On algebras admitting a complete set of near weights, evaluation codes and Goppa codes

On algebras admitting a co mplete set of near w eights, ev alua tion codes and Go pp a codes C ´ ıcero Carv alho 1 and Erc ´ ılio Silv a 2 Abstract. In 1998 Høholdt, v an Lint and Pellik aan introduced the concept of a “weigh t function” defined on a F q -algebra and used it to construct linear co des, obtaining a mong them the alge braic-ge o metric (AG) co des supported on one p o int. Later, in 19 99, it was pr ov ed by Matsumoto that all codes pro duced using a w eig ht function are actually A G co des supp orted on one p oint. Recen tly , “near weigh t f unctions” (a generaliza tion of weigh t functions), also defined on a F q -algebra , were introduced to study co des suppo r ted on tw o p oints. In this paper we show that an algebra admits a s e t o f m nea r weigh t functions having a co mpatibility pro pe r ty , namely , the set is a “ complete set”, if and only if it is the r ing of regular functions o f a n a ffine geometrically irreducible algebr aic curve defined over F q whose po ints at infinity hav e a total of m rationa l br anches. Then the co des pro duced using the near weigh t functions are exactly the A G co des suppor ted on m p oints. A formula for the minimum distance of these co des is presented with examples which show that in some situations it compares b etter than the usua l Goppa b ound. Index terms. near weight functions, ev a lua tion co des , algebra ic geo metric co de s 1 In tro du ction In 1981 V.D. Goppa show ed ho w to use algebraic curve s to pro duce error correcting co des (v. [6]), and his construction op ened a new area of researc h in co ding theory . After a decade of studies, researc hers start ed to w onder if it w as p ossible to find a simpler wa y to pro duce these (so called) algebraic-geometric, or Goppa, co des, one of the earliest attempt b eing made by Bla h ut ([1]). In 1998 Høholdt et al. (v. [5]) presen ted a simple construction for error correcting co des, using an F -algebra R and what they called a weight function on R , their construction clearly pro ducing algebraic-geometric co des supp orted on one p oint. The theory presen ted in [5] w as recen t ly generalized (v. [12] and [2]) by replacing w eigh t functions by other 1 Univ ersidade F ederal de Uberlˆ andia, F aculdade de Matem´ atica, Av. J.N. de ´ Avila 2160, 38408 -100 Ub erlˆ andia – MG, Brazil. email: cicero@ufu.br. Research pa r tially suppo rted by F APE MIG - grant CEX AP Q-471 6-5.0 1 /07 2 Univ ersidade F ederal do ABC, CMCC, Rua Santa Ad ´ elia 166, 09210- 170 Santo Andr ´ e – SP , Brazil. email: ercilio @ufab c.edu.br. 1 functions on R , called ne a r w eights . In the presen t work we study sp ecially algebras that admit m near w eight f unctions with the pro p ert y o f b eing “ a complete set” (see Definition 2.2) . W e will characterize them as b eing the ring of regular f unctions of an affine g eometrically irreducible alg ebraic curv e whose p oin ts at infinit y ha ve a total of m rationa l branc hes, from this w e conclude that the co des o bta ined from suc h algebras using the complete set of near w eigh t functions are exactly the algebraic-geometric co des supp orted o n m p oints (th us generalizing results in [10] and [11 ]). In what follo ws we will denote by N 0 the set of nonnegative in tegers. Let F be a field and R b e a comm utat iv e ring that con tains F , i.e. an F -algebra. Giv en a function ρ : R → N 0 ∪ {−∞} let U ρ := { f ∈ R | ρ ( f ) ≤ ρ (1) } and M ρ := { f ∈ R | ρ ( f ) > ρ (1) } . Definitions 1.1 W e call ρ a ne ar o r der f unction on R (or n- o rder f o r short) if f or an y f , g ∈ R w e ha v e: (N0) ρ ( f ) = −∞ ⇔ f = 0; (N1) ρ ( λf ) = ρ ( f ) ∀ λ ∈ F ∗ ; (N2) ρ ( f + g ) ≤ max { ρ ( f ) , ρ ( g ) } ; (N3) if ρ ( f ) < ρ ( g ) then ρ ( f h ) ≤ ρ ( g h ); if mo r eo ver h ∈ M ρ then ρ ( f h ) < ρ ( g h ); (N4) if ρ ( f ) = ρ ( g ) and f , g ∈ M ρ then there exists λ ∈ F ∗ suc h that ρ ( f − λg ) < ρ ( f ). An n-order function is called a ne ar weight (or n-w eigh t for short) if it also satisfies the follo wing condition. (N5) ρ ( f g ) ≤ ρ ( f ) + ρ ( g ) and equalit y holds when f , g ∈ M ρ . A trivial w a y to define a n n-order ρ on R is to set ρ (0) := −∞ and ρ ( f ) = 1 for a ll f ∈ R , f 6 = 0 , then w e ha v e M ρ = ∅ and U ρ = R . W e w ant to av oid suc h functions, s o w e s a y that an n-order ρ is trivial if M ρ = ∅ and from now o n w e w ork only with nontrivial n-order functions. F rom (N3) it follows that M ρ do es not hav e ze ro divisors, w e also g et the follo wing results (cf. [2, Lemma 4]). Lemma 1.2 L et ρ b e an n-or der on R , then: i) the elem ent λ in ( N4) is uniquely determi n e d; ii) if ρ ( f ) 6 = ρ ( g ) then ρ ( f + g ) = max { ρ ( f ) , ρ ( g ) } . Notation. In the next sections w e deal with subsets of N m 0 , and will use t he follo wing conv en tions: w e denote by 0 the m -tuple ha ving all en tries equal to zero; 2 when we write a ∈ N m 0 it’s to b e understo o d that the en tries of the m -tuple a are a := ( a 1 , . . . , a m ) (similarly for b , c ∈ N m 0 ); w e write sometimes a i ∈ N m 0 , b eing then understo o d that a i = ( a i 1 , . . . , a im ). Also, for i ∈ { 1 , . . . , m } we denote b y e i the m -tuple t ha t has all entries equal to zero, except the i -th entry , whic h is equal to 1. W e add m -tuples and m ultiply them by nonnegative integers in the usual w ay . 