ONE-POINT CODES USING PLACES OF HIGHER DEGREE

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1 ONE-POINT CODES USING PLACES OF HIGHER DEGREE GRETCHEN L. MATTHEWS AND TODD W. MICHEL DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC U.S.A. Abstact. In (IEEE Tans. Infom. Theoy 48 no. 2 (2002), ), Xing and Chen show that thee exist algebaic geomety codes fom the Hemitian function field ove F q 2 constucted using F q 2-ational divisos which ae impovements ove the much-studied one-point Hemitian codes. In this pape, we constuct such codes by using a place P of degee > 1. This motivates a study of gap numbes and pole numbes at places of highe degee. In fact, the code paametes ae estimated using the Weiestass gap set of the place P and elating it to the gap set of the -tuple of places of degee one lying ove P in a constant field extension of degee. 1. Intoduction An algebaic geomety (AG) code ove F q is defined using two divisos G and D of a function field F/F q. Typically, the diviso G is taken to be a multiple of a single place of F of degee one and D := Q 1 + +Q n is suppoted by n distinct places of degee one diffeent fom the place in the suppot of G. Such a code is called a onepoint code. It has been shown that bette AG codes may be obtained by allowing the diviso G to be moe geneal (see [6], [4], [9], [2]). In paticula, Xing and Chen have shown that thee exist F q 2-ational divisos G of the Hemitian function field ove F q 2 such that C L (D, G) has bette paametes than the compaable one-point Hemitian code [9]. In this pape, we conside the special case whee G = αp and P is a place of F of degee geate than one. Such codes may be thought of as one-point codes defined using places of highe degee. This constuction diffes fom that of genealized AG codes, o XNL codes, poposed by Niedeeite, Xing, and Lam [10] in which the diviso D, athe than G, may be suppoted by places of highe degee. To study one-point codes constucted using places of highe degee, we fist conside the Weiestass gap set of a place of highe degee. In doing so, it is helpful to examine the Weiestass gap set of an -tuple of places of degee one in a constant field extension. This allows one to use theoy that has been developed in applications of Weiestass gap sets of -tuples of places to codes [4], [2]. As a esult, we obtain explicit constuctions fo one-point codes using places of highe degee that have bette paametes than the compaable classical one-point code fom the same function field. Moeove, we define specific codes with paametes compaable to (and, at times, bette than) those found in [9]. This poject was suppoted by NSF DMS

2 2 GRETCHEN L. MATTHEWS AND TODD W. MICHEL This pape is oganized as follows. Fist, we eview the notation used thoughout the pape. In Section 2, the Weiestass gap set of a place of highe degee is discussed. Constant field extensions ae utilized to bette undestand this set. In Section 3, applications to codes ae consideed. Finally, examples ae given to illustate these methods. Notation. Unless stated othewise, we will use notation as in [8]. Let F/F q be an algebaic function field of genus g > 1. The diviso (esp. pole diviso) of a function f F will be denoted by (f) F (esp. (f) F ), o simply (f) (esp. (f) ) if the context is clea. Let Ω denote the set of ational diffeentials of F/F q. The diviso of a diffeential η Ω will be denoted by (η) and the esidue of η at a place P will be denoted by es P (η). Given a diviso A of F, let L(A) := {f F \ {0} : (f) A} {0} and Ω(A) := {η Ω \ {0} : (η) A} {0}. Let l(a) denote the dimension of the vecto space L(A) ove F q. The Riemann- Roch Theoem states that l(a) = deg A + 1 g + l(w A) whee W is any canonical diviso of F. Moeove, if the degee of A is at least 2g 1, then l(w A) = 0 and so l(a) = deg A + 1 g. As usual, a code of length n, dimension k, and minimum distance d (esp. at least d) is called an [n, k, d] (esp. [n, k, d]) code. We sometimes wite d(c) to mean the minimum distance of the code C. The set of positive integes is denoted by N and the set of nonnegative integes is denoted by N 0. Let G be a diviso of F/F q and let D = Q 1 + +Q n be anothe diviso of F/F q whee Q 1,..., Q n ae distinct places of F of degee one, each not contained in the suppot of G. The algebaic geomety (AG) code C Ω (D, G) is defined by C Ω (D, G) := {(es Q1 (η),..., es Qn (η)) : η Ω(G D)} and is an [n, l(g D) l(g) + deg D, deg G (2g 2)]-code. 2. Gaps at places of highe degee Let F/F q be an algebaic function field of genus g > 1, and let P be a place of F of degee. Define the Weiestass semigoup of the place P by H(P ) := {α N 0 : f F with (f) = αp } and the Weiestass gap set of the place P by G(P ) := N 0 \ H(P ). Elements of the set G(P ) ae often efeed to as gaps at the place P. One can easily check that the set H(P ) is an additive submonoid of N 0. It is also easy to see that given α N 0, α H(P ) if and only if l(αp ) l((α 1)P ). Recall that the Weiestass Gap Theoem states that given a place P of F of degee one, thee ae exactly g gaps at P and each element of the Weiestass gap set lies in the inteval [1, ]. In the next two popositions, we conside analogous esults fo places of degee possibly geate than one. Poposition 2.1. Let P be a place of F/F q of degee. If α >, then α H(P ). Moeove, G(P ) [1, ].

