Minimum distance functions of graded ideals and Reed-Muller-type codes

详细信息    查看全文

• 作者：José ; Martí ; nez-Bernal ^{jmb@math.cinvestav.mx" class="auth_mail" title="E-mail the corresponding author} ; Yuriko Pitones ^{ypitones@math.cinvestav.mx" class="auth_mail" title="E-mail the corresponding author} ; Rafael H. Villarreal ; ^{vila@math.cinvestav.mx" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Primary ; 13P25 ; secondary ; 14G50 ; 94B27 ; 11T71
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：221
• 期：2
• 页码：251-275
• 全文大小：552 K

文摘

We introduce and study the minimum distance function of a graded ideal in a polynomial ring with coefficients in a field, and show that it generalizes the minimum distance of projective Reed–Muller-type codes over finite fields. This gives an algebraic formulation of the minimum distance of a projective Reed–Muller-type code in terms of the algebraic invariants and structure of the underlying vanishing ideal. Then we give a method, based on Gröbner bases and Hilbert functions, to find lower bounds for the minimum distance of certain Reed–Muller-type codes. Finally we show explicit upper bounds for the number of zeros of polynomials in a projective nested cartesian set and give some support to a conjecture of Carvalho, Lopez-Neumann and López.