2 Co des from ne ar w eigh t s In this section w e sho w ho w to construct co des from algebras that admit a complete set of n-w eights, and give a lo w er b ound for their minim um distance. W e b egin b y in tro ducing the concept of normalized n-orders. Definition 2.1 Let ρ b e an n- order function, w e define the no rmalization ρ ′ of ρ as b eing the function ρ ′ : R → N 0 ∪ {−∞} defined by ρ ′ (0) = −∞ , ρ ′ ( f ) := 0 if f ∈ U ρ \ { 0 } and ρ ′ ( f ) := ρ ( f ) if f ∈ M ρ . F rom the pro of of [2, Prop osition 1] w e kno w that ρ ′ is an n-order, U ρ ′ = U ρ and M ρ ′ = M ρ . F rom no w on w e w or k only with normalized n-orders. If ρ is an n-w eight then from (N5) we see that U ρ is a subalgebra of R . In this section w e will show how to construct linear co des f r om F -algebras and a set o f n-weigh ts whic h hav e a compatibility pro p ert y which w e define now. Let { ρ 1 , . . . , ρ m } b e a set o f (no n trivial, normalized) n-we igh t s. Definition 2.2 W e sa y that { ρ 1 , . . . , ρ m } is a c omplete se t of n-weights for R if ∩ m i =1 U ρ i = F and fo r all k ∈ { 1 , . . . , m } we hav e that N 0 \ ρ k ( ∩ 1 ≤ i ≤ m ; i 6 = k U ρ i ) is a finite set. Let R b e an F -algebra that admits { ρ 1 , . . . , ρ m } as a complete set o f n-w eights. Giv en a = ( a 1 , . . . , a m ) ∈ N m 0 w e define L ( a ) := { f ∈ R : ρ i ( f ) ≤ a i ∀ i = 1 , . . . , m } . F rom (N0),(N1 ) and (N2) we get that L is an F -vec tor subspace of R . Lemma 2.3 F or any k ∈ { 1 , . . . , n } we get L ( a ) ⊂ L ( a + e k ) ; mor e over dim( L ( a + e k ) / L ( a )) ≤ 1 . 3 Pro of: Assume that f , g ∈ L ( a + e k ) \ L ( a ), from (N4) w e kno w that there exists λ ∈ F ∗ suc h that ρ k ( f − λg ) ≤ a k , and from (N1) and (N2) w e get ρ i ( f − λg ) ≤ a i for all i ∈ { 1 , . . . , m } \ { k } . Th us f = λg + h with h ∈ L ( a ) hence f = λg as elemen ts of L ( a + e k ) / L ( a ).  Since L ( 0 ) = F we get as a corollary of the a b ov e lemma tha t L ( a ) is a n F -v ector space of finite dimension for an y a ∈ N m 0 . F or t he remainder of this section, w e will assume that F is a finite field. Let ϕ : R → F n b e a surjectiv e mor phism o f F - algebras and let a ∈ N m 0 . W e will denote b y C ( a ) the co de ϕ ( L ( a )) a nd we w ant to determine a low er b ound for the minim um distance o f C ( a ) ⊥ , in a w ay similar to that whic h ha s b een done b y Høholdt et alli in the case where m = 1 (cf. [5 , Section 4]). Definition 2.4 Let k ∈ { 1 , . . . , m } , and define N k ( a ) as a set of pairs of functions { ( f k , 1 , g k , 1 ) , . . . , ( f k ,ℓ k , g k ,ℓ k ) } suc h that: a) f k ,i , g k ,i ∈ L ( a + e k ) for all i = 1 , . . . , ℓ k ; b) ρ k ( f k ,i ) + ρ k ( g k ,i ) = a k + 1; c) ρ k ( f k , 1 ) < · · · < ρ k ( f k ,ℓ k ) (hence ρ k ( g k , 1 ) > · · · > ρ k ( g k ,ℓ k )); d) given s ∈ { 1 , . . . , ℓ k − 1 } w e ha v e f k ,s g k ,r ∈ L ( a ) for all r = s + 1 , . . . , ℓ k . W e will write ν k ( a ) := # N k ( a ). No w, consider the matrices M and N , where the first ℓ k ro ws of M are ϕ ( f k , 1 ) , . . . , ϕ ( f k ,ℓ k ), the first ℓ k columns o f N are ϕ ( g k , 1 ) , . . . , ϕ ( g k ,ℓ k ), and w e complete the ro ws of M and the columns of N in a w a y suc h that r ank ( M ) = r ank ( N ) = n . Let y = ( y 1 , . . . , y n ) ∈ F n and let D ( y ) := ( a i j ) n × n where a i j = 0 if i 6 = j and a i i = y i for i = 1 , . . . , n . Since r ank ( M ) = r ank ( N ) = n w e get r ank ( M D ( y ) N ) = w t ( y ); moreo ver if r , s ∈ { 1 , . . . , ℓ k } then ( M D ( y ) N ) r,s = y · ( ϕ ( f k ,r ) ∗ ϕ ( g k ,s )), where · is the usual inner pro duct in F n and ∗ is the usual comp o nen twis e pro duct that mak es F n an F -algebra. Prop osition 2.5 If y ∈ C ( a ) ⊥ \ C ( a + e k ) ⊥ then r ank ( M D ( y ) N ) ≥ # N k ( a ) . Pro of: W e ha v e a lr eady noted that ( M D ( y ) N ) r,s = y · ϕ ( f k ,r g k ,s ) for all r , s ∈ { 1 , . . . , ℓ k } . F rom definition 2.4 ( d) we get that the ℓ k × ℓ k minor at t he upp er left corner of M D ( y ) N is a low er tr ia ngular matrix. Since f k ,s g k ,s ∈ L ( a + e k ) \ L ( a ) from Lemma 2.3 w e get dim L ( a + e k ) = dim L ( a ) + 1 hence y · ϕ ( f k ,s g k ,s ) 6 = 0 for all s = 1 , . . . , ℓ k .  