On the Weight Hierarchy of Codes Coming From Semigroups With Two Generators
The weight hierarchy of one-point algebraic geometry codes can be estimated by means of the generalized order bounds, which are described in terms of a certain Weierstrass semigroup. The asymptotical behavior of such bounds for r ≥ 2 differs from that of the classical Feng-Rao distance (r=1) by the...
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| creator | Delgado, Manuel Farran, Jose I. Garcia-Sanchez, Pedro A. Llena, David |
| description | The weight hierarchy of one-point algebraic geometry codes can be estimated by means of the generalized order bounds, which are described in terms of a certain Weierstrass semigroup. The asymptotical behavior of such bounds for r ≥ 2 differs from that of the classical Feng-Rao distance (r=1) by the so-called Feng-Rao numbers. This paper is addressed to compute the Feng-Rao numbers for numerical semigroups of embedding dimension two (with two generators), obtaining a closed simple formula for the general case by using numerical semigroup techniques. These involve the computation of the Apéry set with respect to an integer of the semigroups under consideration. The formula obtained is applied to lower bounding the generalized Hamming weights, improving the bound given by Kirfel and Pellikaan in terms of the classical Feng-Rao distance. We also compare our bound with a modification of the Griesmer bound, improving this one in many cases. |
| doi_str_mv | 10.1109/TIT.2013.2285217 |
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The asymptotical behavior of such bounds for r ≥ 2 differs from that of the classical Feng-Rao distance (r=1) by the so-called Feng-Rao numbers. This paper is addressed to compute the Feng-Rao numbers for numerical semigroups of embedding dimension two (with two generators), obtaining a closed simple formula for the general case by using numerical semigroup techniques. These involve the computation of the Apéry set with respect to an integer of the semigroups under consideration. The formula obtained is applied to lower bounding the generalized Hamming weights, improving the bound given by Kirfel and Pellikaan in terms of the classical Feng-Rao distance. 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The asymptotical behavior of such bounds for r ≥ 2 differs from that of the classical Feng-Rao distance (r=1) by the so-called Feng-Rao numbers. This paper is addressed to compute the Feng-Rao numbers for numerical semigroups of embedding dimension two (with two generators), obtaining a closed simple formula for the general case by using numerical semigroup techniques. These involve the computation of the Apéry set with respect to an integer of the semigroups under consideration. The formula obtained is applied to lower bounding the generalized Hamming weights, improving the bound given by Kirfel and Pellikaan in terms of the classical Feng-Rao distance. We also compare our bound with a modification of the Griesmer bound, improving this one in many cases.</description><subject>AG codes</subject><subject>Algebra</subject><subject>Applied sciences</subject><subject>Arrays</subject><subject>Coding, codes</subject><subject>Conductors</subject><subject>Decoding</subject><subject>Electrical engineering</subject><subject>Erbium</subject><subject>Exact sciences and technology</subject><subject>Feng-Rao numbers</subject><subject>Frequency modulation</subject><subject>Generators</subject><subject>Goppa-like bounds</subject><subject>Hamming weight</subject><subject>Information theory</subject><subject>Information, signal and communications theory</subject><subject>numerical semigroups</subject><subject>order bounds</subject><subject>Signal and communications theory</subject><subject>Telecommunications and information theory</subject><subject>weight hierarchy</subject><issn>0018-9448</issn><issn>1557-9654</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2014</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNo9kE1Lw0AQhhdRsFbvgpcF8Zi6n9ndoxT7gYUejPQYtptJk9Jk626K9N-b0tLLDMM87ww8CD1TMqKUmPdsno0YoXzEmJaMqhs0oFKqxKRS3KIBIVQnRgh9jx5i3PajkJQN0NeyxV0FeAX1purwrIZgg6uO2Jd47AuIfW3qdoMnwTf4G5p6E_xhH_Gq7iqc_Xk8hbbPdD7ER3RX2l2Ep0sfop_JZzaeJYvldD7-WCSOp7xLTOEcAxBgrJJKQaqlWnNga1MWlheaM-0Ic4pS6ZwQyjhrOdVMpKUuU2X5EL2e7-6D_z1A7PKtP4S2f5lToYTWQkrSU-RMueBjDFDm-1A3NhxzSvKTsrxXlp-U5RdlfeTtcthGZ3dlsK2r4zXHNBVGMdZzL2euBoDrOk2l1Frzf2vJc00</recordid><startdate>201401</startdate><enddate>201401</enddate><creator>Delgado, Manuel</creator><creator>Farran, Jose I.</creator><creator>Garcia-Sanchez, Pedro A.</creator><creator>Llena, David</creator><general>IEEE</general><general>Institute of Electrical and Electronics Engineers</general><general>The Institute of Electrical and Electronics Engineers, Inc. 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| subjects | AG codes Algebra Applied sciences Arrays Coding, codes Conductors Decoding Electrical engineering Erbium Exact sciences and technology Feng-Rao numbers Frequency modulation Generators Goppa-like bounds Hamming weight Information theory Information, signal and communications theory numerical semigroups order bounds Signal and communications theory Telecommunications and information theory weight hierarchy |
| title | On the Weight Hierarchy of Codes Coming From Semigroups With Two Generators |
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