On the order bounds for one-point AG codes

The order bound for the minimum distance of algebraic geometry codes was originally defined for the duals of one-point codes and later generalized for arbitrary algebraic geometry codes. Another bound of order type for the minimum distance of general linear codes, and for codes from order domains in particular, was given in [H. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields and their Applications 14 (2008), pp. 92-123]. Here we investigate in detail the application of that bound to one-point algebraic geometry codes, obtaining a bound $d^*$ for the minimum distance of these codes. We establish a connection between $d^*$ and the order bound and its generalizations. We also study the improved code constructions based on $d^*$. Finally we extend $d^*$ to all generalized Hamming weights.


Introduction
Algebraic geometry codes, or AG codes, over the finite field F q with q elements are constructed from a (projective, non-singular, geometrically irreducible) algebraic curve X |F q and two rational divisors with disjoint support, D = P 1 + · · · + P n and G . The code C(D, G) is defined as the image of the Riemann-Roch space L(G) by the evaluation at D map ev D : L(G) → F n q , ev D (f ) = (f (P 1 ), . . . , f (P n )), see Section 3 or [3,10,14]. The divisor G is often taken as a multiple of a single point, G = mQ, with Q ∈ supp(D). In this case C(D, G) = C(D, mQ) is called one-point code.
Given a code C(D, mQ) the first task is to compute its parameters: length, dimension and minimum distance. The length is obviously n = deg(D). In order to compute the dimension an important role is played by the Weierstrass semigroup at Q, Knowing H * is equivalent to knowing the dimension of all codes C(D, mQ). It is clear that H * consists of n elements, that H * ⊂ H and that for m < n, m ∈ H * if and only if m ∈ H.
Regarding the minimum distance d = d(C(D, mQ)) the simplest estimate is given by the Goppa bound, d ≥ n − m. The Goppa bound does not give the true minimum distance in many cases. For example, it does not give any information when m ≥ n. This problem can be solved by using the improved Goppa bound, d ≥ n − m + γ a+1 , where a = ℓ(mQ − D) is the abundance of C(D, mQ). The drawback of this improved bound is that it is based on the gonality sequence (γ i ) of the curve X , see [11], which is difficult to compute.
Besides uniform bounds, some of the most interesting known bounds for d are of order type. These bounds are based on obtaining different estimates for different subsets of codewords. They are successful if for each subset we can find estimates better than a uniform bound for all codewords, see [4]. The original order bound d ORD (also called Feng-Rao bound) was introduced by Feng and Rao in [7] and by Høholdt, van Lint and Pellikaan in [10]. It usually gives very good results, but it has the disadvantage that it can only be applied to the duals of one-point codes, which are not one-point codes in general. A nice generalization of this bound for arbitrary AG codes was given by Beelen [2] and later improved by Duursma, Kirov and Park in a sequence of articles [4,5,6].
Another bound of order type for general linear codes was given in [1]. This bound was applied to order domain codes and to one-point codes in particular. In the present work, we investigate in detail the case of one-point codes, obtaining a bound d * . This bound was already present in [1] (Proposition 37) but here we state it explicitly, by showing how to compute d * from the set H * defined above. Besides we investigate the connection to the order bound. We show that d * is a special case of the Beelen and Duursma-Kirov-Park generalized bouds. Since it can happen that the generalized order bounds give different results than the original one, we also investigate the connection of d * to the original order bound d ORD . We show that when both can be applied -namely when the dual of a onepoint code is isometric to a one-point code-then both coincide. Furthermore we investigate how to construct improved codes from d * and how to extend d * to all generalized Hamming weights. These problems have never been treated in the aforementioned works of Beelen and Duursma-Kirov-Park. Thus the main purpose of this article is not to present a new or better bound, but (i) to make the conection between the Andersen-Geil bound and the order bounds for AG one-point codes, (ii) to emphasize the possibility of manage the order bound entirely in the language of one-point evaluation codes and Weierstrass semigroups; (iii) to study how to construct improved codes; and (iv) to extend d * to all generalized Hamming weights.