4 Definition 2.6 Let a , b ∈ N m 0 b e suc h that a i ≤ b i for all i = 1 , . . . , m . W e call a p ath fr om a to b a finite sequence of m -t uples P := ( a 0 , a 1 , ..., a r ), where a i ∈ N m 0 for all i ∈ { 0 , . . . , r } , a 0 = a , a r = b and for any i ∈ { 0 , . . . , r − 1 } w e ha ve a i +1 = a i + e p ( i ) for some p ( i ) ∈ { 1 , . . . , m } whic h is called the step plac e of a i ∈ P . Lemma 2.7 L et a ∈ N m 0 , then ther e exists b ∈ N m 0 such that dim C ( b ) = n an d a i ≤ b i for al l i ∈ { 1 , . . . , m } . Pro of: Since ϕ is surjectiv e there are f 1 , . . . , f n ∈ R suc h that { ϕ ( f 1 ) , . . . , ϕ ( f n ) } is a basis for F n , so it suffices to take b i := a i + max { ρ i ( f 1 ) , . . . , ρ i ( f n ) } , where i ∈ { 1 , . . . , m } and set b := ( b 1 , . . . , b m ).  As a conse quence of the ab o v e results we hav e the following b o und for the minim um distance of C ( a ) ⊥ . Corollary 2.8 L et a ∈ N m 0 and let b ∈ N m 0 b e such that a i ≤ b i for al l i ∈ { 1 , . . . , m } and dim C ( b ) = n . Given a p a th P = ( a 0 , . . . , a r ) fr om a to b the minimum distanc e of C ( a ) ⊥ is b ounde d fr om b elow by min { ν p ( i ) ( a i ) | i = 0 , . . . , r − 1 } . A t first glance a ma jor dra wbac k of the ab ov e result is that it de p ends on finding b ∈ N m 0 suc h that dim ϕ ( L ( b )) = n , while we w o uld like a b ound that do es not dep end on the kno wledge of ϕ . The f o llo wing considerations sho w that w e do not hav e to find suc h b in order to calculate a b ound. Let k ∈ { 1 , . . . , m } , from (N5) we get tha t S k := ρ k ( ∩ 1 ≤ i ≤ m ; i 6 = k U ρ i ) is a subsemi- group of N 0 and since { ρ 1 , . . . , ρ m } is a complete set for R we get #( N 0 \ S k ) < ∞ (i.e. S k is a num erical semigroup). Observ e also that giv en a ∈ N m 0 and t 1 , t 2 ∈ S k suc h that t 1 + t 2 = a k + 1 then taking f 1 , f 2 ∈ ∩ 1 ≤ i ≤ m ; i 6 = k U ρ i suc h that t i = ρ k ( f i ), i = 1 , 2, w e get ( f 1 , f 2 ) ∈ N k ( a ) (of course also ( f 2 , f 1 ) ∈ N k ( a ), if t 2 6 = t 1 ). Lemma 2.9 L et S b e a numeric al subsemigr oup of N 0 of genus g , let c b e the c onductor of S and let u ∈ N 0 . I f N := { ( a, b ) : a, b ∈ S \ { 0 } ; a + b = 2 c + u } then # N = 2 ( c − g ) + u − 1 . Pro of: W e hav e 2( c − 1 − g ) pairs ( a, b ) ∈ N suc h that either 1 ≤ a ≤ c − 1 or 1 ≤ b ≤ c − 1. And we hav e u + 1 pairs ( a, b ) ∈ N with c ≤ a, b ≤ c + u .  Let a ∈ N m 0 and let ( a i ) i ∈ N 0 ∈ N m 0 b e a sequence of m -tuples suc h that a = a 0 , a i +1 = a i + e p ( i ) for some p ( i ) ∈ { 1 , . . . , m } and lim i →∞ a ij = ∞ , f or all j ∈ 5 { 1 , . . . , m } and all i ∈ N 0 . F rom (the pro of of ) Lemma 2.7 we know that there exists r ∈ N 0 suc h that dim C ( a r ) = n . F or k ∈ { 1 , . . . , m } , let c k b e the conductor of S k , to calculate the b ound indicated in Corollary 2 .8 w e should calculate ν p ( i ) ( a i ) for all i ∈ { 0 , r − 1 } , but w e observ e that: a) if ν k ( a ) =: h > 2( c k − g k ) − 1 then set u := h − 2( c k − g k ) + 1; we shall calculate ν k ( a i ) at most for m -tuples a i suc h that a ik ≤ 2 c k + u − 1 (in fact, if a i k + 1 > 2 c k + u w e will g et ν k ( a i ) > h ); b) if ν k ( a ) ≤ 2( c k − g k ) − 1 then w e shall calculate ν k ( a i ) at most f o r m - tuples a i suc h t ha t a i k ≤ 2 c k − 1 (in fact, if a i k + 1 > 2 c k then ν k ( a i ) > 2( c k − g k ) + 1). Th us we do not hav e to kno w r to calculate the b ound. The next result show s that geometric Goppa co des supp orted in m p oin ts are instances of t he co des w e described ab o v e. Theorem 2.10 L et X b e a nonsi n gular, ge ometric al ly irr e ducible, pr oje ctive alge- br aic curve define d over F , an d let G := P m i − 1 a i Q i and D := P 1 + · · · + P n b e d ivisors on X such that supp ( G ) ∩ s upp ( D ) = ∅ and P i is a r ational p o i n t, for al l i = 1 , . . . , n (henc e the Gopp a c o de C L ( D , G ) is the set of m -tuples ( h ( P 1 ) , . . . , h ( P m )) , wher e h ∈ L ( G ) ) . Then taking R := ∩ Q ∈X ; Q 6 = Q 1 ,...,Q m O Q , wher e O Q is the lo c al ring at Q ∈ X , and defining ϕ ( f ) := ( f ( P 1 ) , . . . , f ( P n )) ther e exists a c omplete set of m ne ar weig hts on R such that C L ( D , G ) = C ( a ) , wher e a := ( a 1 , . . . , a m ) . Pro of: Observ e that R is the F -subalgebra of F ( X ) consisting of the functions regular on X ′ := X \ { Q 1 , . . . , Q m } . Denoting by v k the discrete v aluation of F ( X ) a sso ciated to Q k ( k ∈ { 1 , . . . , m } ), one easily ch ec ks that the function ρ k : R → N 0 ∪ {∞} defined b y ρ k (0) = −∞ , ρ k ( f ) = 0 if v k ( f ) ≥ 0 and ρ k ( f ) = − v k ( f ) if v k ( f ) < 0, for all f ∈ R \{ 0 } is a n n- w eight for all k ∈ { 1 , . . . , m } . W e ha ve U ρ k = R ∩ O Q k and M k = R \ O Q k for all k ∈ { 1 , . . . , m } . More- o ver, since ∩ Q ∈X O Q = F (b ecause X is geometrically irreducible) and S k := ρ k ( ∩ 1 ≤ i ≤ m ; i 6 = k U ρ i ) = ρ k ( ∩ Q ∈X , Q 6 = Q k O Q ) is the W eierstrass semigroup at Q k for all k ∈ { 1 , . . . , m } (hence it has finite gen us) w e get that { ρ 1 , . . . , ρ m } is a complete set o f n- w eights fo r R . Denoting b y M P i the maximal ideal of O P i w e get that R / ( M P i ∩ R ) ∼ = O P i / M P i for all i ∈ { 1 , . . . , n } (see e.g. [13, Prop. I I I.2.9]), hence F n ∼ = R / ( M P 1 ∩ R ) × · · · × R / ( M P n ∩ R ) and from the Chinese Remainder Theorem ϕ is an epimorphism. W e also ha v e L ( a ) = { f ∈ R | − v k ( f ) ≤ a k , k = 1 , . . . , m } = { f ∈ F ( X ) ∗ | div( f ) + P m i =1 a i Q i ≥ 0 } ∪ { 0 } = L ( G ) hence C ( a ) = C L ( D , G ).  