 16 . No w sp ecializing (7) to ζ = γ ∈ F 4 \ F 2 one deduces that λ ( u ) = λ ( v ) b ecause γ + τ u ( γ ) = γ + τ 1 ( γ ) = 1 . This proves the claim. By the previous par agraph | P | ( a − 1) = | Q \ P | = [ I : λ − 1 ( P )] − 1 = a − 1 and hence P = 1. Therefore J = 0 and m = a , b ecause J has co dimension 1 as an F p a -subspace o f K . If a = 2 then C is trivial as affine-inv ariant co de, b y Example 5. Here we use that b = 1, hence the defining s e t of C is a union of 2- cyclotomic classe s mo dulo 3. Thus a = 4 and so G a,b = F ∗ 16 ⋊ Gal ( F 16 / F 2 ). Let σ ∈ Gal ( F 16 / F 2 ) be the F rob enius automor phis m. Since T ◦ λ is surjectiv e there exist u, v ∈ I such that λ ( u ) = ( γ 1 , σ ) a nd λ ( v ) = ( γ 2 , σ 2 ), for some γ i ∈ F ∗ 16 , i = 1 , 2. Since P = 1 we hav e that λ ( u + v ) = ( γ 3 , σ 3 ) for some γ 3 ∈ F 16 . Using (5) we co nclude that every elemen t of F 16 is a ro ot o f the p olynomial p ( X ) = X + γ 1 X 2 + γ 2 X 4 + γ 3 X 8 , which yields the desired contradiction. Corollary 12. L et C b e a non-trivial affine-invariant c o de of length p m and G b e a finite gr oup. (a) If a ( C ) = m , a/ b is c oprime with p and C is a left G -c o de t h en G is isomorphi c t o the p -elementary ab elian gr oup of or der p m . (b) If a ( C ) = m and C is a G -c o de then G is isomorphic to the p - element a ry ab elian gr oup of or der p m . Pr o of. If a ( C ) = m then GL( K F p a ) = GL( K K ) ≃ K ∗ , a group of o rder coprime with p . Ther efore, if α : I → GL( K K ) satisfies condition (4) then α ( x ) = 1 fo r every x ∈ I . Now (b) follows from statement (b) of Theor em 11. The pro of of (a) is similar using statement (a) of Theo r em 11 and the fact that the order of G a,b is | GL( K F p a ) | a/b , which is coprime with p under the assumptions of (a). 9 Remark 13. F or every α : I → G a,b satisfying (4), consider I α acting on I by conjugation inside I ⋊ G a,b . Then the map π : ( x, α ( x ) − 1 ) 7→ x is a bijectiv e 1-co cycle, i.e . π ( g h ) = π ( g ) + g ( π ( h )), for every g , h ∈ I α . Gr oups acting on ab elian gro ups with bijectiv e 1-co cycles hav e re ceiv ed the attent ion of s e veral authors b y its connections with the set-theo retical so lutions of the Y ang-Ba xter equation [CJR, ESS]. 3 A class of group co de structures Theorem 11 describ e s a ll the (left) group co de str uctures of a non-trivial affine- inv ar ian t co de. The mo st o b v ious one is o btained for α the tr ivial map x 7→ 1. In this case I α ≃ I and this yields the p -elementary ab elian gr oup str ucture men tioned in the introduction. In this section we exhibit a family of o ther group co de structures on a fixed a ffin e- inv aria n t code C of length p m under the assumption that a ( C ) 6 = m . W e keep the notation o f the previous sections, that is F a nd K = F p m are finite