The paper is structured in 5 sections: In Section 2 we briefly recall the bound for the minimum distance of linear codes from [1] as well as the main facts and definitions we need. We introduce the bound d * for one-point codes in Section 3, where we also show the connection with the generalized order bounds of Beelen and Duursma-Kirov-Park. We also deal with improved codes, whose construction becomes now very easy. Some worked examples where we show how to compute d * are included. In Section 4 we compare the bound d * to the strict order bound (that is the original order bound d ORD with respect to the evaluation map ev D ), showing that when both can be applied then they give the same result. Furthermore, we continue our study of improved codes. Finally in Section 5 we extend d * to all generalized Hamming weights.

The bound from [1] for the minimum distance of linear codes
For the convenience of the reader, we begin with a brief explanation of some results from [1]. Let B = {b 1 , . . . , b n } be a basis of F n q . We consider the codes C 0 = (0), and for i = 1, . . . , n, Associated to these codes we consider the (valuation-like) map ν : F n q → {0, . . . , n} defined by ν(v) = min{i : v ∈ C i }.
for all i = j, then equality holds.
For c ∈ F n q , c = 0, we consider the space V (c) = {v ∈ F n q : supp(v) ⊆ supp(c)} = {v * c : v ∈ F n q }, where the component-wise product is defined as usual: v * c = (v 1 c 1 , . . . , v n c n ). Clearly dim(V (c)) = wt(c), where wt(c) denotes the weight of c. Now consider in {1, . . . , n} 2 the order (r, s) Since we can write c = This bound can be applied to an arbitrary linear code C, just by including it into an increasing chain of codes C 1 ⊂ · · · ⊂ C k−1 ⊂ C ⊂ C k+1 ⊂ · · · ⊂ C n = F n q . Such a chain is quite natural for one-point codes.

3.
A bound for the minimum distance of one-point codes 3.1. The bound. Let X be a (projective, non-singular, geometrically irreducible algebraic) curve of genus g defined over the finite field F q . We construct one-point codes from X in the usual way. Let Q, P 1 , . . . , P n be different rational points in X . Let v = −v Q , where v Q is the valuation at Q, and consider the spaces L(mQ) and the algebra L(∞Q) = ∪ r=0,1,... L(rQ). Let D = P 1 +· · ·+P n and ev = ev D : L(∞Q) → F n q be the evaluation map at D. The one-point codes C(D, mQ) arising from X , D and Q are defined as the images of the sets L(mQ) by ev, that is C(D, mQ) = ev(L(mQ)). Note that C(D, (n + 2g − 1)Q) = F n q , hence we can restrict ourselves to 0 ≤ m ≤ n + 2g − 1. Let C = C(D, mQ). We shall apply to C the bound from Section 2 with respect to the sequence of codes C 1 ⊂ C 2 ⊂ · · · ⊂ C n , obtained from the sequence (C(D, mQ)) m=0,...,n+2g−1 by deleting the repeated codes. Thus the map ν can be written as From now on, unless explicitly said, we restrict ourselves to codes with length n > 2g + 2.
Note that it is not true in general that ν(ev(f )) = dim(C(D, v(f )Q)) because ev only depends on the points P 1 , . . . , P n , and thus ev(f ) might be equal to ev(g) with g ∈ C(D, (v(f )−1)Q). For example, take a non-constant function f ∈ L(∞Q).
Let H = H(Q) = {h 1 = 0 < h 2 < . . . } be the Weierstrass semigroup of Q. As we know, this is a numerical semigroup of finite genus g. Let l 1 , . . . , l g be the gaps of H. Let us consider the set H * defined in the Introduction, namely It is clear that H * consists of n elements. Let us write H * = {m 1 , . . . , m n }. It is also clear that H * ⊂ H and for m < n it holds that m ∈ H * if and only if m ∈ H. The following results may be useful for computing H * . Remember that for a divisor E, ℓ(E) stands for the dimension of L(E).
and only if both kernels are equal.
Thus, for m ≥ n, and since ℓ((n + 2g − 1)Q − D) = g and H has g gaps, we conclude that g elements of {n, . . . , n + 2g − 1} belong to H * while the other g elements do not. Proof. If D ∼ nQ then n ∈ H * and hence, according to Corollary 3.3, n = n + h 1 , . . . , n + h g ∈ H * . The statement follows by cardinality reasons.
. , ev(f n )} is a basis of F n q and the sequence of codes (C i ) is given by Our sequence (C(D, m i Q)) does not contain the code C 0 = (0). If we want to include it (see Section 4 for example) we simply take m 0 = −1 and C(D, m 0 Q) = (0).