6 No w w e presen t examples whic h sho w that when applied to a g eometric Goppa co de, the b o und for the minim um distance found ab ov e may b e b etter than the usual Goppa b ound. Examples 2.11 Let X b e the hermitian curv e giv en b y Y 3 Z + Y Z 3 − X 4 = 0 and defined ov er the field F 3 2 . T ak e Q 1 , Q 2 and Q 3 to b e t hr ee distinct rational p oints , sa y the p oints in the interse ction of X , the op en set Z 6 = 0 and the line X = 0, let a := ( a 1 , a 2 , a 3 ) ∈ N 3 0 and denote by C L ( D , G a ) the geometric Goppa co de asso ciated to the divisors G a := a 1 Q 1 + a 2 Q 2 + a 3 Q 3 and D = P 1 + · · · + P n , where P 1 , . . . , P n are distinct rational p oin ts, different from Q 1 , Q 2 and Q 3 . The gen us of X is 3 and the so-called G oppa b ound for the co de C L ( D , G a ) ⊥ is d a := deg G a − (2 g − 2 ) = P 3 i =1 a i − 4. Note that S i is the semigroup generated b y 3 and 4, so the conductor is 6, for all k = 1 , 2 , 3 . T o find a b ound a s describ ed in Corolla r y 2.8 it is useful to know the set { ( ρ 1 ( f ) , ρ 2 ( f ) , ρ 3 ( f )) ∈ N 3 0 | f ∈ R } , whic h in this case is exactly the W eierstrass semigroup W a sso ciated to { Q 1 , Q 2 , Q 3 } , i.e. the set S = { ( n 1 , n 2 , n 3 ) ∈ N 3 0 | div ∞ ( f ) = n 1 Q 1 + n 2 Q 2 + n 3 Q 3 } , where div ∞ ( f ) denotes the p ole divisor of f . Suc h semigroups hav e b een m uc h studied in the last decade (see e.g. [9], [8 ], [7], [3]), and in [7] w e find an explicit description of a generating set for this semigroup, so that w e may decide if an elemen t of N 3 0 is or is not in S . Th us, given a ∈ N 3 0 w e pro ceed as follo ws. F or k = 1 , 2 , 3 w e calculate ν k ( a ), if ν k ( a ) > 2(6 − 3 ) − 1 = 5 then set A k := 2 · 6 + ( ν k ( a ) − 5) − 1, if ν k ( a ) ≤ 5 the w e set A k := 2 · 6 − 1 = 11. Let r := P 3 i =1 ( A i − a i ) and consider the path P from a to ( A 1 , A 2 , A 3 ) giv en b y ( a 0 , . . . , a r ) where a 0 = a , a r = ( A 1 , A 2 , A 3 ), a i = a + i e 1 , for i ∈ { 1 , . . . , A 1 − a 1 } , a A 1 − a 1 + j = a + ( A 1 − a 1 ) e 1 + j e 2 , for j ∈ { 1 , . . . , A 2 − a 2 } , and a A 1 − a 1 + A 2 − a 2 + k = a + ( A 1 − a 1 ) e 1 + ( A 2 − a 2 ) e 2 + k e 3 , f o r k ∈ { 1 , . . . , A 3 − a 3 } . F rom the considerations that precede these examples w e get that δ a := min { ν p ( i ) ( a i ) | i = 0 , . . . , r − 1 } is a b ound f or the minim um distance of C L ( D , G a ) ⊥ . Let ρ ( N k ( a )) := { (( ρ 1 ( f k ,i ) , ρ 2 ( f k ,i ) , ρ 3 ( f k ,i )) , ( ρ 1 ( g k ,i ) , ρ 2 ( g k ,i ) , ρ 3 ( g k ,i ))) | i = 1 , . . . , ℓ k } , ta king a = (2 , 1 , 1) we hav e ν 1 ( a ) = 2 (with ρ ( N 1 ( a )) = { ((0 , 0 , 0) , (3 , 0 , 0)) , ((3 , 0 , 0 ) , (0 , 0 , 0)) } ), ν 2 ( a ) = 2 (with ρ ( N 2 ( a )) = { ((0 , 0 , 0) , (0 , 3 , 0)) , ((0 , 3 , 0) , (0 , 0 , 0 )) } ), a nd ν 3 ( a ) = 3 (with ρ ( N 3 ( a )) = { ((0 , 0 , 0) , (0 , 2 , 2)) , ((1 , 1 , 1) , (1 , 1 , 1 )) , ((0 , 2 , 2) , (0 , 0 , 0)) } ). Th us ( A 1 , A 2 , A 3 ) = (11 , 11 , 11 ) and insp ecting W w e get that δ a = 2, while d a = 0. In the table b elow w e presen t results for this and other v alues of a . 7 a ( ν 1 ( a ) , ν 2 ( a ) , ν 3 ( a )) ( A 1 , A 2 , A 3 ) δ a d a (2 , 1 , 1) (2,2,2) (11,11,11) 2 0 (1 , 2 , 1) (2,2,2) (11,11,11) 2 0 (1 , 1 , 2) (2,2,2) (11,11,11) 2 0 (2 , 2 , 1) (2,2,3) (11,11,11) 2 1 (2 , 1 , 2) (2,3,2) (11,11,11) 2 1 (1 , 2 , 2) (3,2,2) (11,11,11) 2 1 (2 , 2 , 2) (3,3,3) (11,11,11) 2 2 (3 , 2 , 2) (4,4,4) (11,11,11) 3 3 (2 , 3 , 2) (4,4,4) (11,11,11) 3 3 (2 , 2 , 3) (4,4,4) (11,11,11) 4 3 T able 1: Bounds for δ a and d a ; co de C ( a ) ⊥ ; curv e Y 3 Z + Y Z 3 − X 4 = 0, defined o ver F 9 . W e also presen t a similar table, con ta ining examples of co des f r o m the hermi- tian curv e given b y Y 4 Z + Y Z 4 − X 5 = 0 a nd defined ov er the field F 16 . Aga in, w e t a k e Q 1 , Q 2 and Q 3 to b e three distinct ra tional p oints of the curv e, now S i is the semigroup generated b y 4 and 5, so the conductor is 12, f o r i = 1 , 2 , 3; the gen us of the curv e is 6. a ( ν 1 ( a ) , ν 2 ( a ) , ν 3 ( a )) ( A 1 , A 2 , A 2 ) δ a d a (1 , 2 , 3) (2,2,2) (23,23,23) 2 -4 (3 , 1 , 3) (2,2,2) (23,23,23) 2 -3 (3 , 2 , 3) (2,2,2) (23,23,23) 2 -2 (3 , 3 , 3) (2,2,2) (23,23,23) 2 -1 (4 , 3 , 2) (2,2,2) (23,23,23) 2 -1 (4 , 3 , 3) (2,2,2) (23,23,23) 2 0 (4 , 4 , 3) (2,2,3) (23,23,23) 2 1 T able 2: Bounds for δ a and d a ; co de C ( a ) ⊥ ; curv e Y 4 Z + Y Z 4 − X 5 = 0; defined o ver F 16 . 3 Algebras with near w eigh t s and algeb raic curv e s In this section w e presen t a characterization for a lgebras whic h admit a complete set o f n- w eights. 