on the minimum distance of castle codes

Finite Fields and Their Applications 20 (2013) 55–63

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Finite Fields and Their Applications

www.elsevier.com/locate/ffa

On the minimum distance of Castle codes

Wilson Olaya-León a,1, Carlos Munuera b,∗,2

a Escuela de Matemáticas, Universidad Industrial de Santander, AA 678 Bucaramanga, Colombiab Dept. of Applied Mathematics, University of Valladolid, Avda. Salamanca SN, 47014 Valladolid, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 June 2012Revised 21 November 2012Accepted 7 December 2012Available online 13 December 2012Communicated by W. Cary Huffman

MSC:94B2794B65

Keywords:Error-correcting codesAG codesCastle curvesCastle codesMinimum distanceOrder bound

Castle codes are algebraic geometry one-point codes on Castlecurves. This family contains some of the most important AG codesamong those studied in the literature to date. The minimumdistance of these codes can be bounded by using the order-likebound d∗, which is known to be equivalent to the classical orderbound when both can be applied. In this paper we computed∗ for some Castle codes, including those related to semigroupsgenerated by two elements and telescopic semigroups. In particularwe compute the bound d∗ in full for Suzuki codes.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

Let X be an algebraic, absolutely irreducible nonsingular curve of genus g defined over the finitefield Fq . Let P1, P2, . . . , Pn be n rational distinct points of X , set D = P1 + P2 + · · ·+ Pn and considera rational divisor G on X whose support is disjoint from D . Associated to the triple (X , D, G) we canconsider the algebraic geometry (AG) code C(X , D, G) (or simply C(D, G)), which is the image of the

* Corresponding author.E-mail addresses: [email protected] (W. Olaya-León), [email protected] (C. Munuera).

1 This work was written in part during a visit of the first author to the Dept. of Algebra, University of Valladolid, partiallyfunded by Foundation Carolina, Spain.

2 This work was supported in part by Junta de Castilla y León under grant VA065A07 and by Spanish Ministry for Scienceand Technology under grants MTM2007-66842-C02-01 and MTM 2007-64704.

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56 W. Olaya-León, C. Munuera / Finite Fields and Their Applications 20 (2013) 55–63

evaluation map

ev : L(G) → Fnq, ev( f ) = (

f (P1), . . . , f (Pn)),

where L(G) is the Riemann–Roch space of rational functions f such that f = 0 or ( f ) + G � 0. Oftenwe take G as a multiple of a single point, G = mQ with Q �= Pi for all i. In this case the code is calledone-point. Computing the parameters of C(D, G) is in general a difficult problem. With respect to itsminimum distance, a special attention deserves the so-called order bound. It was first introduced byFeng and Rao in a language close to that of affine variety codes. Høholdt, van Lint and Pellikaan,see [8], introduced order domains to facilitate the Feng–Rao bound. After this formulation the boundis also called ‘order bound’. One important example of order domains is

⋃∞m=0 L(mQ ), where Q is a

rational point on a curve X . Recall that both, Feng–Rao and order bounds apply to duals of one-pointAG codes (and not to one-point codes themselves). Miura independently developed many of the sameideas for duals of one-point codes [16]. He also formulated the Feng–Rao bound for any linear codedefined by means of its parity check matrix [17,15,18].

The order bound has been generalized to all AG codes by Beelen [2] and by Duursma, Kirov andPark [3,4]. These generalizations are heavily based on algebraic geometry methods. Another gener-alization to all linear codes described by means of generator matrices, was given by Andersen andGeil [1]. This paper primarily treats linear codes, but also the cases of codes from order domains,one-point AG codes in particular, and affine variety codes are studied in detail. The particular appli-cation of this bound to one-point codes was also studied by Geil, Munuera, Ruano and Torres in [5],where the authors used the name d∗ for this case. They showed that when both d∗ and the Høholdt–van Lint–Pellikaan order bound can be applied then both give the same result (which is not truefor the Beelen and Duursma–Kirov–Park bounds). Here we recall again that the original order boundapplies to the duals of one-point codes, while d∗ applies to one-point codes themselves.

On the other hand, recently Lee, Bras-Amoros and O’Sullivan presented a unique decoding algo-rithm for one-point AG codes coming from plane Miura–Kamiya curves Cab [10]. They show that thisalgorithm decodes up to one half of a new bound dLBS . Geil, Matsumoto and Ruano [12,13] extendedthe above algorithm to work by performing list-decoding on any one-point AG code C(X , D,mQ )

whenever D ∼ nQ , where n is the code length. They also showed that dLBS is the same bound as d∗ .As we shall see later, the condition D ∼ nQ is very close to imposing that X is a Castle curve.