3 ONE-POINT CODES USING PLACES OF HIGHER DEGREE 3 Poof. Set s = (2g 1) and suppose α > = +s. Then α > (α 1) 2g 1 + s, which implies l(αp ) > l((α 1)P ) by the Riemann- Roch Theoem. Hence, α H(P ). Clealy, 0 H(P ) as (a) = 0P fo a F q. Thus, G(P ) [1, ]. It emains to show that if 2g 1 then H(P ). Suppose α = = +s whee > 1. Note that 0 < s 1. Then α > 2g 1 which implies l(αp ) = α + 1 g. By Cliffod s Theoem, l(w (α 1)P ) ( s 1) < ( 1) fo any canonical diviso W. It follows that l(αp ) l((α 1)P ) = α + 1 g (α 1) 1 + g l(w (α 1)P ) l(w (α 1)P ) > 1 2 ( 1) > 0 as > 1. Theefoe, α H(P ) and so G(P ) [1, Poposition 2.2. Let P be a place of F/F q of degee. Then g = l(ip ) l((i 1)P ) s whee s = (2g 1). i=0 Poof. By the Riemann-Roch Theoem, l ( 1] = [1, ]. P ) = g + s. Then ( 2g 1 0 = l( 1P ) l(0p ) l(p ) l(2p ) l Hence, g + s = l ( ) P = g + s. ) P l( 1P ) = i=0 l(ip ) l((i 1)P ). Notice that if one takes = 1 in the above poposition, then this shows that the numbe of gaps at P is the genus of F, as s = 0 and 0 l(ip ) l((i 1)P ) 1. Howeve, if P is a place of degee > 1, then it is not necessaily the case that l(ip ) l((i 1)P ) {0, } fo each i N. Lewittes [5] has shown that g = (l(ip ) l((i 1)P )). Fom the poof of Poposition 2.1, we see that (l(ip ) l((i 1)P )) = 0 fo all i >. Thus, Lewittes esult can be impoved to give g = (l(ip ) l((i 1)P )). Next, we show how the Weiestass semigoup of an -tuple of places of degee one in a constant field extension may be used to study that of a place of degee. Let F := F q F/F q be a constant field extension of F/F q of degee. Then the place P splits completely in F. Hence, thee ae distinct places P 1,..., P P F of degee one lying ove the place P : F := F q F P 1,..., P F P By definition, the conom of αp is Con F /F (αp ) = αp αp fo all α N. Accoding to [8, Theoem III.6.3], l(con F /F (αp )) = l(αp )