8 Lemma 3.1 L et R b e an F -algebr a a n d ρ an n-w eight. L et f , g ∈ R b e such that ρ ( f ) > 0 , ρ ( g ) = 0 , g / ∈ F and ρ ( f g ) < ρ ( f ) . T h en for any λ ∈ F ∗ we h ave ρ ( f ( g + λ )) = ρ ( f ) and ρ ( g + λ ) = 0 . Pro of: Let λ ∈ F ∗ , then ρ ( f ( g + λ )) = ρ ( f g + λf )) ≤ max { ρ ( f g ) , ρ ( f ) } . Since ρ ( f g ) < ρ ( f ) w e get ρ ( f ( g + λ )) = ρ ( f ) . W e also ha ve g + λ ∈ U ρ since U ρ is an F -subalgebra of R .  Let R b e an F - algebra whic h admits a (no t nec essarily complete) se t of n- w eights { ρ 1 , . . . , ρ m } . Let ρ : R \ { 0 } → N m 0 b e the map defined b y ρ ( f ) := ( ρ 1 ( f ) , . . . , ρ m ( f )) and let S ρ 1 ,...,ρ m = S := ρ ( R \ { 0 } ). W e will alw a ys assume that if the field F is finite then #( F ) ≥ m . Definition 3.2 Let a i ∈ N m 0 , with i = 1 , . . . , r . W e define the le a s t upp er b ound of a 1 , . . . , a r as b eing the m -tuple lub( a 1 , . . . , a r ) := ( b 1 , . . . , b m ) where b j = max { a j 1 , . . . , a j r } for all j = 1 , . . . , m . Prop osition 3.3 L et a 1 , . . . , a r ∈ S , then lub( a 1 , . . . , a r ) ∈ S . F urthermor e, if f 1 , . . . , f r ∈ R ar e such that ρ ( f i ) = a i for a l l i ∈ { 1 , . . . , r } then ther e ex i s ts f ∈ R , f = P r i =1 λ i f i , wher e λ 1 , . . . , λ r ∈ F such that ρ ( f ) = lub( a 1 , . . . , a r ) . Pro of: Since lub( a 1 , . . . , a j ) = lub(lub( a 1 , . . . , a j − 1 ) , a j ) for all j = 2 , . . . , r it suffices to prov e the case where r = 2. Let f , g ∈ R b e suc h tha t ρ ( f ) = a 1 and ρ ( g ) = a 2 . If a 1 = a 2 then the result is trivial, so w e will assume that a 1 6 = a 2 , a fortio ri f 6 = λg for a ll λ ∈ F ∗ . If #( { j | a 1 j = a 2 j } ) = m − 1 then lub( a 1 , a 2 ) ∈ { a 1 , a 2 } , so w e assume that #( { j | a 1 j = a 2 j } ) ≤ m − 2. Let i ∈ { 1 , . . . , m } , if ρ i ( f ) 6 = ρ i ( g ) then ρ i ( f + λg ) = max { ρ i ( f ) , ρ i ( g ) } for all λ ∈ F ∗ ; if ρ i ( f ) = ρ i ( g ) = 0 then fo r a ll λ ∈ F ∗ w e get ρ i ( f + λg ) = 0; if ρ i ( f ) = ρ i ( g ) 6 = 0 then there exists a unique λ i ∈ F ∗ suc h that ρ i ( f − λ i g ) < ρ i ( f ), hence for all λ ∈ F ∗ , λ 6 = − λ i w e get ρ i ( f + λg ) = ρ i ( f ). Since #( F ) − 1 > m − 2 there exists λ ∈ F ∗ suc h t ha t ρ i ( f + λg ) = max { ρ i ( f ) , ρ i ( g ) } for all i ∈ { 1 , . . . , m } .  Lemma 3.4 L et a an d b b e distinct elements of S and supp ose that a j = b j for some j ∈ { 1 , . . . , m } . Then ther e exists c ∈ S such that: i) c i = max { a i , b i } for i 6 = j a n d a i 6 = b i ; ii) c i ≤ a i for al l i 6 = j and a i = b i ; iii) c j = a j = 0 or c j < a j . 9 Pro of: Let f , g ∈ R suc h that ρ ( f ) = a and ρ ( g ) = b . If a j = b j = 0 then it suffices to take c = ρ ( f + g ). If a j = b j > 0 then f , g ∈ M ρ j and there exists λ ∈ F ∗ suc h t ha t ρ j ( f − λ g ) < a j , so we take c = ρ ( f − λg ) .  Let 4 b e the (partial) ordering in N m 0 giv en b y the relation a 4 b if a i ≤ b i for all i ∈ { 1 , . . . , m } . Prop osition 3.5 L et a ∈ S , then the fol lowing assertions a r e e quivalent: i) a is a minimal elemen t of the set { c ∈ S | c k = a k } for some k ∈ { 1 , . . . , m } such that a k > 0 ; ii) a is a minimal element of the set { c ∈ S | c i = a i } for al l i ∈ { 1 , . . . , m } s uch that a i > 0 . Pro of: A ssume that a is a minimal of the set { c ∈ S | c k = a k } for s ome k ∈ { 1 , . . . , m } a nd supp ose tha t a is not a minimal of the set { c ∈ S | c j = a j } for some j ∈ { 1 , . . . , m } . Then there exists b ∈ S suc h that b 4 a , b 6 = a and b j = a j , furthermore, from the hy p othesis w e m ust hav e b k < a k . F rom Lemma 3 .4 there exists c ∈ S suc h that c i ≤ max { a i , b i } for all i ∈ { 1 , . . . , m } , c k = a k and c j < a j , so a is not a minimal of t he set { c ∈ S | c k = a k } , a con tr a diction.  Definition 3.6 If a ∈ S is a minimal elemen t o f the set { c ∈ S | c k = a k } for some k ∈ { 1 , . . . , m } w e say that a is a minim al o f S (cf. [7, Section 2]). W e will denote b y Γ the set of all minimals. Observ e that 0 and the p oin t s of S whic h ha ve all en tries but one equal to zero are minimals. Theorem 3.7 The set S i s a subsem igr oup of N m 0 . Pro of: Let a , b ∈ S and let f , g ∈ R b e suc h that ρ ( f ) = a and ρ ( g ) = b . Set c := ρ ( f g ), for i ∈ { 1 , . . . , m } w e ha ve c i ≤ a i + b i and equalit y holds whenev er a i > 0 and b i > 0, hence a + b = lub( a , b , c ).  W e assume from no w on that { ρ 1 , . . . , ρ m } is a complete set of n- w eights f or R ; the next result sho ws that Γ generates the semigroup S under the op eration lub . Lemma 3.8 L et a ∈ S and let r b e the numb er of nonzer o entries of a , then ther e exist a 1 , . . . , a r ∈ Γ such that a = lub ( a 1 , . . . , a r ) . 10 Pro of: Let a ∈ S \ Γ and let Λ ⊂ { 1 , . . . , m } b e the set of indexes i for whic h a i > 0; from Pro p osition 3.5 a is not a minimal in any set { b ∈ S | b i = a i } with i ∈ Λ, then for all i ∈ Λ there exists b i ∈ Γ suc h that b i 4 a and b ii = a i , so w e ha ve a = lub( b i ; i ∈ Λ).  