On the automorphism groups of some AG-codes based on C a,b curves
aa r X i v : . [ c s . I T ] A p r ON THE AUTOMORPHISM GROUPS OF SOME AG-CODESBASED ON C a,b CURVES
T. SHASKA AND H. WANG
Abstract.
We study C a,b curves and their applications to coding theory.We show how C a,b curves can be used to construct MDS codes and focuson some C a,b curves with extra automorphisms, namely y = x + 1 , y = x − x, y − y = x . The automorphism groups of such codes are determinedin most characteristics. Introduction
In the design of new codes algebraic geometry codes (AG-codes), also known asGoppa codes, play an important part and have been well studied in the last fewdecades. In designing such codes an important fact is the number of points of thealgebraic curve over a finite field. Hence, it is natural that the algebraic curvesthat have been used so far are curves for which such number of points can becomputed. Is there a ”nice” family of curves which can be used to construct goodcodes? Hermitian curves have been used successfully by many authors in additionto hyperelliptic curves and other families of curves. The most natural curves aresuperelliptic curves as shown in [5, 8, 11, 16, 21]In this paper we investigate a family of curves which belongs to superellipticcurves, namely the C a,b curves. C ab curves are algebraic curves with very interestingarithmetic properties. There are algorithms suggested which count the number ofpoints of these curves using the Monsky-Washnitzer cohomology; see [7]. In thispaper we study how these properties can be used in constructing good algebraicgeometry codes.In Section 2, we give a brief introduction to algebraic geometry codes (AG-codes). This is well-known material. As a standard reference we use [9]. Manyother excellent resources exist.In Section 3, we briefly define C ab curves. Such curves have degree a, b coversto P . The existence of certain divisors makes these curves useful in coding theory.For more details on AG-codes and quantum AG-codes one can check [8, 15, 16, 19].In Section 4, we study the locus of genus g , C a,b curves for fixed a, b . Suchcurves have degree a, b covers to P . Such covers are classified according to theramification structure. We assume that the cover has the largest possible modulidimension. This determines a ramification structure σ . The simplest case of C a,b curves are hyperelliptic curves and superelliptic curves. Such curves have beenstudied in detail by many authors and are well understood. They are the mainclasses of curves being used in coding theory and cryptography. Mathematics Subject Classification.
Primary: 11T71, 14G50, Secondary: 94A05, 14Q05.
Key words and phrases. C a,b curves, AG-codes, automorphism groups, superelliptic curves. Denote the moduli spaces of these maximal moduli dimension degree a, b cover-ings by M a and M b respectively and let g = ( a − b − M a , M b arealgebraic varieties of M g (not necessarily irreducible). The locus of C a,b curves in M g is the intersection M a ∩ M b . Studying this locus is the focus of section 3.In the last section we use C a,b curves of genus 3 to construct AG-codes. Suchcodes are MDS codes. We focus on some genus 3 C ab curves with extra automor-phisms, namely y = x + 1 , y = x − x, y − y = x . The automorphism groupsof such codes are determined for some characteristics.This is an updated version of a small note from 2006. Connections to superellipticcurves are added and an updated list of references. Notation:
Throughout this paper X will denote a smooth, projective curve definedover some field F . By Aut ( X ) we denote the group of automorphisms of X definedover ¯ F . By C we will denote a linear code. The permutation automorphism group of C will be denoted by PAut ( C ), the monomial automorphism group by MAut ( C ),and the automorphism group by ΓAut ( C ). F q denotes a finite field of q elements.2. Preliminaries
Let F q be a finite field of size q and X a genus g ≥ F q . Let F be the function field of X and P , . . . , P n be points of multiplicity on in X We take divisors D = P + · · · + P n and G such that supp( G ) ∩ supp( D ) = ∅ . Inaddition L ( G ) denotes the Riemann-Roch space for the divisor G . The algebraicgeometry code C L ⊆ F nq is defined by C L ( D, G ) = { ( f ( P ) , . . . , f ( P n )) | f ∈ L ( G ) } ⊆ F nq The following linear map is called the evaluation map ϕ : (cid:26) L ( G ) → F nq f ( f ( P ) , . . . , f ( P n )) . Thus, the code is given by C L ( D, G ) = ϕ ( L ( G )) . It is a linear code [ n, k, d ] with parameters k = dim G − dim( G − D ) ,d ≥ n − deg G. The following result is well known, see [23, Thm. II.2.3] among many others.
