Let S be a numerical semigroup. For each s_i in S , let ni(s_i) denote the number of pairs (t,u) such that t+u=s_i; it is well-known that there exists an integer m such that the sequence ni(s_i) is non-decreasing for i > m. The problem of finding m is solved only in special cases. By way of a suitable parameter t, we improve the known bounds for m and in several cases we determine m explicitly. In particular we give the value of m when the Cohen Macaulay type of the semigroup is three or when the multiplicity is less than or equal to six. When S is the Weierstrass semigroup of a family {C_i} of one-point algebraic geometry codes, these results give better estimates for the order bound on the minimum distance of the codes {C_i}.
On the order bound of one point algebraic geometry codes
ONETO, ANNA;TAMONE, GRAZIA
2009-01-01
Abstract
Let S be a numerical semigroup. For each s_i in S , let ni(s_i) denote the number of pairs (t,u) such that t+u=s_i; it is well-known that there exists an integer m such that the sequence ni(s_i) is non-decreasing for i > m. The problem of finding m is solved only in special cases. By way of a suitable parameter t, we improve the known bounds for m and in several cases we determine m explicitly. In particular we give the value of m when the Cohen Macaulay type of the semigroup is three or when the multiplicity is less than or equal to six. When S is the Weierstrass semigroup of a family {C_i} of one-point algebraic geometry codes, these results give better estimates for the order bound on the minimum distance of the codes {C_i}.
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