Giv en j ∈ { 1 , . . . , m } let H j := { a ∈ N 0 | ∃ f ∈ ∩ m i =1; i 6 = j U ρ i suc h t ha t ρ j ( f ) = a } ( i.e. a ∈ H j if and only if there exists a ∈ S having a ll en tries equal to zero, except the j -th entry , whic h is equal to a ). Then H j is a semigroup whic h has finite gen us (since { ρ 1 , . . . , ρ m } is a complete set of n-w eights). Lemma 3.9 L et a ∈ Γ an d let Λ = { j | a j > 0 } ⊂ { 1 , . . . , m } . If #Λ ≥ 2 then a j / ∈ H j for al l j ∈ Λ . Pro of: Let j ∈ Λ a nd a ssume by means of absurd that a j ∈ H j ; let b ∈ N m 0 b e the m - tuple hav ing all en t r ies equal to zero except the j -th, whic h is equal to a j . Then b ∈ S , b 4 a and b 6 = a , hence a / ∈ Γ.  Let ˜ Γ := { a ∈ Γ | a has at least tw o nonzero en tries } , an easy but imp ortant consequenc e o f t he a b ov e lemma is the follo wing. Corollary 3.10 The set ˜ Γ is fin i te. Pro of: Let G j b e the set of gaps of H j , then #( G j ) < ∞ for all j ∈ { 1 , . . . , m } and fr o m the lemma ab ov e ˜ Γ ⊂ G 1 × · · · × G m .  F or eac h a ∈ Γ let f a ∈ R b e suc h that ρ ( f a ) = a , and let B := { f a ∈ R ; | a ∈ Γ } . Prop osition 3.11 The set B sp ans R as an F -ve ctor sp ac e. Pro of: W e w an t to sho w tha t an y f ∈ R \ { 0 } is a finite linear com binat ion ov er F of elemen t s of B , and we do this b y induction o n the num b er o f nonzero entries of a := ρ ( f ). If this num b er is zero then f ∈ F ∗ and is a multiple of f 0 ∈ F ∗ . Assume that a has r nonzero en tries, with r ≥ 1, and fo r simplicit y , let’s assume that these entries ar e a 1 , . . . , a r . F rom Lemma 3.8 there are a 1 , . . . , a r ∈ Γ suc h that a = lub( a 1 , . . . , a r ) and from Prop osition 3.3 there are λ 1 , . . . , λ r ∈ F suc h that g := P r i =1 λ i f a i satisfies ρ ( g ) = ρ ( f ). Since ρ 1 ( f ) = ρ 1 ( g ) = a 1 > 0 there is λ ∈ F ∗ suc h that ρ 1 ( f − λg ) < a 1 , moreov er ρ j ( f − λg ) ≤ a j for all j ∈ { 2 , . . . , m } . If f = λg w e are done, o therwise w e rep eat t he pro cess, starting with f − λg this time, until we get either that f is a linear combination of finite elemen ts of B or 11 that the m -tuple o btained by applying the function ρ to f min us a finite linear com bination of elemen t s of B has less than r nonzero elemen ts (b ecause the first en try is certainly zero); either w ay we ’re done.  Prop osition 3.12 R is a finitely gener ate d algebr a over F . Pro of: Le t i ∈ { 1 , . . . , m } , we know that the semigroup H i ⊂ N 0 has finite gen us, hence it is finitely generated, so let H i = h a i 1 , . . . , a ir i i . F or eac h a ij with i ∈ { 1 , . . . , m } and j ∈ { 1 , . . . , r i } there is a ij ∈ Γ ha ving all entries equal to zero, except the i -th en t ry which is equal to a ij . Th us if a ∈ Γ \ ˜ Γ, i.e. if a has only one p ositiv e en t ry , whic h is in the i -th p osition fo r some i ∈ { 1 , . . . , m } , then for certain α 1 , . . . , α r i ∈ N 0 w e hav e ρ ( f α 1 a i 1 · . . . · f α r i a ir i ) = a (recall that f a ij ∈ B and are suc h that ρ ( f a ij ) = a ij for all j ∈ { 1 , . . . , r i } ) and w e can tak e f a := f α 1 a i 1 · . . . · f α r i a ir i . Since ˜ Γ is a finite set, the result follo ws from the ab ov e prop osition.  Theorem 3.13 L et f ∈ R , f 6 = 0 , then dim F R / ( f ) < ∞ . Pro of: W e ma y assume that f ∈ R \ F . W e also assume m ≥ 2 (for m = 1 t he pro of is in [10]). F rom Prop osition 3.11 w e ha v e that the set B := { f a ∈ R / ( f ) | a ∈ Γ } spans R / ( f ) as a v ector space ov er F , and since ˜ Γ is finite, it suffices to sho w that for all a ∈ Γ \ ˜ Γ, except ma yb e fo r a finite n umber, we ma y tak e f a ∈ ( f ) . Th us, we will show that for a n y i ∈ { 1 , . . . , m } there exists n i ∈ N 0 suc h that for all n ≥ n i with n ∈ H i w e may find s ∈ ( f ) suc h that ρ i ( s ) = n and ρ j ( s ) = 0 for all j ∈ { 1 , . . . , m } , j 6 = i . F or simplicity , let’s take i = 1; w e will consider t wo cases. In the first case, w e assume that ρ j ( f ) = 0 for all j = 2 , . . . , m , hence ρ 1 ( f ) > 0 (since f / ∈ F ). Let ℓ 1 b e the largest gap of H 1 and set d 1 := ρ 1 ( f ), if a 11 , . . . , a 1 r 1 are generators for H 1 , then using the notation of the preceding pro o f , for any n > ℓ 1 + d 1 w e may find α 1 , . . . , α r 1 ∈ N 0 suc h that ρ 1 ( f α 1 a 11 · . . . · f α r 1 a 1 r 1 f ) = n and ρ j ( f α 1 a 11 · . . . · f α r 1 a 1 r 1 f ) = 0 for all j = 2 , . . . , m . In the second case w e assume that there exists j ∈ { 2 , . . . , m } suc h t ha t ρ j ( f ) > 0. Let g ∈ R b e suc h that ρ 1 ( g ) > 0 a nd ρ i ( g ) = 0 f o r all i ∈ { 2 , . . . , m } (suc h g exists b ecause the gen us of H 1 is finite, moreo v er g / ∈ F ), then ρ j ( f g ) ≤ ρ j ( f ) and there exists λ ∈ F suc h that ρ j ( f g − λf ) = ρ j ( f ( g − λ ) < ρ j )( f ). W e ha v e g − λ ∈ M ρ 1 but for all i ∈ { 2 , . . . , m } , since g ∈ U ρ i w e hav e g − λ ∈ U ρ i and ρ i ( f ( g − λ )) ≤ ρ i ( f ). By rep eating this pro cess we may find h ∈ M ρ 1 suc h that ρ i ( hf ) ≤ ρ i ( f ) for all 12 i ∈ { 2 , . . . , m } and ρ j ( hf ) = 0; rep eating ev en further we find t ∈ M ρ 1 suc h tha t ρ i ( tf ) = 0 for all i ∈ { 2 , . . . , m } (observ e that ρ 1 ( tf ) > 0 since if ρ 1 ( tf ) = 0 then tf ∈ ∩ m i =1 U ρ i = F , and a fortior i f ∈ F , a contradiction). Let ℓ 1 b e the largest gap in H 1 and set d 1 := ρ 1 ( tf ); g iv en n > ℓ 1 + d 1 let u ∈ M ρ 1 b e suc h that ρ 1 ( u ) = n − d 1 and ρ i ( u ) = 0 for all i ∈ { 2 , . . . , m } , then ρ 1 ( utf ) = n and ρ i ( utf ) = 0 for all i ∈ { 2 , . . . , m } . This completes the pro of.  