Thus from the bound in Section 2 we get a bound for one-point codes as follows. For i = 1, . . . , n, consider the sets and thus m ∈ H * according to Corollary 3.3. Thus the sets Λ * i can also be written as Λ We call this inequality the d * bound for one-point codes. Let us remember that the classical bound on the minimum distance of an code is given by the Goppa estimate d(C(D, mQ)) ≥ d G (C(D, mQ)) := n − m. d * improves the Goppa bound as the next result shows (see also Proposition 37 in [1]). The first element in H \ H * is denoted by π = π(H). Note that π ≥ n.
Proof. For the first statement it suffices to show that #(H * \ Λ * r ) ≤ m r for all r. Since and this follows from the fact that #(H \ (m r + H)) = m r (see [10], Lemma 5.15). If m i + l g < π, then all elements in H \ (m i + H) are smaller than π and hence H * \ Λ 3.2. d * and the generalized order bounds of Beelen and Duursma-Kirov-Park. The bound d * can also be obtained from the generalized order bounds of Beelen and Duursma-Kirov-Park. Let us show first how to get d * from the Beelen generalized order bound d B stated in [2]. Let m i ∈ H * and consider the code C(D, m i Q). The Beelen bound applies to the duals of evaluation codes. Thus, let W be a canonical divisor with simple poles and residue 1 at all points P ∈ supp(D) and let G = D + W − m i Q. It is well known that C(D, m i Q) = C(D, G) ⊥ (see [14]). By using the notation as in [2], for r = 0, 1, 2, . . . , consider the divisors Note that all the divisors F (r) , F In our case, for all r = 0, 1, . . . , we have H(Q, F According to the Rieman-Roch theorem, for an integer m it holds that 1 Let us show briefly how to obtain d * from the generalized order bound of Duursma, Kirov and Park. Consider again the code C(D, m i Q). In the formulation of [4,5,6] On the other hand, the choice of the sets Λ * i (instead of the counting made in the Beelen and Duursma-Kirov-Park bounds) has some technical advantages. Firstly it does not involve more divisors that the ones naturally associated to the code C(D, mQ). And secondly, in contrast to what happens with those bounds, d * allows us to study improved codes very easily. Also it allows us to extend the same idea to all generalized Hamming weights (see Section 5). In fact, for these two problems d * works even better than the original order bound d ORD . As discussed in Section 4, d * extends exactly d ORD to onepoint codes.
3.3. Improved codes. Let δ be an integer, 0 < δ ≤ n. In the same way as the order bound allows us to construct codes with designed minimum distance δ and dimension as large as possible, see [10], the bound d * shows how to construct similar codes from sequences (C(D, m i Q)), see [1]. Specifically, given δ let us consider the improved code , and the discussion before Theorem 3.6, it is clear that the minimum distance of C(D, Q, δ) is at least δ.
The sequence (Λ * i ) is said to be monotone for δ if for every i, j such that #Λ * i ≥ δ and #Λ * j < δ we have that i < j. If (Λ * i ) is monotone for δ it is clear that C(D, Q, δ) is a usual one-point code, so improved codes only improve one-point codes for those δ for which the sequence is not monotone. In this case the code C(D, Q, δ) depends on the choice of the set {f 1 , . . . , f n }. In fact, if #Λ * i = δ and #Λ * j < δ for some j < i, then v(f i + f j ) = v(f i ) but in general ev(f j ) ∈ C(D, Q, δ), hence ev(f i + f j ) ∈ C(D, Q, δ). Thus we have a collection of improved codes with designed distance δ, depending on the collection of sets {f 1 , . . . , f n }.

Worked examples.
We compute H * for some examples.