fields o f characteristic p and I is the underlying additiv e group of K . Let C be a no n-trivial affine-inv aria n t co de inside F I and set a = a ( C ) and b = b ( C ). Let f and χ be as follows · f : K → K is an F p a -linear ma p , · χ : K → F p a is a n additive map (i.e. F p -linear) , · χ 6 = 0 6 = f , f 2 = 0 and χ ◦ f = 0 . (8) Observe that there are χ and f sa tisfying (8) if and only if a < m . Consider the map α = α χ,f : I − → G a,b x 7→ 1 + χ ( x ) · f F or x, y ∈ I α ( α ( y )( x )) = 1 + [ χ ( x ) + χ ( χ ( y ) · f ( x ))] · f = 1 + χ ( x ) · f = α ( x ) and hence α ( α ( y )( x )) α ( y ) = α ( x ) α ( y ) = (1 + χ ( x ) · f ) (1 + χ ( y ) · f ) = 1 + χ ( y ) · f + χ ( x ) · f = α ( x + y ) . Therefore α satisfie s condition (4). Since α ( x ) − 1 = 1 − χ ( x ) f , b y Theo rem 11, C is a left I χ,f -co de, where I χ,f = { ( x, 1 − χ ( x ) f ) : x ∈ I } . Moreov er , using the notation of Theorem 11, we hav e β ( x, y ) = α ( x ) − 1 ( y ) − y = − χ ( x ) f ( y ) and hence, if χ is F p a -linear then C is I χ,f -co de. Our next go al consis t in describing the s tr ucture of I χ,f . F or that we nee d to intro duce some group construc tio ns from some vector spaces. 10 Notation 1 4. Given an F p -ve ctor sp ac e V and line ar map µ : V → F p a , we define the gr oup V µ = V × V with the fol lowing pr o duct : ( v 1 , w 1 )( v 2 , w 2 ) = ( v 1 + v 2 − µ ( w 1 ) w 2 , w 1 + w 2 ) , for v 1 , w 1 , v 2 , w 2 ∈ V . If U is an additive sub gro up of F p a then we c onsider the fol lowing action on V µ : u ( v , w ) = ( v − uw , w ) , ( u ∈ U, v , w ∈ V ) . The c orr esp onding semidir e ct pr o duct is denote d by V µ ⋊ U and its elements by ( v 1 , v 2 ; u ) , with v 1 , v 2 ∈ V and u ∈ U . Lemma 15. If ( x, f ) a nd ( y , g ) b elong to I ⋊ G a,b then ( x, f )( y , g ) = ( y , g )( x, f ) ⇔  x + f ( y ) = y + g ( x ) and f g = g f . Pr o of. Straig h tfor w ard. Theorem 16. L et χ and f b e as in (8) and c onsider K as an F p -ve ctor sp ac e. Then (a) I χ,f is ab elian if and only if χ is F p a -line ar and ker( χ ) ⊆ ker( f ) . In this c ase ker( χ ) = k er( f ) . (b) If I χ,f is non-ab elian then the c enter of I χ,f is { ( z , 1) : z ∈ ker( f ) ∩ ker( χ ) } . (c) If p is o dd then I χ,f has exp onent p . If p = 2 then the exp onent of I χ,f is 4 . (d) Le t V = Im( f ) , Z a c omplement of V in ker( χ ) ∩ k er( f ) and U a c omplement of ker( χ ) ∩ k er( f ) in ker ( f ) . Then I χ,f ≃ Z × ( V χ ◦ g ⋊ χ ( U )) . wher e g : V → K is an additive map satisfying f ◦ g = 1 V and g f (ker( χ )) ⊆ ker( χ ) . Pr o of. (a) Using Lemma 15 , it is easy to se e that ( x, α ( x ) − 1 ) a nd ( y , α ( y ) − 1 ) commute if and only if χ ( x ) f ( y ) = χ ( y ) f ( x ). Using this and the ass umpt ion χ 6 = 0, o ne easily follows that if I χ,f is a b elian then ker( χ ) ⊆ k er( f ). F urther more, if γ ∈ F p