W e hav e alr eady observ ed that R is a domain, and w e will denote b y K its field of fr actions. Lemma 3.14 K is an algebr a ic function field of one variable over F . Pro of: L et f ∈ R , f 6 = 0, from Theorem 3.13 w e know that R / ( f ) is an F -vec tor space of finite dimension, furthermore, all ideals o f R/ ( f ) are F -subspaces hence R / ( f ) is an artinian ring, s o dim Krull R / ( f ) = 0 . T aking f ∈ R \ F , fr o m [4, Corollary 13.11] w e ha ve dim Krull R = dim Krull R / ( f ) + 1; o n the other ha nd from [4, Theorem A, page 2 23] w e get tr deg F K = dim Krull R = 1.  Corollary 3.15 The algebr a R is the affine c o or dinate rin g of an (irr e ducible) algebr aic curve. Pro of: It is an immediate consequenc e of Prop osition 3.12 a nd the a b ov e lemma.  Let i ∈ { 1 , . . . , m } , from the pro o f of Theorem 3.13 w e get that if f ∈ R \ { 0 } then there exists g ∈ M ρ i suc h that g f ∈ M ρ i , henc e if a, b ∈ R \ { 0 } and g 1 a, g 2 b ∈ M ρ i with g 1 , g 2 ∈ M ρ i then ( g 1 g 2 ) a, ( g 1 g 2 ) b ∈ M ρ i . Definition 3.16 Let i ∈ { 1 , . . . , m } and let v i : K → Z ∪ {∞} b e the function defined by setting v i (0) := ∞ and v i ( a/b ) := ρ i ( g b ) − ρ i ( g a ), where a, b, ∈ R \ { 0 } and g ∈ M ρ i is suc h tha t g a, g b ∈ M ρ i . Observ e that v i ( a/b ) do es not dep end on the c hoice of g b ecause if h ∈ M ρ i is suc h that ha, hb ∈ M ρ i then ρ i ( g b ) − ρ i ( g a ) − ( ρ i ( hb ) − ρ i ( ha )) = ρ i ( g bha ) − ρ i ( g ahb ) = 0, fo r all i ∈ { 1 , . . . , m } ; a similar reasoning sho ws that if a ′ /b ′ = a/b , with a, a ′ , b, b ′ ∈ R \ { 0 } then v i ( a/b ) = v i ( a ′ /b ′ ). Lemma 3.17 L et i ∈ { 1 , . . . , m } . a) The function v i : K → Z ∪ {∞ } is a discr ete valuation o f the function field K | F ; b) I f f ∈ R then v i ( f ) ≥ 0 when f ∈ U ρ i and v i ( f ) = − ρ i ( f ) whe n f ∈ M ρ i . 13 Pro of: Given f , g ∈ K \ { 0 } it is easy to c hec k that v i ( f g ) = v i ( f ) + v i ( g ) and that v i ( f ) = 0 if f ∈ F ∗ . Since H i has finite gen us, we kno w that fo r a sufficien tly large n ∈ N there ar e f , g ∈ M ρ i suc h that ρ i ( f ) = n , ρ i ( g ) = n + 1, hence v i ( f /g ) = 1 . Let f = a/b, g = c/d ∈ K , with a, c ∈ R and b, d ∈ R \ { 0 } , and let h 1 , h 2 ∈ M ρ i suc h that h 1 a, h 1 b, h 2 c, h 2 d ∈ M ρ i , then v i ( f + g ) = v i (( ad + bc ) /bd ) = ρ i ( h 1 h 2 bd ) − ρ i ( h 1 h 2 ad + h 1 h 2 bc ) ≥ min { ρ i ( h 1 h 2 bd ) − ρ i ( h 1 h 2 ad ) , ρ i ( h 1 h 2 bd ) − ρ i ( h 1 h 2 bc ) } = min { ρ i ( h 1 b ) − ρ i ( h 1 a ) , ρ i ( h 2 d ) − ρ i ( h 2 c ) } = { v i ( f ) , v i ( g ) } . No w let f ∈ R \ { 0 } and g ∈ M ρ i b e suc h that g f ∈ M ρ i , if f ∈ U ρ i then from (N5) and the fact that ρ i is normalized w e get v i ( f / 1) = ρ i ( g ) − ρ i ( g f ) ≥ − ρ i ( f ) = 0; o n the other ha nd, if f ∈ M ρ i then v i ( f / 1) = ρ i ( g ) − ρ i ( g f ) = − ρ i ( f ).  This sho ws that eve ry n- w eight ρ i on R defines a v aluatio n v i of the function field K | F . Thes e are distinct v a lua tions (e.g. for a sufficien tly la rge n ∈ N we ma y find f i ∈ M ρ i for all i ∈ { 1 , . . . , m } suc h t ha t v i ( f i ) = − n and v j ( f i ) ≥ 0 f o r all j ∈ { 1 , . . . , m } \ { i } ). W e denote by P i the place asso ciat ed to the v alua t io n v i and b e O P i the corresp onding v aluation ring ( i ∈ { 1 , . . . , m } ). Prop osition 3.18 F or al l i ∈ { 1 , . . . , m } the plac e P i has de gr e e one (a fortiori, F is the ful l field o f c onstants of K ). Pro of: Let i ∈ { 1 , . . . , m } , w e m ust prov e that the inclusion map F → O P i /P i is surjectiv e. Let f = a/b ∈ O P i , where a, b ∈ R , let g ∈ M ρ i suc h that g a, g b ∈ M ρ i and a ssume that v i ( f ) = 0. Then ρ i ( g b ) = ρ i ( g a ) a nd there exists a unique λ ∈ F ∗ suc h tha t ρ i ( g a − λg b ) < ρ i ( g b ). Let h ∈ M ρ i b e suc h that h ( a − λb ) , hb ∈ M ρ i , then v i ( a/b − λ ) = ρ i ( hb ) − ρ i ( h ( a − λb )) = ρ i ( hg b ) − ρ i ( hg ( a − λb )), so from ρ i ( g b ) − ρ i ( g a − λg b ) > 0 and prop ert y (N3) w e get ρ i ( hg b ) − ρ i ( hg ( a − λb )) > 0, whic h completes the pro of.  