The original order bound is often hard to compute and the above generalizations are sometimesmuch harder, so that they have been determined in very few examples. In this paper we study theproblem of computing d∗ for one-point codes C(X , D,mQ ) coming from Castle curves. These curveswill be defined in Section 2.1. In particular we shall show that when H(Q ) is generated by twoelements, then d∗ can be obtained in a very simple way. This result is analogous to one obtainedby Høholdt, van Lint and Pellikaan for the classical order bound (Theorem 5.30 of [8]). To illustratethese results, we study the case of Hermitian codes, obtaining a characterization of the minimumdistance easier than any obtained to date (Example 3). The case of telescopic semigroups is alsoconsidered, obtaining similar but incomplete results. We pay particular attention to the case of Castlecodes coming from Suzuki curves, for which d∗ is computed in full.

2. Some preliminaries

In this section we briefly recall some basic notations and facts we shall use throughout this work.For all facts concerning algebraic geometry and AG codes we refer to [20].

2.1. Castle curves and codes

Castle curves and codes were introduced in [19]. Let X be an algebraic curve of genus g definedover Fq and let Q be a rational point on X . The Weierstrass semigroup of Q is defined as

H = H(Q ) ={−νQ ( f ): f ∈

⋃m∈N

L(mQ )

}= {0 = ρ1 < ρ2 < · · ·},

0

W. Olaya-León, C. Munuera / Finite Fields and Their Applications 20 (2013) 55–63 57

where N0 is the set on nonnegative integers and νQ is the valuation at Q . H is said to be symmetricif it has conductor c = 2g . Lewittes [11] and later Geil and Matsumoto [6], used H to give an upperbound on the number of rational points on X , as

#X (Fq) − 1 � #(

H \ (qH+ + H

))� qρ2,

where qH+ + H = {qα + β: α,β ∈ H, α �= 0}. The curve X is called Castle if there is a rational pointQ for which: (i) H(Q ) is symmetric, and (ii) we get equality in the Lewittes–Geil–Matsumoto bound,#X (Fq) − 1 = qρ2. Many well known curves are Castle, including rational, Hermitian, generalizedHermitian, Norm–Trace, Suzuki, Ree, and many others.

A Castle code is an one-point AG code C(X , D,mQ ) coming from a Castle curve X , where thedivisor D is the sum of all n = qρ2 rational points on X different from Q . For the properties of thesecodes we refer to [19]. In particular, in that article it is shown that D ∼ nQ , which is the conditionrequired in [12,13] to work.

2.2. The bound d∗

The bound d∗ on the minimum distance of one-point AG code was introduced in [1] and [5]. Con-sider a code C = C(D,mQ ), where D = P1 + · · · + Pn , and let H be the Weierstrass semigroup of Q .Associated to C we can consider the set H∗ = H∗(D, Q ) = {m ∈ N0: C(D,mQ ) �= C(D, (m − 1)Q )}.It is clear that H∗ has n elements, H∗ = {m1,m2, . . . ,mn} ⊂ H , and dim(C(D,mi Q )) = i. For Castlecodes we have

H∗ = H \ (n + H) = (H \ (n +N0)

) ∪ (n + Gaps(H)

)(1)

where Gaps(H) = {l1, . . . , lg} is the set of gaps of H . For i = 1, . . . ,n, let us consider the sets

Λ∗i = {

m ∈ H∗: m − mi ∈ H} = (mi + H) ∩ H∗.

The bound d∗ on the minimum distance of Ci = C(D,mi Q ) is defined as

d∗i = min

{#Λ∗

t : t � i}

and it holds that d(Ci) � d∗i . It should be noted that d∗ is always at least as good as the Goppa bound

dG(C(D,mi Q )) = n − mi , that is d∗i � dG(C(D,mi Q )).

3. Computing d∗ for Castle codes

Let (X , Q ) be a Castle curve of genus g over Fq . Write X (Fq) = {Q , P1, P2, . . . , Pn}. For simplicity,from now on we shall assume n > c = 2g . Keeping the same notations as in the previous section, letD = P1 + P2 +· · ·+ Pn , H = {ρ1,ρ2, . . .} be the Weierstrass semigroup at Q , Gaps(H) = {l1, . . . , lg} theset of gaps of H and H∗ = {m1, . . . ,mn} as in (1). An element of H will be called a pole number. Notethat ρi � i + g − 1 with equality when i > g . Furthermore mi = ρi when i � n − g and mi = n + li+g−nwhen i > n − g . Our purpose is to compute the bound d∗

i on the minimum distance of Castle codesCi = C(D,mi Q ).