a and x, y ∈ K then χ ( γ y ) f ( x ) = χ ( x ) f ( γ y ) = γ χ ( x ) f ( y ) = γ χ ( y ) f ( x ). Using that f 6 = 0 o ne deduce s that χ is F p a -linear. Ther efore, ker( χ ) = ker( f ) bec ause ker( χ ) has co dimension 1 in K F p a and ker ( f ) is a pr oper subspace of K F p a . Conv ersely , assume that χ is F p a -linear and ker( χ ) ⊆ ker ( f ). Then the equality holds a s a bov e. Let v ∈ K \ ker( χ ) a nd for every x, y ∈ K write x = w x + β x v a nd y = w y + β y v with w x , w y ∈ ker( χ ) and β x , β y ∈ F p a . Then 11 χ ( x ) f ( y ) = β x β y χ ( v ) f ( v ) = χ ( y ) f ( x ) and hence  x, α ( x ) − 1  and  y , α ( y ) − 1  commute. (b) Ass ume that I χ,f is non-ab elian. B y Lemma 15, ( z , α ( z ) − 1 ) b elongs to the cent er of I χ,f if and only if χ ( x ) f ( z ) = χ ( z ) f ( x ), for every x ∈ K . In particular, if z ∈ k er( f ) ∩ ker( χ ) then ( z , α ( z ) − 1 ) = ( z , 1) b elongs to the center of I χ,f . Con versely , let v = ( z , α ( z ) − 1 ) b elong to the cen ter o f I χ,f . If x 1 ∈ K \ ker( χ ) and x 2 ∈ K \ k er( f ) then fro m the equalities χ ( x 1 ) f ( z ) = χ ( z ) f ( x 1 ) and χ ( x 2 ) f ( z ) = χ ( z ) f ( x 2 ) o ne deduces that z ∈ ker ( χ ) if and only if z ∈ k er( f ). So it is enoug h to show that f ( z ) = 0 or χ ( z ) = 0. By (a), either χ is no t F p a - linear or there is y ∈ ker( χ ) \ ker( f ). In the latter c ase, 0 = χ ( y ) f ( z ) = χ ( z ) f ( y ) and hence χ ( z ) = 0 . In the for mer c a se, there is γ ∈ F p a and x ∈ K with χ ( γ x ) 6 = γ χ ( x ). Ho wev er , χ ( γ x ) f ( z ) = χ ( z ) f ( γ x ) = γ χ ( z ) f ( x ) = γ χ ( x ) f ( z ) and hence f ( z ) = 0. (c) Let x ∈ K and v = ( x, α ( x ) − 1 ) = ( x, 1 − χ ( x ) f ) ∈ I χ,f . A stra igh tforward calculation s ho ws that v p = − p − 1 X i =0 iχ ( x ) f ( x ) , 1 ! =  p ( p − 1 ) 2 χ ( x ) f ( x ) , 1  . Hence, if p is o dd then v p = (0 , 1) and if p = 2 then v 2 ∈ I , so that v 4 = 1 . By means of contradiction, assume that p = 2 and the exp onent of I χ,f is 2 . Then χ ( x ) f ( x ) = 0, for every x ∈ K . In particular, f ( x ) = 0 for every x ∈ K \ ker( χ ). If w ∈ k er( χ ) and x ∈ K \ k er( χ ) then f ( w ) = f ( x + w ) − f ( x ) = 0. This shows that f = 0, a contradiction. (d) The exis tence of the map g follows b y sta nda rd linear alg ebra a r gumen ts. Let W = g ( f (ker( χ ))) and W ′ = g ( H ), where H is a complement o f f (ker( χ )) in V . Clearly the restriction maps f : W ⊕ W ′ → V and g : V → W ⊕ W ′ are mutually inv erse to each other. Since f ◦ g = 1 V , we hav e W ∩ k er( f ) = 0. F urthermore , dim(ker( χ )) = dim(k er( χ ) ∩ ker( f )) + dim f (ker( χ )) = dim(ker( χ ) ∩ ker( f )) + dim( W ). This shows that ker ( χ ) = W ⊕ (ker( χ ) ∩ ker( f )). W e claim that K = W ′ ⊕ (ker( χ ) +ker( f )). Indeed, if x ∈ W ′ ∩ (ker( χ ) + ker ( f )) then x = u + v = g ( h ) for some