Channel coding is the branch of Information Theory which studies the noise that can occur in data transmitted through a channel. Algebraic Coding Theory is the part of Channel Coding which studies the possibility to detect and correct errors using algebraic and geometric techniques. Nowadays, the best performing linear codes are known to be mostly algebraic geometry codes, also named Goppa codes, which arise from an algebraic curve over a finite field, by the pioneering construction due to V. D. Goppa. The best choices for curves on which Goppa's construction and its variants may provide codes with good parameters are those with many rational points, especially maximal curves attaining the Hasse-Weil upper bound for the number of rational points compared with the genus of the curve. Unfortunately, maximal curves are difficult to find. The best known examples of maximal curves are the Hermitian curve, the Ree curve, the Suzuki curve, the GK curve and the GGS curve. In the present thesis, we construct and investigate algebraic geometry codes (shortly AG codes), their parameters and automorphism groups.

Intersections of Algebraic Curves and their link to the weight enumerators of Algebraic-Geometric Codes / Bonini, Matteo. - (2019), pp. 1-110.

### Intersections of Algebraic Curves and their link to the weight enumerators of Algebraic-Geometric Codes

#### Abstract

Channel coding is the branch of Information Theory which studies the noise that can occur in data transmitted through a channel. Algebraic Coding Theory is the part of Channel Coding which studies the possibility to detect and correct errors using algebraic and geometric techniques. Nowadays, the best performing linear codes are known to be mostly algebraic geometry codes, also named Goppa codes, which arise from an algebraic curve over a finite field, by the pioneering construction due to V. D. Goppa. The best choices for curves on which Goppa's construction and its variants may provide codes with good parameters are those with many rational points, especially maximal curves attaining the Hasse-Weil upper bound for the number of rational points compared with the genus of the curve. Unfortunately, maximal curves are difficult to find. The best known examples of maximal curves are the Hermitian curve, the Ree curve, the Suzuki curve, the GK curve and the GGS curve. In the present thesis, we construct and investigate algebraic geometry codes (shortly AG codes), their parameters and automorphism groups.
##### Scheda breve Scheda completa Scheda completa (DC)
2019
31
Mathematics
Sala, Massimiliano
Rinaldo, Giancarlo
Inglese
Settore MAT/02 - Algebra
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Tipologia: Tesi di dottorato (Doctoral Thesis)
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Tipologia: Tesi di dottorato (Doctoral Thesis)
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11572/368573`
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