4 4 GRETCHEN L. MATTHEWS AND TODD W. MICHEL fo all α N. Hence, (1) l(αp ) l((α 1)P ) = l(αp αp ) l((α 1)P (α 1)P ). It is vey natual to elate the Weiestass semigoup H(P ) to that of the - tuple (P 1,..., P ). Given m distinct places Q 1,..., Q m of degee one of F, the Weiestass semigoup H(Q 1,..., Q m ) of the m-tuple (Q 1,..., Q m ) is defined by { } m H(Q 1,..., Q m ) = α N m 0 : f F with (f) = α i Q i, and the Weiestass gap set G(Q 1,..., Q m ) of the m-tuple (Q 1,..., Q m ) is defined by G(Q 1,..., Q m ) = N m 0 \ H(Q 1,... Q m ). Poposition 2.3. Let P be a place of degee of a function field F/F q and P 1,..., P be the extensions of P in the constant field extension F of F of degee. Given α N, α H(P ) if and only if (α,..., α) H(P 1,..., P ). Poof. Suppose α H(P ). Then thee is a function f F with diviso (f) F = A αp whee P / supp A. By [8, Poposition III.1.9], (f) F = Con F /F (A) αp 1 αp which implies (α,..., α) H(P 1,..., P ). Suppose (α,..., α) H(P 1,..., P ). Then l(αp αp ) = l(αp (α 1)P i + + αp ) + 1 fo all 1 i. Fom (1), it follows that l(αp ) l((α 1)P ). Consequently, α H(P ). Fom the above esult, α G(P ) if and only if (α,..., α) G(P 1,..., P ). It tuns out that much moe is tue. Notice that α N is an element of the Weiestass gap set G(P 1,..., P ) if and only if thee exists j, 1 j, such that ( ) (2) l α i P i = l (α j 1)P j + α i P i.,i j In [4] and late in [2], the authos conside those elements of the Weiestass gap set G(P 1,..., P ) with all possible symmety. Moe pecisely, they conside α N satisfying (2) fo all j, 1 j. Such elements of the Weiestass gap set ae called pue gaps. The set of pue gaps of the -tuple (P 1,..., P ) is denoted by G 0 (P 1,..., P ). In [2, Lemma 2.5], it is shown that α G 0 (P 1,..., P ) if and only if l ( α ip i ) = l ( (α i 1)P i ). Poposition 2.4. Let P be a place of degee of a function field F/F q and P 1,..., P be the extensions of P in the constant field extension of F of degee. Suppose that α,..., α + t G(P ). Then [α, α + t] G 0 (P 1,..., P ). Poof. Suppose thee exists v [α, α + t] such that v / G 0 (P 1,..., P ). Then ( ) l v i P i l (v j 1)P j + v i P i,i j

5 ONE-POINT CODES USING PLACES OF HIGHER DEGREE 5 fo some j, 1 j. Then l((α 1)P ) = l ( (α 1)P i) l (v j 1)P j + ),i j v ip i < l ( v ip i ) l ( (α + t)p i) = l((α + t)p ) which is a contadiction as α,..., α + t G(P ) implies that l((α 1)P ) = l(αp ) = = l((α + t)p ). Example 2.5. Conside the function field F := F q (x, y)/f q whee y q + y = x q+1. Let P be a place of F of degee two. We claim that the Weiestass semigoup of the place P is H(P ) = q 1, q, q + 1 { t whee a 1,..., a t := c ia i : c i N 0 }. Thee ae exactly two places of degee one, P 1 and P 2, in the constant field extension F := F q 2F lying ove P. Note that F /F q 2 = F q 2(x, y)/f q 2 is the Hemitian function field ove F q 2. It is well known that the Weiestass semigoup of any place of F /F q 2 of degee one is q, q + 1. In [6, Theoem 3.7], the Weiestass semigoup of any pai of places of F /F q 2 of degee one is detemined. Next, we use these two facts to find H(P ). Clealy, accoding to Poposition 2.3, H(P ) = {α N 0 : (α, α) H(P 1, P 2 )} = {α H(P 1 ) : (α, α) H(P 1, P 2 )} {α G(P 1 ) : (α, α) H(P 1, P 2 )} = q, q + 1 {α G(P 1 ) : (α, α) H(P 1, P 2 )}. By [6, Theoem 3.7], α G(P 1 ) and (α, α) H(P 1, P 2 ) implies α q 1. Thus, H(P ) = q, q + 1 q 1 q 1, q, q + 1. Now suppose that α = a(q 1) + bq + c(q + 1) with a, b, c N 0. By [6, Theoem 3.4] and Poposition 2.3, a(q 1) H(P ). Since H(P 1 ) = H(P 2 ) = q, q + 1, Poposition 2.3 implies that bq + c(q + 1) H(P ). It follows that α H(P ) as H(P ) is closed unde addition. Theefoe, H(P ) = q 1, q, q One point codes using places of highe degee In this section, we conside AG codes C Ω (D, αp ) ove F q whee P is a place of F/F q of degee geate than one. The next lemma illustates how a code of this fom elates to the code C Ω (Con F /F (D), Con F /F (αp )). Lemma 3.1. Let G and D := Q Q n be divisos of F/F q whee Q 1,..., Q n ae distinct places of degee one of F, none of which ae contained in the suppot of G. Set := max {deg P : P supp G}. Let F := F F q /F q be a constant field extension of F/F q of degee. Then the two codes C Ω (D, G) and C Ω (Con F /F (D), Con F /F (G)) have the same length. The dimension of C Ω (D, G) (ove F q ) is equal to that of C Ω (Con F /F (D), Con F /F (G)) (ove F q ). The minimum distance of C Ω (D, G) is at least that of C Ω (Con F /F (D), Con F /F (G)).