Example 3.8. (Codes on Castle curves) A curve X defined over F q is said to be Castle if there is a rational point Q such that the Weierstrass semigroup at Q, H = H(Q), is symmetric and qh 2 +1 = #X (F q ) (where h 2 is the first nonzero element of H). If D is the sum of all rational points of X except Q, the codes C(D, mQ) are called Castle codes, see [13]. It is simple to see that for Castle curves we have D ∼ nQ, hence H * ∩{n, . . . , n+2g −1} = {n + l 1 , . . . , n + l g } according to Proposition 3.4. In Section 4 we shall see that, being the semigroup H symmetric, we have H * = H \ (n + H). Recall that the family of Castle codes includes Hermitian, generalized Hermitian, Norm-trace, Suzuki, Ree and many of the most known codes. To study a concrete example, let us consider the Suzuki curve X over F 8 (see [13] again). This curve has genus g = 14 and 65 rational points. A plane model of X is given by the equation . This model is nonsingular except at the point (0 : 1 : 0). Being this singularity uni-branched, the unique point Q lying over (0 : 1 : 0) is rational. Let us consider the codes C(D, mQ), where D is the sum of all rational points of X except Q.  14,13,14,10,14,8,13,10,10,9,9,6,9,8,4,6,5,5,4, 6, 5, 3, 2, 3, 3, 2, 1, 1). This sequence is monotone for δ = 3, 5, 6, 9, 13, 14, 18, 20, 21. For example the code C(D, 70Q) has dimension 55 and distance at least 4 (that is d * (55) = 4), whereas C(D, Q, 4) has dimension 57.
Let us study the rational points of X . Firstly there is just one point Q over x = ∞.
Note that 2g − 1 = 97 ∈ H and so H is not symmetric. In order to construct codes from this curve let us consider the divisors D ′ = R 1 + R 2 , and for α ∈ F 16 , α = 0, 1 According Once H * is known we can compute the dimensions of all codes C(D, mQ) and apply Theorem 3.6 to estimate the minimum distances. Note that for large m we do not obtain good parameters. In fact, as D ′′ α ∼ 15Q, for all m < n, m multiple of 15, the true minimum distance of C(D, mQ) equals the Goppa estimate. In particular the minimum distance distance of C(D, 210Q) is d = 2. The bound d * gives d ≥ 2 for m = 224 (that is, for dimension k ≤ 175) and hence all codes C(D, mQ), m = 210, . . . , 224 have true minimum distance d = 2.
In order to obtain codes with better parameters (that is, better minimum distance) the usual approach is to consider another divisor G. We shall show that this goal can also be accomplished by taking a slightly different D. Consider the codes C(D ′′ , mQ) of length n ′′ = 210. Then the function from which the codeword of weight 2 arises belongs to the kernel of the evaluation map. The set H * = H * (D ′′ , Q) can be now computed by using Corollary 3.4, and H * ∩{n ′′ , . . . , n ′′ +2g−1} = {n ′′ +l 1 , . . . , n ′′ +l g }, where l 1 , . . . , l g are the 49 gaps of H. It is not necessary to apply the bound d * to see that the minimum distance of these codes is larger for m ≥ n ′′ . For example, from the improved Goppa bound we know that the minimum distance of C(D ′′ , 210Q) satisfies d ≥ n ′′ − 210 + γ 2 = γ 2 , where γ 2 is the usual gonality of X , see [11] . It is not easy to compute γ 2 , but at the first sight we have γ 2 ≥ #X (F 16 )/#P 1 (F 16 ), hence γ 2 ≥ 13 (so γ 2 = 13 or 14) and d ≥ 13 as well. 4. Relating the bounds d * and d ORD As we noted above, in some cases the generalized order bounds may give different results than the original order bound, see [2] Example 8. Likewise, also the Andersen-Geil bound, from which we have obtained d * , can be very different from the original order bound, see Example 51 of [1]. In this Section we shall compare d * and the original order bound d ORD . This comparison can be done over sequences of one-point codes such that their duals are also one-point. We can slightly relax this condition by imposing that the duals are isometric to one-point codes.

4.1.
The isometry-dual condition. Let C, D, be two linear codes in F n q and let x ∈ (F * q ) n be an n-tuple of non-zero elements. We say that C and D are isometric according to x (or simply x-isometric) if the map χ x : F n q → F n q given by χ x (v) = x * v satisfies χ x (C) = D. Note that χ x is a true linear isometry for the Hamming distance, hence isometric codes have the same parameters. The dual of a code C is denoted by C ⊥ . Proof. Let c ∈ C and d = χ Let us recall that we have fixed a basis B = {b 1 , . . . , b n } of F n q and the associated codes Definition 4.2. A sequence of codes (C i ) i=0,...,n is said to satisfy the isometry-dual condition if there exists x ∈ (F * q ) n such that C i is x-isometric to C ⊥ n−i for all i = 0, 1, . . . , n.