u ∈ ker( χ ), v ∈ ker ( f ) a nd h ∈ H . So, f ( x ) = f ( u ) = f g ( h ) = h , and then h ∈ f (ker( χ )) ∩ H = { 0 } . Th us x = 0. Moreov er , dim K = dim( V ) + dim(ker( f )) = dim( V ) + dim(k er( f ) + k er( χ ))+ dim(k er( χ ) ∩ k er( f )) − dim(k er( χ )) = dim( V ) − dim( f (ker( χ )))+ dim(k er( f ) + k er( χ )) = dim( H ) + dim(k er( f ) + ker( χ )) = dim( W ′ ) + dim(ker( f ) + k er( χ )) . This pr o ves the c la im. Then K = Z ⊕ V ⊕ W ⊕ W ′ ⊕ U and the pro duct in I χ,f is given b y ( x 1 , 1 − χ ( x 1 ) f )( x 2 , 1 − χ ( x 2 ) f ) = ( x 1 + x 2 − χ ( w ′ 1 + u 1 ) f ( w 2 + w ′ 2 ) , 1 − χ ( x 1 + x 2 ) f ) 12 for x 1 = z 1 + v 1 + w 1 + w ′ 1 + u 1 and x 2 = z 2 + v 2 + w 2 + w ′ 2 + u 2 , with z i ∈ Z, v i ∈ V , w i ∈ W , w ′ i ∈ W ′ and u i ∈ U . W e conclude that the map I χ,f → Z × ( V χ ◦ g ⋊ χ ( U )) given by ( z + v + w + w ′ + u, 1 − χ ( w ′ + u ) f ) 7→ ( z , ( v , f ( w + w ′ ); χ ( u ))) is a bijection. The fact that this map is a gr oup ho momorphism follows by straightforward co mputatio ns. As it was mentioned ab ove, if χ is F p a -linear then C is I χ,f -co de. In this case one can obtain a mo r e friendly descriptio n of I χ,f . Let F a = ( F p a ) 1 , i.e. F a = F p a × F p a with the pro duct ( x 1 , y 1 )( x 2 , y 2 ) = ( x 1 + y 1 + x 1 y 2 , x 2 + y 2 ) (see Notation 14)l. F or a n F p a -vector space V conside r the following action of F a on V × V : ( x, y ) · ( u, v ) = ( u − y v, v ) , ( x, y ∈ F p a , u, v ∈ V ) . (9) Let ( V × V ) ⋊ F a denote the corr e sponding semidirect pro duct and deno te its elements with ( v 1 , v 2 ; x 1 , x 2 ) for v i ∈ V and x i ∈ F p a . Corollary 17. L et χ and f b e as in (8) and assume that χ is F p a -line ar. L et u denote the r ank of f . Then we have: (a) If ker ( χ ) = ker( f ) and p is o dd then I χ,f is p -elementary ab elian. (b) If ker ( χ ) = ker( f ) and p = 2 then I χ,f is a dir e ct pr o duct of m − 2 a c opies of gr oups of or der 2 and a c opies of cyclic gr oups of or der 4 . (c) If ker ( f ) 6⊆ ker( χ ) then I χ,f is isomorphic to the gr oup F m a − 2 u − 1 p a ×  F u p a × F u p a  ⋊ F p a  . (d) If ker ( f ) ( ker ( χ ) then I χ,f is isomorphic to the gr oup F m a − 2 u p a ×  F u − 1 p a × F u − 1 p a )  ⋊ F a  . Pr o of. (a) and (b). Assume that ker( χ ) = ker( f ). Then I χ,f is ab elian, by statement (a ) o f Theorem 1 6 . If p is o dd then I χ,f is elementary ab elian by statement (c) of the same propo sition. Suppose that p = 2. Then I χ,f is a direct pro duct of cyclic gr o ups of order 2 or 4, b y statement (d ) of Theor em 16. If x ∈ I then v = ( x, 1 − χ ( x ) f ) has or der ≤ 2 if and only if (0 , 1) = v 2 = ( − χ ( x ) f ( x ) , 1 ) if and only if x ∈ k er( χ ). Thus, if I χ,f is a dir ect pr oduct of k copies o f g roups of or der 2 and l copies of cyclic groups of order 4 then k + l = a dim F 2 a (ker( χ )) = m − a a nd m = k + 2 l . Solving this tw o equations we deduce that k = m − 2 a and l = a . In the re ma inder of the pr oof w e use the notation of Theorem 16 and its pro of. So I χ,f ≃ Z × ( V χ ◦ g ⋊ χ ( U )). Notice that Z, V , U , W and W ′ can b e 13 selected as F p a -subspaces of K . Since ker( χ ) ha s co dimension 1 in K F p a and U ⊕ W ′ is a complement of ker( χ ) in K , either U = 0 or W ′ = 0 . (c) Suppo se that ker( f ) 6⊆ ker( χ ). Then W ′ = 0, χ ( U ) = F p a and χ ◦ g = 0. By statement (d) of Theorem 16, I χ,f ≃ Z × (( V × V ) ⋊ F p a ). F urthermor e V ≃ F u p a and Z ≃ F m a − 2 u − 1 p a . (d) Assume now that ker( f ) ( ker( χ ). Then U = 0 a nd W and W ′ are F p a - subspaces o f K of dimensions u − 1 and 1 resp ectiv ely . Using this one deduces that X = { ( v 1 , v 2 ) : v 1 , v 2 ∈ f ( W ) } a nd X ′ = { ( v ′ 1 , v ′ 2 ) : v ′ 1 , v ′ 2 ∈ f ( W ′ ) } are subgroups of V χ ◦ g with X ∩ X ′ = 1 a nd X normal in V χ ◦ g . Then V χ ◦ g = X ⋊ X ′ , so tha t I χ,f ≃ Z × ( X ⋊ X ′ ). Mor e over X = f ( W ) × f ( W ), b ecause χ ◦ g v anishes on f ( W ). On the o ther hand, the ma p φ : X ′ → F a , given by φ (( v 1 , v 2 )) = ( χg ( v 1 ) , χg ( v 2 )), is a group isomorphism. Hence the action of X ′ on X by conjugation yie lds and ac tion of F a on X via φ . It is easy to see that this a c tion is precisely the action defined in (9). So I χ,f ≃ Z × (( f ( W ) × f ( W )) ⋊ F a ). Finally , Z ≃ F m a − 2 u p a and f ( W ) ≃ F u − 1 p a . In the remainder of the section we fix a non-trivia l a ffine- in v aria n t code C with a = a ( C ) 6 = m a nd show how to obtain F p a -linear maps χ and f satisfying (8) and yielding all the cases of Cor ollary 17. F or that w e start with an a rbitrary non-zer o linear form χ of K F p a and construct a n endomorphism f of K F p a satisfying the conditions o f (8). The existence o f the endomorphism f in all the cases is clear. T o obtain an ab elian gr oup code structure on C with a g iv en no n-zero linear form χ we just need a n endomorphism f of K F p a with ker( f ) = ker( χ ) and 0 6 = f ( v ) ∈ k er( χ ) for a given v ∈ K \ ker( χ ). If m > 2 a then it is alwa ys possible to o bt ain a no n-abelian g roup code structure on C . In fact, for every p ositiv e integer u with 2 u ≤ m a − 1 ther e are endomorphisms f 1 and f 2 of K F p a , satisfying the conditions of (8) such that I χ,f 1 and I χ,f 2 are as in sta temen ts (c) and (d) of Corolla r y 1 7 resp ectiv ely . Indeed, in this case, the dimension of ker( χ ) as F p a -vector space is m a − 1 ≥ 2 u . Thu s we ha ve K = ker( χ ) ⊕ X and ker( χ ) = Z 1 ⊕ V ⊕ W 1 = Z 2 ⊕ V ⊕ W 2 , for F p a - subspaces X, Z 1 , Z 2 , V , W 1 and W 2 of K , where dim F p a ( X ) = 1, dim F p a ( V ) = dim F p a ( W 1 ) = u and dim F p a ( W 2 ) = u − 1. Then we can co nstruct the desired endomorphisms f 1 and f 2 of K b y setting f i ( W i ⊕ X ) = V fo r i = 1 , 2, k er( f 1 ) = Z 1 ⊕ V ⊕ X a nd ker( f 2 ) = Z 2 ⊕ V . Observe tha t, in case (d), we have u > 1. On the other hand, if u = 1 then ker( f 2 ) = k er( χ ) a nd hence I χ,f 2 is a belian. How ever if m = 2 a then it is not poss ible to obtain a non-ab elian group co de structure by the following result. Corollary 18. L et C b e a non-t ri vial affine-invariant c o de of length p m . Then the fol lowi ng c onditions ar e e quivalent. (a) C is a G -c o de for some non-ab elian gr oup G . (b) 2 a ( C ) < m . 14 F u rthermor e, if 2 a ( C ) ≥ m and C is a G -c o de then either G ≃ I or p = 2 and G is a dir e ct pr o duct of a c opies of cyclic gr oups of or der 4 . Pr o of. (b) implies (a ) is a consequence of the ar g umen ts giv en b efore the corol- lary . (a) implies (b) Let a = a ( C ) and a ssume that 2 a ≥ m , so that m is either a or 2 a and G is a group such that C is a G -co de. By Theorem 11, G is isomorphic to I α , for α : I → GL( K F p a ) a map satisfying c o ndition (4) such that β ( x, y ) = α ( x ) − 1 ( y ) − y is F p a -bilinear. If α ( x ) = 1 for every x then G ≃ I and s o G is ab elian as wan ted. This happ ens, for example, if m = a b ecause in this cas e the order of GL( K F p a ) = K ∗ is co prime with p and the order of α ( x ) is a p -th p o wer. Assume now that α ( x ) 6 = 1 for some x ∈ I . Then m = 2 a a nd, by Sylow’s Theo rem, we may assume that α ( x ) b elongs to a prescr ibed Sylow p -subgro up of GL( K F p a ). F o r example, we may fix a basis u 1 , u 2 of K F p a and assume that α ( x )( u 1 ) = u 1 and α ( x ) ( u 2 ) − u 2 = χ ( x ) u 1 for some χ ( x ) ∈ F p a (see the pro of of L e mm a 10). Let f b e the F p a -linear endomor phis m of K given by f ( u 1 ) = 0 and f ( u 2 ) = u 1 . Then α ( x ) − 1 = 1 − χ ( x ) f and β ( x, y ) = − χ ( x ) f ( y ). Since β is F p a -linear, χ is F p a -bilinear. F urthermore , 1 + ( χ ( x ) + χ ( y )) f = 1 + χ ( x + y ) f = α ( x + y ) = α ( α ( y )( x )) α ( y ) = (1 + χ ( x + χ ( y ) f ( x )) f )(1 + χ ( y ) f ) = 1 + ( χ ( x ) + χ ( y )) f + χ ( y ) χ ( f ( x )) f and we conclude that χ ◦ f = 0, i.e. χ and f satisfy the conditions of (8 ) and G ≃ I χ,f . Mor eo ver dim F p a (ker( χ )) = 1 and so ker( f ) = f ( K ) = F p a v 1 = ker( χ ). W e conclude that I χ,f is ab elian, by sta temen t (a) of Theore m 16. Finally , the last statement is a co nsequence of sta temen ts (a) and (b) of Corollar y 17. References [BCh] T.P . Berger and P . Charpin, The Permutation gr oup of affine-invariant extende d cyclic c o des , IEE E T r ans. Inform. Theory 42 (1996 ) 2194-2 209. [BRS] J.J. Bernal, ´ A. del R ´ ıo and J.J. Sim´ on, An intrinsic al descri ption of gr oup c o des , Des. Co des, Crypto. (to app ear). [CJR] F. Ced´ o , E . 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