6 6 GRETCHEN L. MATTHEWS AND TODD W. MICHEL Poof. Clealy, Con F /F (D) = Q Q n as each Q i has degee one, and the dimension of C Ω (D, G) is l(g D) l(g)+deg D = l(con F /F (G) Con F /F (D)) l(con F /F (G))+deg Con F /F (D), the dimension of C Ω (Con F /F (D), Con F /F (G)). Let d denote the minimum distance of C Ω (D, G). Suppose that η Ω(G D) and that the weight of (es Q1 (η),..., es Qn (η)) is equal to d. Without loss of geneality, we may assume that (η) G (Q Q d ). Then Con F /F ((η)) is a canonical diviso of F /F q and Con F /F ((η)) Con F /F (G) Con F /F (Q 1 + +Q d ) which implies that C Ω (Con F /F (D), Con F /F (G)) has a codewod of weight d. The pue gap set of a pai of places of degee one is used to define codes with minimum distance geate than the usual lowe bound in [4]. This is genealized to -tuples of places of degee one in [2]. These esults togethe with those in Section 2 will be applied to obtain bette bounds on the minimum distance of codes defined using elements of the gap set of a place of highe degee. Fo convenience, we include hee the two esults fom [2] that we will use. Lemma 3.2. [2, Theoem 3.3] Let Q 1,..., Q n, P 1,..., P m be distinct places of F/F q such that deg Q i = 1 fo each i, 1 i n. Set D := Q Q n. Suppose G := m (a i + b i 1)P i whee a, b G 0 (P 1,..., P m ). If the code C Ω (D, G ) is nontivial, then it has minimum distance at least deg G (2g 2) + m. Lemma 3.3. [2, Theoem 3.4] Let Q 1,..., Q n, P 1,..., P m be distinct places of F/F q such that deg Q i = 1 fo each i, 1 i n. Set D := Q Q n. Suppose G := m (a i + b i 1)P i whee v G 0 (P 1,..., P m ) fo all v N m 0 such that a v b. If the code C Ω (D, G ) is nontivial, then it has minimum distance at least deg G (2g 2) + m (b i a i + 1). We can modify Lemma 3.2 to get an analog of [3, Theoem 1] fo codes defined using places of highe degee. Theoem 3.4. Let P be a place of degee and Q 1,..., Q n be distinct places of F/F q of degee one such that Q i P fo each i, 1 i n. Set D := Q 1 + +Q n. Suppose G := (α + β 1)P whee α, β G(P ). Then C Ω (D, G) has minimum distance at least deg G (2g 2) +. Poof. Let P 1,..., P be the extensions of P in the degee constant field extension F of F. By Poposition 2.4, (α,..., α), (β,..., β) G 0 (P 1,..., P ) as α, β G(P ). Take G = Con F /F (G) = (α + β 1)P i and D = Con F /F (D) in Lemma 3.2. Then the minimum distance of C Ω (D, G ) satisfies d (C Ω (D, G )) deg G (2g 2) + = deg G (2g 2) +. Fom Lemma 3.1 it follows that the minimum distance of C Ω (D, G) is at least deg G (2g 2) +. Next, we modify [2, Theoem 3.4] to obtain a esult simila to [3, Theoem 4] fo codes defined using places of highe degee. Theoem 3.5. Let P be a place of degee and Q 1,..., Q n be distinct places of F/F q of degee one such that Q i P fo each i, 1 i n. Set D := Q 1 + +Q n. Suppose G := (α + (α + t) 1)P whee α,..., α + t G(P ) with t 0. Then C Ω (D, G) has minimum distance at least deg G (2g 2) + (t + 1).