Let us study the case of AG codes. We consider the sequence of codes (C(D, m i Q)) i=0,...,n arising from the curve X and the associated set H * = {m 1 , . . . , m n }. In addition let m 0 = −1 and C(D, m 0 Q) = (0). If (C(D, m i Q)) satisfies the isometry-dual condition then both d * and the order bound d ORD can be used to estimate the minimum distance of these codes. Let us remember that we are assuming that n > 2g + 2. Remember also that the dual of C(D, mQ) is C(D, D + W − mQ), where W is a canonical divisor with simple poles and residue 1 at every point in supp(D) (see [14]). (a) The sequence (C(D, m i Q)) i=0,...,n satisfies the isometry-dual condition.
Proof. Let us consider the divisor E = (n + 2g − 2)Q − D and for an integer m write m ⊥ = n + 2g − 2 − m. ((a)⇔(b)) Assume that the sequence (C(D, m i Q)) i=0,...,n satisfies the isometry-dual condition. Let m be such that 2g ≤ m ≤ n − 2 (since n > 2g + 2, such an m does exist). Then 2g ≤ m ⊥ ≤ n − 2 and hence m, m ⊥ ∈ H * . In particular dim(C(D, mQ)) + dim(C(D, m ⊥ Q)) = n. Since the sequence (C(D, m i Q)) i=0,...,n satisfies the isometry-dual condition we have that C(D, D + W − mQ) = C(D, mQ) ⊥ is isometric to C(D, m ⊥ Q). This implies that the divisors D + W − mQ and m ⊥ Q are equivalent (see [12]). Then W ∼ (m + m ⊥ )Q − D = E and this divisor is canonical. Conversely, if E is a canonical divisor then there is a rational function f such that E + div(f ) = W .
In particular f has neither poles nor zeros in supp(D). Let x = ev D (f ). Then we have  Over the field F 8 , X has 24 rational points (the maximum allowed by Weil-Serre bound) and a rich geometrical structure. Codes coming from this curve are usually constructed by using the divisors G = m(Q 1 + Q 2 + Q 3 ), where Q 1 = (1 : 0 : 0), Q 2 = (0 : 1 : 0) and Q 3 = (0 : 0 : 1), since this choice has some technical advantages (see [3], [8], [10]). However, one-point codes over X can also be considered. Let Q = Q 2 , D ′ = Q 1 + Q 3 , D ′′ = P 1 + · · · + P 21 be the sum of all rational points except Q 1 , Q 2 , Q 3 and let D = D ′ + D ′′ . It is easy to see that div(x) = 3Q 3 − 2Q 2 − Q 1 and div(y) = 2Q 1 + Q 3 − 3Q 2 . Then div(xy) = Q 1 + 4Q 3 − 5Q 2 and div(x 2 y) = 7Q 3 − 7Q 2 . Then the Weierstrass semigroup H = H(Q) is generated by 3,5 and 7. In particular {1, y, xy, y 2 , x 2 y, . . . } is a basis of L(∞Q). In order to compute H * = H * (D, Q) we can proceed as in Example 3.9. By considering the morphism φ = y, φ : X → P 1 (F 8 ) of degree 3, we observe that D ′′ ∼ 21Q. This fact leads us to consider the codes C(D ′ , mQ) of length 2 and the set H * (D ′ , Q). Since x 2 y is the first non constant function in the above basis for which Q 1 is not a zero, we deduce that H   Thus for isometry-dual sequences the set H * is symmetric in the sense that for an integer m it holds that m ∈ H * if and only if n + 2g − 1 − m ∈ H * (and conversely this property implies the isometry-dual condition). It follows that n + 2g − 1 − m i = m n−i+1 . We must not confuse this kind of symmetry with the symmetry of the semigroup H. Let us remember that a semigroup H of genus g is called symmetric if 2g − 1 ∈ H or equivalently (since its largest gap l g satisfies l g ≤ 2g − 1) if l g = 2g − 1. For symmetric semigroups it holds that m ∈ H if and only if l g − m ∈ H, see [10]. When the Weierstrass semigroup H = H(Q) is symmetric, (2g−2)Q is a canonical divisor, hence the isometry-dual property is equivalent to D ∼ nQ. Since in this case the condition n + 2g − 1 − m ∈ H is equivalent to m − n ∈ H, or m ∈ n + H, then the set H * is given by Let us return to the general case of H, where it might not be symmetric. The symmetrical description of H * given by Lemma 4.7 allows us to write H * in the following way Proof. We have l 1 , . . . , l g ∈ H * . In the same way, if l is a gap of H then n + 2g − 1 − (n + 2g − 1 − l) = l ∈ H and hence n + 2g − 1 − l ∈ H * . Furthermore, since l g < n, then l g < n + 2g − 1 − l g and hence #{l 1 , . . . , l g , n + 2g − 1 − l g , . . . , n + 2g − 1 − l 1 } = 2g. By cardinality reasons we get the result.