W e denote b y P ( K ) the set of places of t he function field K | F . F or P ∈ P ( K ) w e write O P for the correspo nding v a luation ring; let S ( R ) := { P ∈ P ( K ) | R ⊂ O P } . Prop osition 3.19 S ( R ) = P ( K ) \ { P 1 , . . . , P m } . Pro of: First w e observ e that, for all i ∈ { 1 , . . . , m } we hav e P i / ∈ S ( R ), since R ⊂ O P i w ould imply M ρ i = ∅ , a contradiction with t he fa ct that ρ i is non- t r ivial. Supp ose by means o f absurd that P ( K ) \ ( S ( R ) ∪ { P 1 , . . . , P m } ) 6 = ∅ . Then, from the Stro ng Appro ximation Theorem (see [13 , Thm. I.6 .4 ]) w e kno w that for all 14 j ∈ N there exists f j ∈ K suc h that v i ( f j ) = j , for all i ∈ { 1 , . . . , m } and f j ∈ O Q for all Q ∈ S ( R ), thus f j ∈ ∩ Q ∈S ( R ) O Q =: ¯ R , the in tegral closure o f R in K . Let W := { x ∈ ¯ R | v i ( x ) > 0 ∀ i = 1 , . . . , m } , observ e that W is an F - v ector space and also W ∩ R = { 0 } : in fact, if x ∈ W ∩ R then ρ i ( g ) − ρ i ( g x ) > 0 for some g ∈ M ρ i , th us ρ i ( g x ) < ρ i ( g ) and fr o m (N5 ) either ρ i ( x ) = 0 fo r all i ∈ { 1 , . . . , m } or x = 0, since ∩ m i =1 U ρ i = F and x ∈ W w e m ust ha v e x = 0. Th us dim F W ≤ dim F ¯ R / R and this last dimension is finite (see e.g. [10 , Lemma 8]) , but { f 1 , . . . , f n } ⊂ W is a linearly indep enden t set ov er F for all n ∈ N .  Corollary 3.20 R is an F -algebr a admitting a c omplete set of m n-weights if and only if R is the ring of r e gular functions of an affine ge ometric al ly irr e ducible algebr aic curve, wh o se p oints in the closur e have a total of r br anches, al l of them c orr esp onding to r ational plac es in the field of r ational functions of the curve. Pro of: The “only if ” part is a consequence of the ab ov e results. As for the “if ” part let X b e the affine curv e and X b e its closure, if Y is the normalization of X a nd η : Y → X is the normalization morphism then there are m r a tional p oints Q 1 , . . . , Q m in the inv erse image b y η of the set X \ X . No w w e pro ceed as in theorem 2 .10; thus w e observ e t ha t R = ∩ Q ∈X O Q , where O Q is the lo cal ring at Q ∈ X and denoting b y v k the discrete v aluatio n of F ( X ) asso ciated to Q k ( k ∈ { 1 , . . . , m } ) w e define the function ρ k : R → N 0 ∪ {∞} by setting ρ k (0) := −∞ , ρ k ( f ) := 0 if v k ( f ) ≥ 0 and ρ k ( f ) := − v k ( f ) if v k ( f ) < 0, for all f ∈ R \ { 0 } , one ma y ch ec k that ρ k is an n-w eight for all k ∈ { 1 , . . . , m } . F rom ∩ m k =1 U ρ k = R ∩ ( ∩ m k =1 O Q k ) = F and the fact that S k := ρ k ( ∩ 1 ≤ i ≤ m ; i 6 = k U ρ i ) = ρ k ( ∩ Q ∈X O Q ) is the W eierstrass semigroup a t Q k for all k ∈ { 1 , . . . , m } w e get that { ρ 1 , . . . , ρ m } is a complete set o f n-weigh ts fo r R .  Theorem 3.21 L et R b e an F -algebr a that a d mits a c omplete set of m n-wei g h ts, let ϕ : R → F n b e a surje ctive morphism of F -a lgebr as and a ∈ N m 0 , then C ( a ) is an alg e b r aic-ge ometric Gopp a c o de C L ( D , G ) with G supp orte d on m p oints. Pro of: F rom the hypothesis on R w e know that there is a geometrically irre- ducible, pro jectiv e, nonsingular curv e Y and p oin ts Q 1 , . . . , Q m suc h that R = ∩ P ∈Y \{ Q 1 ,...,Q m } O P . F or i ∈ { 1 , . . . , n } consider the F -a lgebra surjectiv e homomor- phism π i : F n → F defined b y π i ( λ 1 , . . . , λ n ) = λ i , then M i := ( π i ◦ ϕ ) − 1 (0) is a maximal ideal of R . F urtermore, for distinct i, j ∈ { 1 , . . . , n } we get M i 6 = M j since ϕ is surjectiv e and then exists g ij ∈ R suc h that ( π i ◦ ϕ ) ( g ij ) = 0 and 15 ( π j ◦ ϕ )( g ij ) 6 = 0. F rom [13, Prop. II I.2.9] w e get that there a r e P 1 , . . . , P n ∈ Y suc h t ha t P i / ∈ { Q 1 , . . . , Q m } , M i = M P i ∩ R (where M P i is the maximal ideal of O P i ) for all i = 1 , . . . , n . W e also get F ≃ R / M i ≃ O P i / M P i for all i = 1 , . . . , n hence P 1 , . . . , P n are ratio nal p oin t s o f Y and we ma y rewrite ϕ as the morphism o ver O P 1 / M P 1 × · · · × O P n / M P n defined b y ϕ ( f ) = ( f + M P 1 , . . . , f + M P n ). Let G := a 1 Q 1 + · · · + a m Q m , then L ( G ) ⊂ R and L ( G ) = { f ∈ R : v i ( f ) + a i ≥ 0 for all i = 1 , . . . , m } = { f ∈ R : − v i ( f ) ≤ a i whenev er v i ( f ) < 0 , i = 1 , . . . , m } = { f ∈ R : − v i ( f ) ≤ a i whenev er f ∈ M ρ i , i = 1 , . . . , m } = { f ∈ R : ρ i ( f ) ≤ a i whenev er f ∈ M ρ i , i = 1 , . . . , m } = { f ∈ R : ρ i ( f ) ≤ a i , i = 1 , . . . , m } = L ( a ) , hence C ( a ) = C L ( D , G ), where D = P 1 + · · · + P n .  References [1] R. E. Blahut, “AG co des without AG”, presen ted at 1992 IEEE Infor ma t io n Theory W orkshop, Salv ador da Bahia, Brazil, June 1992. [2] C. Carv a lho , C. Mu ˜ nuera, E. Silv a and F. T orres, “Near orders and co des”, IEEE T rans. Inf. Theory , v ol. 53, pt. 5, pp. 1919–19 2 4, 2007 . [3] C. Carv alho and F. T orres, “On Goppa co des and W eierstrass gaps at sev era l p oints ”, Des. Co des Cryptogr. vol. 35, n. 2, pp. 211– 2 25, 200 5. [4] D. 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