Proposition 3.1. For mi � n − c, we have d∗i = n − mi.

Proof. If mi + c � n then the conductor of the set mi + H is at most n. By using the expression inEq. (1) we have


Λ∗i = (mi + H) ∩ [(

H \ (n +N0)) ∪ (

n + Gaps(H))]

= [(mi + H) ∩ (

H \ (n +N0))] ∪ (

n + Gaps(H)).

Since (mi + H) ∩ (H \ (n + N0)) = (mi + H) \ (n + N0) and this set has cardinality n − mi − g , weconclude that #Λ∗

i = n − mi , hence d∗i = n − mi . �

Thus, for mi � n − c, the bound d∗ is the same as the Goppa bound, dG(C(D,mi Q )). In whatfollows we shall restrict to the case mi > n − c. For 0 < w < c we shall consider the set D(w) ={(l,m): l ∈ Gaps(H), m ∈ H∗, l − m = w}. The next lemma states, without proof, some elementaryproperties of D(w).

Lemma 3.2. The set D(w) verifies the following properties.

(a) (l,m) ∈ D(w) if and only if (c − 1 − m, c − 1 − l) ∈ D(w).(b) If w ∈ H then D(w) = ∅.(c) #D(c − 1) = 1. If w ∈ Gaps(H) and w �= c − 1 then #D(w) � 2.

Proposition 3.3. For mi > n − c, we have

#Λ∗i =

{n − mi + #D(n − mi) if mi < n,

#D(mi − n) if mi > n.

Proof. Assume n − c < mi < n. Let A := {m ∈ (mi + H) ∩ H∗: m < n}, B := {m ∈ (mi + H) ∩ H∗: n <

m < mi + c} and C := {m ∈ (mi + H)∩ H∗: m � mi + c}, so that Λ∗i = A ∪ B ∪ C . Note that m ∈ A if and

only if m = mi + ρ < n for some ρ ∈ H , or equivalently if ρ < n − mi . Then #A = ωi , where ωi is thenumber of elements of H smaller than n −mi . Analogously, m ∈ C if and only if m = n + l � mi + c forsome l ∈ Gaps(H), iff c − l � n − mi . So, by symmetry, #C = ωi . Then, to prove the first statement itsuffices to show that #B = n−mi −2ωi +#D(n−mi). Note that m ∈ B if and only if m = mi +ρ = n+ lfor some ρ ∈ H with n − mi < ρ < c and l ∈ Gaps(H). This happens iff ρ + c − 1 − l = c − 1 + n − mi .Write this equality as ρ + ρ ′ = ρg+n−mi , where ρ ′ = c − 1 − l ∈ H by the symmetry of H . Accordingto [8, Theorem 5.24] the number of solutions of the equation ρ(1) +ρ(2) = ρg+n−mi with ρ(1), ρ(2) ∈ His n − mi + #D̃(g + n − mi − 1), where D̃(t) = {(x, y): x, y ∈ Gaps(H), x + y = ρt+1}. Since (x, y) ∈D̃(g +n −mi −1) iff (x, c −1− y) ∈ D(n −mi), we have #D̃(g +n −mi −1) = #D(n −mi). There are ωisolutions of ρ(1) + ρ(2) = ρg+n−mi such that ρ(1) < n − mi and other ωi solutions such that ρ(1) � c,so a simple computation proves our claim on #B and thus the first statement of the proposition. Toprove the second statement note that when mi > n, then m ∈ Λ∗

i if and only if m = mi + ρ = n + l, iff(l,ρ) ∈ D(mi − n). �

Then, for n − c < mi < n, the numbers #D(n − m j), i � j � g , determine the improvement of d∗i on

the Goppa bound. In particular we have the following.

Corollary 3.4. If n − mi ∈ H, then #Λ∗i = n − mi and hence d∗

i = n − mi = dG(Ci).

So we can have an improvement on the Goppa bound only when n − mi is a gap of H . Themaximum possible value of this improvement is also obtained as a consequence of Proposition 3.3and Corollary 3.4.

Corollary 3.5. If mi < n then d∗i � min{ρ ∈ H: ρ � n − mi}.

In general, equality does not hold in the above corollary when n − mi is a gap of H , as the nextexample shows.


Example 1. Let us consider the Suzuki curve S of affine equation y8 − y = x10 − x3 over F8, see [7].S has 64 rational affine points plus one point Q at infinity, whose Weierstrass semigroup is H =〈8,10,12,13〉. Then it is a Castle curve. After some computation we have m49 = 62, m50 = 63, henceCorollary 3.5 gives d∗

49 � 8 and d∗50 � 8. A direct computation shows that d∗

49 = d∗50 = 6.