Proposition 4.9. If (C(D, m i Q)) satisfies the isometry-dual condition, then #Λ * Proof. Let L = {l 1 , . . . , l g , n + 2g − 1 − l g , . . . , n + 2g − 1 − l 1 }, and for i = 1, . . . , n, i . Since H * ⊆ H and the sum of two non-gaps is again a non-gap, we have B Then d * can be written for isometry-dual codes as Let us prove now that d * and the strict order bound with respect to the evaluation map ev D , d ORD,ev ([10], Section 4.3), give the same result when applied to codes satisfying the isometry-dual condition. Let m i ∈ H * and let us compute both bounds for C(D, m i Q). If m i < n − l g , according to Proposition 3.7 and Theorem 4.7 in [10], both bounds are equal to Goppa bound.
In order to compute the order bound, we first need the duals of the codes C(D, m r Q). As we know, C(D, m r Q) ⊥ is isometric to C(D, (n + 2g − 2 − m r )Q). Let h s , h s+1 ∈ H be such that h s ≤ n + 2g − 2 − m r < h s+1 . Then C(D, h s Q) = C(D, (n + 2g − 2 − m r )Q) and hence C(D, h s Q) ⊥ is isometric to C(D, m r Q). Note that C(D, m r Q) has dimension r, so C(D, h s Q) has dimension n−r. Furthermore, Lemma 4.7 implies that n+2g−1−m r ∈ H * hence h s+1 = n + 2g − 1 − m r = m n−r+1 and dim C(D, h s+1 Q) = n − r + 1.
For h ∈ H let us consider the set The strict order bound on the minimum distance of C(D, m i Q) together with our previous discussion, imply that where the last two equalities follow from 4.7 and the fact that m n−r+1 = n + 2g − 1 − m r .    It is well known that the minimum distance ofC(D, Q, δ) is at least δ. When the sequence (C(D, m i Q)) is isometry-dual, Proposition 4.11 allows us to writeC(D, Q, δ) in terms of the sets Λ * i 's,C (D, Q, δ) = {ev(f i ) : #Λ * n+1−i ≥ δ} ⊥ . Then it is natural to wonder about the relation between these two improved codes C(D, Q, δ) and C(D, Q, δ). Proof. If C(D, Q, δ) is generated by t vectors thenC(D, Q, δ) is defined by n − t independent parity checks.
If the sequence (Λ * i ) is monotone for δ then C(D, Q, δ) is a one-point code, hence C(D, Q, δ) andC(D, Q, δ) are isometric. Let us study the general case.

Generalized Hamming weights
The same ideas used to obtain the bound d * for the minimum distance can be applied to all generalized Hamming weights (see [1]). Let us remember that given a set D ⊆ F n q , the support of D is defined as supp(D) = v∈D supp(v).
Let C be a code of dimension k. For r = 1, . . . , k, the r-th generalized Hamming weight of C is defined as Proof. Given D, let us consider the space V (D) = {v ∈ F n q : supp(v) ⊆ supp(D)}. Since #supp(D) = dim(V (D)) and supp(D) = supp(c 1 ) ∪ · · · ∪ supp(c r ), we have that V (D) = V (c 1 ) + · · · + V (c r ) and the statement follows from the results in Section 2. Proof. According to Lemma 2.1 (c), every linear subspace D of C i has a basis {c 1 , . . . , c r } such that 1 ≤ ν(c 1 ) < · · · < ν(c r ) ≤ i. Conversely, given vectors {c 1 , . . . , c r } satisfying the above condition, c 1 , . . . , c r is a vector subspace of C i of dimension r. Then the result is a consequence of Lemma 5.1.
This result is easily translated to one-point AG codes. With the notation as in Section 3, we have codes C(D, mQ) and C i = C(D, m i Q). We showed that #{ν(b j * b t ) : t ∈ Λ j }) ≥ #Λ * j . Thus we have This result is similar to the corresponding one for the order bound in [9]. Also similar results to the ones contained in this section can be obtained for improved codes as well.