Lemma 1. If deg G < n , then (1) ϕ : L ( G ) → C L ( D, G ) is injective and C L ( D, G ) is an [ n, k, d ] code with k = dim G ≥ deg G + 1 − g, and d ≥ n − deg G. (2) If in addition g − < deg G < n , then k = deg G + 1 − g. N THE AUTOMORPHISM GROUPS OF SOME AG-CODES BASED ON C a,b CURVES 3 (3) If ( f , . . . , f k ) is a basis of L ( G ) , then M = f ( P ) · · · f ( P n ) ... ... f k ( P ) · · · f k ( P n ) is a generator matrix for C L ( D, G ) . To characterize the dual code of a AG-code we need to look at the originaldefinitions of Goppa by means of differential forms and its relations to the codedefined above. We define the code C Ω ( D, G ) by C Ω ( D, G ) := { (res P ( ω ) , . . . , res P n ( ω )) | ω ∈ Ω F ( G − D ) } ⊆ F nq . The code C Ω ( D, G ), where D and G are as above is equal to the dual of C L ( D, G ) ⊥ . In other words, C L ( D, G ) ⊥ = C Ω ( D, G ). Also, C Ω ( D, G ) = a · C L ( D, H )with H = D − G + ( η ) where η is a differential, v P i ( η ) = − i = 1 , . . . , n , and a = (res P ( η ) , . . . , res P n ( η )). Moreover, C L ( D, G ) ⊥ = a · C L ( D, H ) . The following proposition is cited from [23, Prop. VII.1.2]. It allows to constructdifferentials with special properties that help to construct a self-orthogonal code.
Lemma 2.
Let x and y be elements of F such that v P i ( y ) = 1 , v P i ( x ) = 0 and x ( P i ) = 1 for i = 1 , . . . , n . Then the differential η := x · dyy satisfies v P i ( η ) = − , and res P i ( η ) = 1 for i = 1 , . . . , n . Next we give some standard definitions on automorphism groups of codes, whichwill be the main focus of this paper. A basic reference is [9].The permutation automorphism group of the code C ⊆ F nq is the subgroup of S n (acting on F nq by coordinate permutation) which preserves C . We denote suchgroup by PAut ( C ).The set of monomial matrices that map C to itself forms the monomial auto-morphism group , denoted by MAut ( C ). Every monomial matrix M can be writtenas M = DP where D is a diagonal matrix and P a permutation matrix. Let γ bea field automorphism of F q and M a monomial matrix. Denote by M γ the map M γ : C → C such that ∀ x ∈ C we have M γ ( x ) = γ ( M x ). The set of all maps M γ forms the automorphism group of C , denoted by ΓAut ( C ). It is well known thatPAut ( C ) ≤ MAut ( C ) ≤ ΓAut ( C )Next we will define as admissible a class of curves which have some additionalconditions on their divisors. Definition 1.
A genus g ≥ curve X /F q is called admissible if it satisfies thefollowing properties:i) there exists a rational point P ∞ and two functions x, y ∈ F such that ( x ) ∞ = kP ∞ , ( y ) ∞ = lP ∞ , and k, l ≥ ; T. SHASKA AND H. WANG ii) for m ≥ , the elements x i y j with ≤ i, ≤ j ≤ k − , and ki + lj ≤ m forma basis of the space L ( mP ∞ ) . Next we define
Aut
D,G ( X ) := { σ ∈ Aut ( X ) | σ ( D ) = D and σ ( G ) = G } . With the above notation we have the following:
Theorem 1.