Remark 1. If mi < n then the minimum distance of Ci coincides with the Goppa bound if and onlyif there exists a divisor D ′ ∼ mi Q with 0 � D ′ � D . This can only happen if mi is a pole number(otherwise Q is a base point). Since X is a Castle curve, the existence of such divisor implies D ′′ =D − D ′ ∼ (n −mi)Q , and then the minimum distance of C(D, (n −mi)Q ) is also equal to the estimategiven by the Goppa bound. In conclusion, d(Ci) = dG(Ci) can only happen when both mi and n − mi

are pole numbers. Otherwise d(Ci) > n − mi = dG(Ci).

Example 2 (Codes on the Suzuki curve, continued). Let us consider the Suzuki codes C(D,62Q ),C(D,63Q ) of Example 1. By taking D ′ = div∞(x) ∼ 8Q in the previous remark, we find d(C(D,

56Q )) = d(C43) = d∗(C43) = 8. On the other hand, by using the property that d(C(D,nQ )) � ρ2 (forthis and the previous fact, see [19, Section 2]), we get d(C(D,64Q )) � 8. As the minimum distance ofC(D,mQ ) cannot increase when m does, we conclude that d(C49) = d(C50) = 8. This example showsthe surprising fact that the formula of Corollary 3.5 can provide the true minimum distance of a code,while not coincide with the bound d∗ .

For some particular types of semigroups we have equality in Corollary 3.5, as we shall show later.In what follows we shall restrict to two of these types: semigroups generated by two elements andtelescopic semigroups.

3.1. Semigroups generated by two elements

In this subsection we shall assume that H is the semigroup generated by two elements, H = 〈a,b〉with a < b and gcd(a,b) = 1. We shall also assume n � 2c − a.

Lemma 3.6. Assume that H = 〈a,b〉 and n − c < mi < n. If n − mi = l ∈ Gaps(H), let s be such that ρs−1 <

l < ρs . Then #Λ∗i � ρs .

Proof. It suffices to show #D(l) � ρs − l. We have (lk,m j) ∈ D(l) if and only if lk − m j = l if andonly if m j + c − 1 − lk = c − 1 − l. Let m = c − 1 − l ∈ H∗ . According to [8, Lemma 5.27], there existsa gap in the interval [m − #D(l),m]. Since {l, l + 1, . . . , ρs − 1} ⊂ Gaps(H), by symmetry we have{c − ρs, . . . , c − 1 − l = m} ⊂ H . Then m − #D(l) < c − ρs and hence #D(l) � ρs − l. �Theorem 3.7. If H = 〈a,b〉 is the semigroup generated by two elements, then for mi < n we have d∗

i =min{ρt ∈ H: ρt � n − mi}.

Proof. If n − mi ∈ H then the result follows from Corollary 3.4. Assume n − mi ∈ Gaps(H). If n − mi =c − 1 then, according to item (c) of Lemma 3.2, we have #D(lg) = 1, hence Λ∗

i = c and d∗i = c. If,

otherwise, n − mi = lt �= c − 1, write ρs−1 < n − mi < ρs with a � ρs � c − a by symmetry of H . Thenm j = n − ρs ∈ H∗ , so #Λ∗

j = ρs and Lemma 3.6 concludes the proof. �Let us study now the case mi > n. For t = 1,2, . . . ,a − 1, consider the numbers λt = tb − a. It is

simple to see that all of them are gaps of H . In addition let λ0 = 0.

Lemma 3.8. Assume H = 〈a,b〉. If mi = n + l where l is a gap of H such that λt � l < λt+1 then #D(l) �#D(λt) = a − t.


Proof. If l = λt , then (lk,m j) ∈ D(λt) if and only if lk − m j = λt that is iff m j + c − 1 − lk = c − 1 − λt .Since c − 1 − λt = (a − 1 − t)b, according to [8, Lemma 5.27], there exist a − t pairs satisfying thisequality. Now the proof of the case λt < l < λt+1 is similar to the proof of Proposition 5.28 of [8]. �

As a direct consequence of Lemma 3.8, the following result gives the value of d∗i when mi > n.

Note that in this case we have mi = n + li−n+g .

Theorem 3.9. Assume that H = 〈a,b〉 is the semigroup generated by two elements and let mi > n. If λt �li−n+g < λt+1 then d∗

i = a − t.

Theorems 3.7 and 3.9 together allow the computation of d∗i for all codes Ci . Let us see a well

known example.

Example 3 (Hermitian codes). Let us consider the Hermitian curve H: yq + y = xq+1 of genus g =q(q−1)

2 over Fq2 . It has n = q3 rational affine points plus one point Q at infinity, whose Weierstrasssemigroup is H = 〈q,q + 1〉 (and thus it is a Castle curve). Hermitian codes have been studied bymany authors. In particular, their minimum distances were calculated by Yang and Kumar [21]. Wecan apply the results of this subsection to these codes. To that end, we define a desert of H as amaximal set of consecutive gaps. Let us denote by L1, . . . , Lq−1 the deserts of H . According to theprevious results we have

d∗i =

⎧⎨⎩

n − mi if i � n − g and n − mi ∈ H;

qt if i � n − g and n − mi ∈ Lt ;

q − t if i > n − g and mi − n ∈ Lt .

We refer the reader to [21] to verify that this bound in fact gives the true minimum distance ofHermitian codes. Note that this characterization of the minimum distance is simpler than any of theones existing in the literature. Observe also that we get a [64,53,8] code over F16 which is a newrecord according to MinT tables [14].

3.2. Telescopic semigroups

In this subsection we shall study the case in which H is the semigroup generated by a telescopicsequence (a1,a2, . . . ,ak), see e.g. [8, Definition 5.31]. For i = 1, . . . ,k, let δi = gcd(a1, . . . ,ai). We shallrestrict to the case in which δk−1 > 1, δk = 1 and ak = max(a1/δ1, . . . ,ak/δk). Furthermore, as in theprevious subsection, we assume n � 2c − ρ2, where c is the conductor of H . Note that semigroupsgenerated by two elements are telescopic. Then most proofs in this subsection are similar to thecorresponding ones in the previous subsection and will be omitted.

Lemma 3.10. Assume that H is a telescopic semigroup and n−c < mi � n−(δk−1 −1)ak. If n−mi ∈ Gaps(H),let s be such that ρs−1 < n − mi < ρs . Then #Λ∗

i � ρs .

The proof is similar to the proof of Lemma 3.6, by using Lemma 6.9 of [9]. As a direct consequenceof this lemma we have the following.

Theorem 3.11. Assume that H is a telescopic semigroup. If mi � n − (δk−1 − 1)ak then

d∗i = min{ρt ∈ H: ρt � n − mi}.

For t = 0, . . . , δk−1 − 1, consider the numbers λt = c − 1 − tak . All of them are gaps of H . The proofof the following result is similar to the proof of Lemma 3.8, again by using Lemma 6.9 of [9].


Lemma 3.12. Assume that H is a telescopic semigroup.

(a) #D(λt) = t + 1 for all t = 0, . . . , δk−1 − 1.(b) If mi = n + l where l is a gap verifying λs < l < λs−1 , then #D(l) � #D(λs) = s + 1.

Theorem 3.13. Assume that H is a telescopic semigroup and let mi � n + λδk−1−1 . Write mi = n + li−n+g . Ifλs � li−n+g < λs−1 , then d∗

i = s + 1.

Example 4. In Examples 1 and 2 we have considered the Suzuki curve S over F8. The Weierstrasssemigroup of its unique point Q at infinity is H = 〈8,12,10,13〉. This semigroup is telescopic ofgenus g = 14 and conductor c = 28. Then d∗

i = dG(Ci) = n − mi for mi � n − c = 36. By Example 2, wehave also d(Ci) = dG(Ci) when mi < n and mi is a multiple of 8. The same example shows also thatd(Ci) = 8 for 56 � mi < 64. Finally, the results of this subsection give the bound d∗ for all remaindervalues of mi except 53, 55, 65, 66, 67, 68, 69, 70, 71, 73 and 75 (that is, 53 up to 64 values). Amongthese codes we find three records, according to the MinT tables [14]: a [64,37,� 16], a [64,50,8]and a [64,58,� 4] codes.

The values of mi in the range n − (δk−1 − 1)ak < mi < n + lg − (δk−1 − 1)ak are not covered by theprevious results. In the next subsection we shall study the case of semigroups coming from Suzukicurves in more detail.

3.3. Suzuki codes

Let q = 2q20, where q0 � 2 is a power of 2. The Suzuki curve Sq is defined over Fq by the equation

yq − y = xq0(xq − x

).

This curve has genus g = q0(q − 1) and q2 rational affine points, plus one point Q at infinity see [19].The Weierstrass semigroup H = H(Q ) is generated by the telescopic sequence a1 = q, a2 = q + 2q0,a3 = q + q0, a4 = q + 2q0 + 1. Then δ3 = q0. Furthermore, the following facts hold, see [9, Lemmas 6.4and 6.5].

1. c − 1 = δ3 + (δ3 − 1)(a2 + a4).2. If ρ ∈ H then there exist uniquely determined nonnegative integers x1 � 0, 0 � x2 < δ3, 0 � x3 < 2

and 0 � x4 < δ3 such that ρ has a normal representation ρ = x1a1 + x2a2 + x3a3 + x4a4.

We shall compute d∗ for all values of mi not covered by the previous results. We distinguish twotypes of deserts of H : Those of length smaller than δ3 = q0 are called short, while the others are long.It is simple to see that there are δ3 − 1 long desserts, L(t) := [(t − 1)a4 + 1, ta1 − 1], t = 1, . . . , δ3 − 1.The set of gaps belonging to long (resp. short) deserts is denoted by Gapsl(H) (resp. Gapss(H)). ThusGapsl(H) = ⋃

t=1,...,δ3−1 L(t) . Let us begin by studying the case n − mi ∈ Gapss(H).

Lemma 3.14. If l ∈ Gaps(H) with l < δ3 + (δ3 − 1)a2 then #D(l) � δ3 .

Proof. (lk,m j) ∈ D(l) iff m j + c − 1 − lk = c − 1 − l. Since (δ3 − 1)a4 < c − 1 − l = m, then m ∈ H . Letm = ∑

xiai be the normal representation of m. Then #D(l) �∏

(xi + 1) >∑

xi � δ3 − 1. �Proposition 3.15. If n − mi = l ∈ Gapss(H), let s be such that ρs−1 < l < ρs . Then #Λ∗

i � ρs .

Proof. Since ρs − l < δ3, Proposition 3.3 and Lemma 3.14 imply the result. �As a direct consequence we have the following.


Theorem 3.16. If n − mi ∈ Gapss(H), then

d∗i = min{ρt ∈ H: ρt � n − mi}.

Example 5. Consider the Suzuki curve over F8. Among the 11 values of mi for which d∗ is not deter-mined in Example 4, the above Theorem 3.16 allows the computation of d∗

i for mi = 53,55.

Let us study now the case in which n − mi ∈ Gapsl(H). For 0 < t < δ3 − 1, consider the num-bers λ+

2 (t) = ta2 + δ3, λ+4 (t) = ta4 + δ3 and λ−

4 (t) = ta4 − δ3. In addition let λ+2 (0) = λ+

4 (0) = δ3 andλ−

4 (0) = 0. Clearly all of them, except λ−4 (0) = 0, are gaps and λ+

2 (t), λ+4 (t) belong to long deserts.

Furthermore, for all t it holds that λ−4 (t) < λ+

2 (t) � λ+4 (t) < λ−

4 (t + 1).

Lemma 3.17. Let l, t be two integers such that l is a gap of H and 0 � t < δ3 − 1. The following statementshold.

(a) #D(λ+2 (t)) = #D(λ+

4 (t)) = δ3(δ3 − t).(b) If t �= 0, #D(λ−

4 (t)) = (δ3 + 1)(δ3 − t).(c) If λ−

4 (t) < l < λ+2 (t) then #D(l) � (δ3 + 1)(δ3 − t).

(d) If λ+2 (t) < l < λ+

4 (t) then #D(l) � δ3(δ3 − t) + λ+4 (t) − l.

(e) If λ+4 (t) < l < λ−

4 (t + 1) then #D(l) � δ3(δ3 − t).(f) If λ+

4 (t) < l < (t + 1)a1 then #D(l) � (t + 1)a1 − l.

Proof. (a) and (b) are straightforward computations by using uniqueness of normal representations.(c) (lk,m j) ∈ D(l) iff m j + c − 1 − lk = c − 1 − l. Gaps in this interval are of the form l = λ−

4 (t)+ w andl = λ+

2 (t) − w with 0 < w < δ3 − t . So c − 1 − l = δ3a1 + (δ3 − t − 1)a4 − w = δ3a1 + wa2 + (δ3 − t −1 − w)a4 and c − 1 − l = (δ3 − t − 1)a2 + (δ3 − 1)a4 + w = (δ3 − t − 1 − w)a2 + δ3a1 + a3 + (w − 1)a4respectively. Then #D(l) � (δ3 + 1)(δ3 − t − w)(w + 1) � (δ3 + 1)(δ3 − t) and #D(l) � (δ3 + 1)(δ3 −t − w)2w � (δ3 + 1)(δ3 − t) respectively. (d) Let l = λ+

