Numerical semigroups generated by intervals

Analysis of the Berlekamp-Massey-Sakata algorithm for decoding onepoint codes leads to two methods for improving code rate. One method, due to Feng and Rao, removes parity checks that may be recovered by their majority voting algorithm. The second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic. In this work, formulae are given for the redundancies of Hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance. The results proceed from an analysis of numerical semigroups generated by two consecutive integers. The formula for the redundancy of optimal Hermitian codes correcting a given number of errors answers an open question stated by Pellikaan and Torres in 1999.

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“…, (a − 2)a + a − 2} ⊔ {i : i (a − 1)a}. The second one was proved in [8] and it is given in the next lemma.…”

Section: Semigroups Generated By Two Consecutive Integersmentioning

confidence: 83%

On Semigroups Generated by Two Consecutive Integers and Improved Hermitian Codes

Bras-Amorós¹,

O’Sullivan

2007

IEEE Trans. Inform. Theory

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“…, a + x − 1}. These semigroups can be found, for instance, in [18]. Also, the Feng-Rao numbers of such semigroups are studied in [11].…”

Section: B a Generalization: Semigroups Generated By Intervalsmentioning

confidence: 99%

“…. }, with D(14) = {0, 4, 5, 8, 10, 12, 15, 16, 20}, d = 9, d + 2g − 1 = 20; {0,4,5,8,9,10,13,14,15,18,19, 23}, d = 12, d + 2g − 1 = 23; and Λ \ D(i) for all i > 17. In this last case, D(i) = {0, 4, 5, 8, 9, 10, 12, 13, .…”

mentioning

confidence: 99%

New Lower Bounds on the Generalized Hamming Weights of AG Codes

Bras-Amorós

Lee

Vico-Oton

2014

IEEE Trans. Inform. Theory

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Abstract-A sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized. Then the result is used, through the so-called Feng-Rao numbers, to bound the generalized Hamming weights of algebraic-geometry codes. This is further developed for Hermitian codes and the codes on one of the Garcia-Stichtenoth towers, as well as for some more general families.

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“…Let a and b be two nonnegative integers such that A ¼ /a; a þ 1; y; a þ bS: By [1] we know that xAA if and only if x mod apI x a mb; whence xAA if and only if x mod ap xÀðx mod aÞ a b: Thus xAA if and only if ða þ bÞx mod aða þ bÞpbx: Therefore,…”

Section: Numerical Semigroups That Are Proportionally Modularmentioning

confidence: 99%