Let X /F q be an admissible curve over F q of genus g where l > k .Assume that m ≥ l . Let D = P P ∈ J P where J ⊆ P \{ P ∞ } , P is the set of allrational points of X . If n > max (cid:26) g + 2 , m, k (cid:18) l + k − β (cid:19) , lk (cid:18) k − m − k + 1 (cid:19)(cid:27) , where n = | J | and β = min { k − , r | y r ∈ L ( mP ∞ ) } , then Aut ( C L ( D, mP ∞ )) ∼ = Aut
D,mP ∞ ( X ) . Proof.
See [26] for details (cid:3)
In the next two sections we will see how we can compute the automorphismgroups of certain AG-codes constructed by some superelliptic curves.3.
Introduction to C ab curves In this section, we introduce the notion of C ab curves which constitute a wideclass of algebraic curves including elliptic curves, hyperelliptic curves and superel-liptic curves. They have been studied by many people due to their nice properties.Throughout this section k denotes an algebraically closed field of characteristicnot equal to 2. Let a and b be relatively prime positive integers. Then a curve X defined over k is called an C ab curve if it is a nonsingular plane curve defined by f ( x, y ) = 0, where f ( x, y ) ∈ k [ x, y ] has the form(1) f ( x, y ) = α ,a y a + α b, x b + X ai + bj 1) + 1The degree b cover has d := 2( g − 1) + 2 b − ( b − 1) = ( a − b − 1) + b + 1 = a ( b − 1) + 1branch points. N THE AUTOMORPHISM GROUPS OF SOME AG-CODES BASED ON C a,b CURVES 5 Corollary 1. All hyperelliptic curves are C a,b curves.Proof. Every genus g hyperelliptic curve can be written as y = f ( x ) such thatdeg f = 2 g + 1. Take a = 2 and b = 2 g + 1. (cid:3) Next, we will see superelliptic curves which are even a larger class than that ofhyperelliptic curves, but first the following example. Example 1. Let a = 3 , b = 4 . Then the genus of the curve is g = 3 and we have (2) Y + α X + α X + α X Y + α XY + α X + α Y + α XY + α X + α Y + α = 0 Since the dimension of M is 5 then we should be able to write this curve in a”better” way; see the next section for details. The next proposition will be usefulwhen we construct AG-codes from C ab curves. Proposition 1. Let X be a C ab curve defined by f ( X, Y ) = 0 with f ( X, Y ) ∈ F [ X, Y ] . Then { X i Y j | ≤ j ≤ a − , i ≥ , ai + bj ≤ m } is a basis of a vector space L ( m · ∞ ) over F , where m ∈ Z ≥ . Corollary 2. C a,b curves are admissible curves. Hence we can use the results of the previous section when constructing codesfrom such curves.3.1. Superelliptic curves. There are a special class of C ab curves which are wellunderstood due to the work of many authors [1–4, 6, 11, 13, 18, 25].Let X be a genus g ≥ k , G its automorphism group,and H a subgroup of G of order | H | = m , inside the center Z ( G ), such that thegenus of the quotient space X /H is zero. Such curves are called superelliptic curves and they can be written with the affine equation y m = f ( x ) for some f ∈ k [ x ]; see[3] for more details.The following lemma is an immediate consequence of the definition. Lemma 3. Superelliptic curves are C ab curves. Then we have the following. Corollary 3. Superelliptic curves are admissible curves. In [14] are determined are possible groups of superelliptic curves defined overfields of characteristic = 2. In [20] are determined even the equations for eachgroup. This is not known for algebraic curves in general.3.2. Automorphism groups of C ab curves. Let X be a C ab curve as above.Can we determine the automorphism group of X over k in terms of a, b ? Forgenus g = 2 , C ab curve is hyperelliptic or superelliptic. In general there is noknown algorithm to determine the automorphism group of an algebraic curve. T. SHASKA AND H. WANG Lemma 4. Let X be a genus g = 2 algebraic curve as in Eq. (1) defined over k .Then Aut ( X ) is isomorphic to one of the following:i) p = 3 : Z , V , D , D , GL (3) , ii) p = 5 : Z , Z , V , D , D , GL (3) , .iii) p ≥ : Z , V , D , D , SL (3) . For the case p = 2 see [24] for details. For g = 3 see [14, 20]. The automor-phism groups of superelliptic curves defined over a field k such that char k = 2 aredetermined completely in [14, 20] and the corresponding equations are determinedin [17]. 4. The locus of C , curves in the moduli space M In this section we want to focus on non-hyperelliptic genus 3 curves. Moreprecisely, we want to study the space of C , curves in the moduli space M .Throughout this section all curves are defined over a characteristic zero field.We first give a brief introduction to the Hurwitz spaces and projection of suchspaces on M g . Let X be a curve of genus g and f : X → P be a covering of degree n with r branch points. We denote the branch points by q , . . . , q r ∈ P and let p ∈ P \ { q , . . . , q r } . Choose loops γ i around q i such thatΓ := π ( P \ { q , . . . , q r } , p ) = h γ , . . . , γ r i , γ · · · γ r = 1 . Γ acts on the fiber f − ( p ) by path lifting, inducing a transitive subgroup G ofthe symmetric group S n (determined by f up to conjugacy in S n ). It is called the monodromy group of f . The images of γ , . . . , γ r in S n form a tuple of permutations σ = ( σ , . . . , σ r ) called a tuple of branch cycles of f . We call such a tuple the signature of φ . The covering f : X → P is of type σ if it has σ as tuple of branchcycles relative to some homotopy basis of P \ { q , . . . , q r } .Two coverings f : X → P and f ′ : X ′ → P are weakly equivalent (resp. equivalent ) if there is a homeomorphism h : X → X ′ and an analytic automorphism g of P such that g ◦ f = f ′ ◦ h (resp., g = 1). Such classes are denoted by [ f ] w (resp., [ f ]). The Hurwitz space H σ (resp., symmetrized Hurwitz space H sσ ) is theset of weak equivalence classes (resp., equivalence) of covers of type σ , it carries anatural structure of an quasiprojective variety.Let C i denote the conjugacy class of σ i in G and C = ( C , . . . , C r ). The set ofNielsen classes N ( G, C ) is N ( G, σ ) := { ( σ , . . . , σ r ) | σ i ∈ C i , G = h σ , . . . , σ r i , σ · · · σ r = 1 } The braid group acts on N ( G, C ) as[ γ i ] : ( σ , . . . , σ r ) → ( σ , . . . , σ i − , σ σ i i +1 , σ i , σ i +2 , . . . , σ r )where σ σ i i +1 = σ i σ i +1 σ − i . We have H σ = H τ if and only if the tuples σ , τ are inthe same braid orbit O τ = O σ .Let M g be the moduli space of genus g curves. We have morphisms H σ Φ σ −→ H sσ ¯Φ σ −→ M g [ f ] w → [ f ] → [ X ](3)Each component of H σ has the same image in M g . We denote by L g := ¯Φ σ ( H sσ ) . N THE AUTOMORPHISM GROUPS OF SOME AG-CODES BASED ON C a,b CURVES 7 We say that the covering f or the ramification σ has moduli dimension δ := dim L g .Let a, b be fixed and g = ( a − b − / 2. The generic C a,b curve of genus g hasa degree a cover π a : C a,b → P (resp. degree b cover π b : C a,b → P ).The ramification structure of π a : C a,b → P is ( a, , . . . , 2) where the numberof branch points is d = b ( a − 1) + 1, as discussed in section 3. Let H a denote theHurwitz space of such covers and M a its image in M g , as described above. Then,the dimension of M a ia δ ≤ b ( a − − 2. Similarly, we get that the dimension of M b is δ ≤ a ( b − − 2. Of course, the cover with smallest degree among π a and π b is the one of interest. From now on, we assume that a < b .As mentioned above the goal of this section is to study the space M a,b for fixed a and b , particularly on the case a = 3 and b = 4. Theorem 2. Every genus 3 curve is a C , curve. Moreover, every genus C , curve defined over a field k , is isomorphic to a curve with equation (4) f ( x, y ) = ( x + b ) y + ( cx + d ) y + ( ex + f x ) y + x + kx + lx = 0 . Proof. The case of hyperelliptic curves is obvious. Hence, we focus on non-hyperellipticgenus 3 curves. Let C be a non-hyperelliptic genus 3 curve, P be a Weierstrasspoint on C , and K the function field of C . Then exists a meromorphic function x which has P as a triple pole and no other poles. Thus, [ K : L ( x )] = 3. Consider x as a mapping of C to the Riemann sphere. We call this mapping ψ : C → P andlet ∞ be ψ ( P ). From the Riemann-Hurwitz formula we have that ψ has at most 8other branch points. There is also a meromorphic function y which has P as a poleof order 4 and no other poles. Thus the equation of K is given by(5) f ( x, y ) := γ ( x ) y + γ ( x ) y + γ ( x ) y + γ ( x ) = 0where γ ( x ) , . . . , γ ( x ) ∈ L [ x ] and deg ( γ ) = 4 , deg( γ ) = 0 , deg( γ ) ≤ , deg( γ ) ≤ . The discriminant of f ( x, y ) with respect to yD ( f, y ) := − 27 ( γ γ ) + 18 γ γ γ γ + ( γ γ ) − γ γ − γ γ , must have at most degree 8 since its roots are the branch points of ψ : C → P .Thus, we have deg ( γ γ ) ≤ , deg ( γ γ ) ≤ , deg ( γ γ ) ≤ . If deg ( γ ) = 2 then deg ( γ ) ≤ deg ( γ ) = 0. Thus, deg ( f, x ) = 2. Then, f ( x, y ) = 0 is not the equation of an genus 3 curve. Hence, deg ( γ ) ≤ 1. Clearly, deg ( γ ) ≤ 1. We denote: γ ( x ) := a,γ ( x ) := cx + dγ ( x ) := ex + f,γ ( x ) := gx + hx + kx + lx + m (6)Then, we have f ( x, y ) = ay + ( cx + d ) y + ( ex + f ) y + ( gx + hx + kx + lx + m ) = 0 T. SHASKA AND H. WANG which is obviously an C , curve. This completes the proof of the first statement.Let C be a C , curve defined over k . Then, C is a non-hyperelliptic genus 3curve. Hence, it is isomorphic over k to a curve with equation as in Eq. (4); see[22] for details. This completes the proof. (cid:3) Hence, the space of C , curves correspond to the moduli space M . It is aninteresting problem to see what happens for higher genus g .5. Codes obtained from C a,b curves In this section we will give examples of codes which are constructed based on C ab curves. We will focus on three curves, namely y = x + 1, y = x − x , and y − y = x . All these curves are genus 3 non-hyperelliptic curves. For characteristic p > n, k, d ] code with d = n − k + 1 is called maximum distance separable code oran MDS code.5.1. The curve y = x + 1 . Let X be the curve y = x + 1defined over F q . This is a C , curve of genus 3. For characteristic p = 2 , X is C ⋊ A , which has Gap identity (48 , X over F q by { P , . . . , P n } . Let C = C L ( D, G ),where n + 1 is the number of rational points of X and G = mP ∞ , D = P + · · · P n We have the following result: Theorem 3. For the permutation automorphism group PAut ( C ) , one hasi) If ≤ m < or m > n + 4 then PAut ( C ) ∼ = S n .ii) If n > and ≤ m < n/ then PAut ( C ) ∼ = Aut D,mP ∞ ( X ) .Proof. i) If 0 ≤ m < 3, then from Proposition 1 we know (1 , , · · · , 1) is a basis of thevector space L ( m · P ∞ ) , thus dim G = 1. Since dim( G − D ) ≥ , dim C ≥ 1, togetherwith dim C = dim G − dim( G − D ) we have dim C = 1. Therefore PAut ( C ) ∼ = S n .If m > n + 4, then deg ( G − D ) > g − 2. Thus dim C = dim G − dim( G − D ) = n.C is the full space, therefore PAut ( C ) ∼ = S n .ii) With the notation of definition 1. In this case k = 3 , l = 4 , g = 3 , β =1 , m ≥ 4. For Theorem 1 to hold we need n > max (cid:26) , m, , (cid:18) m − (cid:19)(cid:27) . Since m ≥ 4, 12(1 + m − ) ≤ 24. Thus when n > 24 and 4 ≤ m < n/ C ) ∼ = Aut D,mP ∞ ( X ). (cid:3) It can be seen from the proof of theorem that C L ( D, G ) is a [ n, , n ] MDS codewhen 0 ≤ m < 3, and a [ n, n, 1] MDS code when m > n + 4. N THE AUTOMORPHISM GROUPS OF SOME AG-CODES BASED ON C a,b CURVES 9 Example 2. Let X be defined over F . Take m = 4 . By computation using GAP,we find that C L ( D, G ) is a [8 , , MDS code with a generator matrix α α α α α α α α α α α α 11 1 1 1 1 1 1 1 , where α is a primitive element of F . The permutation automorphism group isPAut ( C L ( D, G )) ∼ = Z . The curve y = x − x . Let X be the curve y = x − x defined over F q . For characteristic p > X is thecyclic group of order 9. Denote the set of affine rational points of X over F q by { P , . . . , P n } . Let C = C L ( D, G ), where n + 1 is the number of rational points of X and G = mP ∞ , D = P + · · · P n We have the following result: Theorem 4. For the permutation automorphism group PAut ( C ) , one hasi) If ≤ m < or m > n + 4 then PAut ( C ) ∼ = S n .ii) If n > and ≤ m < n/ then PAut ( C ) ∼ = Aut D,mP ∞ ( X ) .Proof. i) If 0 ≤ m < 3, then from Proposition 1 we know (1 , , · · · , 1) is a basis of thevector space L ( m · P ∞ ) , thus dim G = 1. Since dim( G − D ) ≥ , dim C ≥ 1, togetherwith dim C = dim G − dim( G − D ) we have dim C = 1. Therefore PAut ( C ) ∼ = S n .If m > n + 4, then deg ( G − D ) > g − 2. Thus dim C = dim G − dim( G − D ) = n.C is the full space, therefore PAut ( C ) ∼ = S n .ii) With the notation of definition 1. In this case k = 3 , l = 4 , g = 3 , β =1 , m ≥ 4. For Theorem 1 to hold we need n > max (cid:26) , m, , (cid:18) m − (cid:19)(cid:27) Since m ≥ (cid:16) m − (cid:17) ≤ 24. Thus when n > 24 and 4 ≤ m < n/ C ) ∼ = Aut D,mP ∞ ( X ). (cid:3) It can be seen from the proof of theorem that C L ( D, G ) is a [ n, , n ] MDS codewhen 0 ≤ m < 3, and a [ n, n, 1] MDS code when m > n + 4. Example 3. Let X be defined over F . Take m = 3 . By computation usingGAP, we find that C L ( D, G ) is a [8 , , code with permutation automorphismgroup [56 , (Gap identity), which is clearly an MDS code. The curve y − y = x . Let X be the curve y − y = x defined over F q . Denote the set of affine rational points of X over F q by { P , . . . , P n } .Let C = C L ( D, G ), where n + 1 is the number of rational points of X and G = mP ∞ , D = P + · · · P n We have the following; Theorem 5. For the permutation automorphism group PAut ( C ) , one hasi) If ≤ m < or m > n + 4 then PAut ( C ) ∼ = S n .ii) If n > and ≤ m < n/ then PAut ( C ) ∼ = Aut D,mP ∞ ( X ) .Proof. i) If 0 ≤ m < 3, then from Proposition 1 we know (1 , , · · · , 1) is a basis of thevector space L ( m · P ∞ ) , thus dim G = 1. Since dim( G − D ) ≥ , dim C ≥ 1, togetherwith dim C = dim G − dim( G − D ) we have dim C = 1. Therefore PAut ( C ) ∼ = S n .If m > n + 4, then deg ( G − D ) > g − 2. Thus dim C = dim G − dim( G − D ) = n.C is the full space, therefore PAut ( C ) ∼ = S n .ii) With the notation of Definition 1. In this case k = 3 , l = 4 , g = 3 , β =1 , m ≥ 4. For Theorem 1 to hold we need n > max (cid:26) , m, , m − (cid:27) . Since m ≥ 4, 12(1 + m − ) ≤ 24. Thus when n > 24 and 4 ≤ m < n/ C ) ∼ = Aut D,mP ∞ ( X ). (cid:3) It can be seen from the proof of theorem that C L ( D, G ) is a [ n, , n ] MDS codewhen 0 ≤ m < 3, and a [ n, n, 1] MDS code when m > n + 4. Example 4. Let X be defined over F . Take m = 6 . By computation using GAP,we find that C L ( D, G ) is a [4 , , code with a generator matrix α α α α α α , where α is a primitive element of F . The permutation automorphism group isisomorphic to the group with GAP identity [24 , . In this casePAut ( C ) ֒ → Aut ( X ) . This code is clearly an MDS code. Concluding remarks It is an open question to determine the automorphism groups of AG-codes ob-tained by C a,b curves, in all characteristics. Moreover, even determining the list ofautomorphism groups of C a,b curves seems to be a non-trivial problem.Furthermore, to determine the locus of C a,b curves, defined over C , in the modulispace M g , where g = ( a − b − , seems an interesting problem in its own right. A C a,b curve has covers of degree a and b to P . One has to take such covers inthe most generic form. The space of C a,b curves in M g will be the intersection ofthe corresponding Hurwitz spaces. To generalize this for any a, b would require acareful analysis of the corresponding Hurwitz spaces.Since the first version of this note, considerable progress has been made with su-perelliptic curves. Such curves are well understood and their automorphism groupsfully determined in all characteristics different from two, due to work of Sanjeewa[14]. Moreover, for all such groups we can determine the equation of the corre-sponding curve [20]. For curve with extra automorphisms such equations can bedetermined over the minimal field of definition due to work of Beshaj/Thompson N THE AUTOMORPHISM GROUPS OF SOME AG-CODES BASED ON C a,b CURVES 11 [6] and Hidaldo/Shaska [10]. Furthermore, due to work of Beshaj such equationsover the minimal field of definition can even be chosen with minimal coefficients[2].It is still unknown whether a precise relation exists between the automorphismgroup of the curve and the automorphism group of the Ag-codes, even in the caseof superelliptic curves whose automorphism groups are well understood. References [1] L. Beshaj, A. Elezi, and T. Shaska, Theta functions of superelliptic curves (201503), availableat .[2] Lubjana Beshaj, Reduction theory of binary forms (201502), available at .[3] Lubjana Beshaj, Valmira Hoxha, and Tony Shaska, On superelliptic curves of level n andtheir quotients, I , Albanian J. Math. (2011), no. 3, 115–137. MR2846162 (2012i:14036)[4] Lubjana Beshaj, Tony Shaska, and Caleb Shor, On Jacobians of curves with superellipticcomponents , Contemporary Mathematics, Volume 629, 2014, pg. 1-14 (201310), available at .[5] Lubjana Beshaj, Tony Shaska, and Eustrat Zhupa (eds.), Advances on superelliptic curves andtheir applications. Based on the NATO Advanced Study Institute (ASI), Ohrid, Macedonia,2014. , Amsterdam: IOS Press, 2015 (English).[6] Lubjana Beshaj and Fred Thompson, Equations for superelliptic curves over their minimalfield of definition , Albanian J. Math. Vol. 8 (2014), no. 1, 3-8 (201405), available at .[7] Jan Denef and Frederik Vercauteren, Counting points on C ab curves using Monsky-Washnitzer cohomology , Finite Fields Appl. (2006), no. 1, 78–102. MR2190188(2007c:11075)[8] A. Elezi and T. Shaska, Quantum codes from superelliptic curves , Albanian J. Math. 5 (2011),no. 4, 175–191 (201305), available at .[9] W. Cary Huffman and Vera Pless, Fundamentals of error-correcting codes , Cambridge Uni-versity Press, Cambridge, 2003. MR1996953[10] R. Hidalgo and T. Shaska, On the field of moduli of superelliptic curves , Algebraic curvesand their fibrations in mathematical physics and arithmetic geometry, 2016.[11] Milagros Izquierdo and Tony Shaska, Cyclic curves over the reals (201501), available at .[12] K. Magaard, T. Shaska, S. Shpectorov, and H. V¨olklein, The locus of curves with prescribedautomorphism group , S¯urikaisekikenky¯usho K¯oky¯uroku (2002), 112–141. Communica-tions in arithmetic fundamental groups (Kyoto, 1999/2001). MR1954371[13] E. Previato, T. Shaska, and G. S. Wijesiri, Thetanulls of cyclic curves of small genus , Alba-nian J. Math. (2007), no. 4, 253–270. MR2367218 (2008k:14066)[14] R. Sanjeewa, Automorphism groups of cyclic curves defined over finite fields of any charac-teristics , Albanian J. Math. Vol. 3, Number 4, 2009, 131-160 (201301), available at .[15] T. Shaska and E. Hasimaj (eds.), Algebraic aspects of digital communications , NATO Sciencefor Peace and Security Series D: Information and Communication Security, vol. 24, IOS Press,Amsterdam, 2009. Papers from the Conference “New Challenges in Digital Communications”held at the University of Vlora, Vlora, April 27–May 9, 2008. MR2605610 (2011a:94002)[16] Tony Shaska, Quantum codes from algebraic curves with automorphisms , Condensed MatterPhysics (2008), no. 2, 383–396.[17] T. Shaska, Subvarieties of the hyperelliptic moduli determined by group actions , Serdica Math.Journal, No. 4, 355-374, 2006 (201302), available at .[18] Rachel Shaska, Equations of curves with minimal discriminant (201407), available at .[19] T. Shaska, W. C. Huffman, D. Joyner, and V. Ustimenko (eds.), Advances in coding theory andcryptography , Series on Coding Theory and Cryptology, vol. 3, World Scientific PublishingCo. Pte. Ltd., Hackensack, NJ, 2007. Papers from the Conference on Coding Theory andCryptography held in Vlora, May 26–27, 2007 and from the Conference on Applications ofComputer Algebra held at Oakland University, Rochester, MI, July 19–22, 2007. MR2435341(2009h:94158) [20] R. Sanjeewa and T. Shaska, Determining equations of families of cyclic curves , Albanian J.Math. VOL 2, NO 3 (2008) (201301), available at .[21] T. Shaska and C. Shor, Weierstrass points of superelliptic curves (201502), available at .[22] T. Shaska and J. Thompson, On the generic curve of genus 3 , Affine algebraic geometry,2005, pp. 233–243. MR2126664 (2006c:14042)[23] Henning Stichtenoth, Algebraic function fields and codes , Second, Graduate Texts in Math-ematics, vol. 254, Springer-Verlag, Berlin, 2009. MR2464941 (2010d:14034)[24] T. Shaska and H. V¨olklein, Elliptic subfields and automorphisms of genus 2 function fields , Al-gebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), 2004, pp. 703–723. MR2037120 (2004m:14047)[25] Tony Shaska, Eustrat Zhupa, and Lubjana Beshaj, The case for superelliptic curves (201502),available at .[26] Stephan Wesemeyer, On the automorphism group of various Goppa codes , IEEE Trans. In-form. Theory (1998), no. 2, 630–643. MR1607734 (99m:94058)(1998), no. 2, 630–643. MR1607734 (99m:94058)