4 (t) − α, note that 0 < α < t . (lk,m j) ∈ D(l)iff m j + c − 1 − lk = c − 1 − l. Since c − 1 − l = (δ3 − 1 − α)a2 + (δ3 − 1 − t + α)a4, we get #D(l) �(δ3 −α)(δ3 − t +α) = δ3(δ3 − t)+α(t −α). (e) Let us consider two cases. If λ+

4 (t) < l < (t + 1)a1 then(lk,m j) ∈ D(l) iff m j + c − 1 − lk = c − 1 − l. Let l(w) = (t + 1)a1 − wδ3 with 0 < w � 2(δ3 − t)− 1. Foreven w we have c −1− l(w) = (δ3 − t −1− w/2)a1 +a3 + (w/2−1)a2 + (δ3 −1)a4, hence #D(l(w)) �δ3(δ3 − t)+δ3(w(δ3 − t − w/2)− (δ3 − t)). For odd w we get c −1− l(w) = (δ3 − t −1− (w −1)/2)a1 +(w − 1)/2a2 + (δ3 − 1)a4 so #D(l(w)) � δ3(δ3 − t) + δ3(w − 1)/2(δ3 − t − 1 − (w − 1)/2). Now if l =l(w) + α with 0 < α < δ3, a similar reasoning as in Lemma 3.17(d) shows that #D(l) > #D(l(w)) + α.The second case to consider is (t +1)a1 < l < λ−

4 (t +1). We can restrict to (t +1)a1 < l < a3 +ta2, sinceother integers in the above range are nongaps. Then (lk,m j) ∈ D(l) iff m j + c − 1 − lk = c − 1 − l. Letl(α) = a3 +ta2 −α with 0 < α < δ3 −t . Thus c−1−l(α) = (δ3 −1)a1 +a3 +(δ3 −1−t −α)a2 +(α−1)a4and #D(l(α)) � δ3(δ3 − t) + δ3α(δ3 − t − 1 − α). Now consider the numbers a3 + ta2 − 2iδ3 ± α with0 < i � t and 0 < α < δ3. Gaps in our interval are of the form l(α) = ia1 + a3 + (t − i)a2 − α andl(α) = ia1 + a3 + (t − i)a4 + α, with 0 < α < δ3 − t + i. Then c − 1 − l(α) = (δ3 − i − 1)a1 + a3 + (δ3 −1− t + i −α)a2 + (α−1)a4 and c −1− l(α) = (δ3 − i)a1 + (α−1)a2 + (δ3 −1− t + i −α)a4, respectively.In both cases we obtain #D(l) � δ3(δ3 − t). (f) Is a consequence of the first part of previous item. �

The above Lemma 3.17 allows us the computation of d∗ for all values in the interval n − (δ3 −1)a4 < mi < n + δ3 + (δ3 − 1)a2.

Theorem 3.18. Let mi < n be such that n − mi ∈ Gapsl(H).

(1) If ta4 < n − mi � λ+2 (t), then d∗

i = λ+2 (t) + δ3(δ3 − t).

(2) If λ+2 (t) < n − mi � λ+

4 (t), then d∗i = λ+

4 (t) + δ3(δ3 − t).(3) If λ+

4 (t) < n − mi < (t + 1)a1 , then d∗i = (t + 1)a1 = min{ρ ∈ H: ρ � n − mi}.


Proof. (1) Write n − mi = λ+2 (t) − w with 0 � w < δ3 − t . Proposition 3.3 and Lemma 3.17, item (c),

imply #Λ∗i � λ+

2 (t) − w + (δ3 + 1)(δ3 − t) � λ+2 (t) + δ3(δ3 − t). (2) Write n − mi = λ+

4 (t) − w with0 � w < t . Proposition 3.3 and Lemma 3.17, item (d), imply #Λ∗

i = n − mi + #D(l) � λ+4 (t) − w +

δ3(δ3 − t) + w � λ+4 (t) + δ3(δ3 − t). (3) Follows from Proposition 3.3 and Lemma 3.17, item (f). �

Note that, according to Theorems 3.16 and 3.18, for all gaps in short deserts and most gaps in longdeserts we have equality in Corollary 3.5. Finally we have the following.

Theorem 3.19. Let mi > n and let us write mi = n + li−n+g .

(1) If λ−4 (t) � li−n+g < λ+

2 (t) for 0 � t � δ3 − 2, then d∗i = (δ3 + 1)(δ3 − t).

(2) If λ+2 (t) � li−n+g < λ−

4 (t + 1) then d∗i = δ3(δ3 − t).

(3) If li−n+g = λ−4 (δ3 − 1) then d∗

i = (δ3 + 1).

Proof. Statements (1) and (2) follow directly from Proposition 3.3 and Lemma 3.17, items (c), (d)and (e). (3) Since c − 1 − λ−

4 (δ3 − 1) = δ3a1, we have Λ∗i = δ3 + 1. Then use (2) with t = δ3 − 2. �

Acknowledgments

The authors wish to thank Professor Tom Høholdt and Thomas Glinski for pointing out some errorsin the first version of this paper. We also wish to thank the referee for many interesting comments.

References

[1] H. Andersen, O. Geil, Evaluation codes from order domain theory, Finite Fields Appl. 14 (2008) 92–123.[2] P. Beelen, The order bound for general algebraic geometric codes, Finite Fields Appl. 13 (2007) 665–680.[3] I. Duursma, R. Kirov, An extension of the order bound for AG codes, in: M. Bras, T. Høholdt (Eds.), Applied Algebra, Algebraic

Algorithms and Error-Correcting Codes, in: Lecture Notes in Comput. Sci., vol. 5527, Springer-Verlag, Berlin, 2009, pp. 11–22.

[4] I. Duursma, S. Park, Coset bounds for algebraic geometric codes, Finite Fields Appl. 16 (2010) 36–55.[5] O. Geil, C. Munuera, D. Ruano, F. Torres, On the order bounds for one-point AG codes, Adv. Math. Commun. 5 (2011)

489–504.[6] O. Geil, H. Matsumoto, Bounding the number of rational place using Weierstrass semigroups, J. Pure Appl. Algebra 213

(2009) 1152–1156.[7] J.P. Hansen, H. Stichtenoth, Group codes on certain algebraic curves with many rational points, Appl. Algebra Engrg. Comm.

Comput. 1 (1990) 67–77.[8] T. Høholdt, J.H. van Lint, R. Pellikaan, Algebraic-geometry codes, in: V. Pless, W.C. Huffman (Eds.), Handbook of Coding

Theory, vol. 1, Elsevier, Amsterdam, 1998.[9] C. Kirfel, R. Pellikaan, The minimum distance of codes in an array coming from telescopic semigroups, IEEE Trans. Inform.

Theory 41 (1995) 1720–1731.[10] K. Lee, M. Bras-Amoros, M.E. O’Sullivan, Unique decoding of plane AG codes via interpolation, IEEE Trans. Inform. Theory 58

(2012) 3941–3950.[11] J. Lewittes, Places of degree one in functions fields over finite fields, J. Pure Appl. Algebra 69 (1990) 177–183.[12] O. Geil, R. Matsumoto, D. Ruano, List decoding algorithms based on Gröbner bases for general one-point AG codes, in:

Proceedings of 2012 IEEE International Symposium on Information Theory, ISIT, 1–6 July 2012, pp. 86–90.[13] O. Geil, R. Matsumoto, D. Ruano, List decoding algorithm based on voting in Gröbner bases for general one-point AG codes,

preprint, arXiv:1203.6127, 2012.[14] MinT, Online database for optimal parameters of (t,m, s)-nets, (t, s)-sequences, orthogonal arrays, linear codes and OOAs,

available at http://mint.sbg.ac.at/.[15] S. Miura, Linear codes on affine algebraic varieties, Trans. IEICE 10 (1998) 1386–1397 (in Japanese).[16] S. Miura, Linear codes on affine algebraic curves, Trans. IEICE 10 (1998) 1398–1421 (in Japanese).[17] S. Miura, PhD thesis, University of Tokyo, 1997 (in Japanese).[18] R. Matsumoto, S. Miura, On the Feng–Rao bound for the L-construction of algebraic-geometry codes, IEICE Trans. Fundam. 5

(2000) 923–927.[19] C. Munuera, A. Sepulveda, F. Torres, Algebraic geometry codes from Castle curves, in: A. Barbero (Ed.), Coding Theory and

Applications, in: Lecture Notes in Comput. Sci., vol. 5228, Springer-Verlag, Berlin, 2008, pp. 117–127.[20] H. Stichtenoth, Algebraic Function Fields and Codes, second edition, Springer-Verlag, Berlin, 2009.[21] K. Yang, P.V. Kumar, On the true minimum distance of Hermitian codes, in: Coding Theory and Algebraic Geometry, in:

Lecture Notes in Comput. Sci., vol. 1518, Springer-Verlag, Berlin, 1992, pp. 99–107.

http://mint.sbg.ac.at/

on the minimum distance of castle codes

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