7 ONE-POINT CODES USING PLACES OF HIGHER DEGREE 7 Poof. Let P 1,..., P be the extensions of P in the degee constant field extension F of F. By Poposition 2.4, [α, α + t] G 0 (P 1,..., P ) as α,..., α + t G(P ). Thus, v G 0 (P 1,..., P ) fo all v such that (α,..., α) v (α + t,..., α + t). Take G = Con F /F (G) = (α+(α+t) 1)P i and D = Con F /F (D) in Lemma 3.3. Then the minimum distance of C Ω (D, G ) satisfies d (C Ω (D, G )) deg G (2g 2) + + t = deg G (2g 2) + + t. Fom Lemma 3.1 it follows that the minimum distance of C Ω (D, G) is at least deg G (2g 2) + (t + 1). Example 3.6. Conside the Hemitian function field F := F 81 (x, y)/f 81 defined by y 9 + y = x 10. Note that F has genus 36 and 730 places of degee one. Using Magma [1], we find that F has a place P of degee 3 with Weiestass gap set G(P ) = {1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 14, 20}. Take α = 14 and β = 20 in the Theoem 3.4. Then G := ( )P = 33P. Let D be the sum of all places of F of degee one othe than P. Then C Ω (D, G) is a [729, 665, 32] code. The one-point code on F of dimension 665 is a [729, 665, 29]- code. The gap set G(P ) may be used to constuct seveal othe codes which have geate minimum distance than the compaable one-point Hemitian code. Example 3.7. Conside the Hemitian function field F := F 49 (x, y)/f 49 defined by y 7 + y = x 8. Note that F has genus 21 and 344 places of degee one. Using Magma [1], we find that F has a place P of degee 3 with Weiestass gap set G(P ) = {1, 2, 3, 4, 5, 9, 10}. Take α = 9 and t = 1 in the Theoem 3.5. Then G := ( )P = 18P. Let D be the sum of all places of F of degee one othe than P. Then C Ω (D, G) is a [343, 309, 20] code. The one-point code on F of dimension 309 is a [343, 309, 14]- code. Note that the existence of an AG code on F with paametes [343, 309, 18] is shown in [9]. Example 3.8. Conside the Hemitian function field F := F 64 (x, y)/f 64 defined by y 8 + y = x 9. Note that F has genus 28 and 513 places of degee one. Using Magma [1], we find that F has a place P of degee 3 with Weiestass gap set G(P ) = {1, 2, 3, 4, 5, 6, 10, 11, 12}. Take α = 10 and t = 2 in the Theoem 3.5. Then G := ( )P = 21P. Let D be the sum of all places of F of degee one othe than P. Then C Ω (D, G) is a [512, 476, 18] code. The one-point code on F of dimension 309 is a [512, 476, 9]- code. Hence, the code constucted using a place of highe degee coects at least twice as many eos as the compaable one-point code fom the same function field. It is woth pointing out that while thee exists a F 64 -ational diviso G of F such that C Ω (D, G ) is a [512, 476, 19]-code [9], it is not clea how to detemine G.

8 8 GRETCHEN L. MATTHEWS AND TODD W. MICHEL Remak 3.9. While Theoem 3.4 and Theoem 3.5 may be thought of as pescibing a method fo constucting one-point codes fom places of highe degee, these esults can also be viewed as specifying a method fo defining -point codes so that the esults of [4], [2] apply. 4. Acknowledgements The authos wish to thank A. Gacia and R. F. Lax fo pointing out the efeence [5]. Some of these esults also appea in the second autho s M. S. thesis [7]. Refeences [1] W. Bosma, J. Cannon, and C. Playoust, The MAGMA algeba system, I: The use language, J. Symb. Comp. 24 (1997), [2] C. Cavalho and F. Toes, On Goppa codes and Weiestass gaps at seveal points, Designs, Codes and Cypt., to appea. [3] A. Gacia, S. J. Kim, and R. F. Lax, Consecutive Weiestass gaps and minimum distance of Goppa codes, J. Pue Appl. Algeba 84 (1993), [4] M. Homma and S. J. Kim, Goppa codes with Weiestass pais, J. Pue Appl. Algeba 162 (2001), [5] J. Lewittes, Genus and gaps in function fields, J. Pue Appl. Algeba 58 (1989), [6] G. L. Matthews, Weiestass pais and minimum distance of Goppa codes, Des. Codes and Cyptog. 22 (2001), [7] T. W. Michel, One-point codes using places of highe degee, M. S. thesis., Clemson Univesity, [8] H. Stichtenoth, Algebaic Function Fields and Codes, Spinge-Velag, [9] C. P. Xing and H. Chen, Impovements on paametes of one-point AG codes fom Hemitian codes, IEEE Tans. Infom. Theoy 48 no. 2 (2002), [10] C. P. Xing, H. Niedeeite, and K. Y. Lam, A genealization of algebaic geomety codes, IEEE Tans. Infom. Theoy 45 (1999),

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},

$ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},$ ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability