Evaluation codes and their basic parameters
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Evaluation codes and their basic parameters Delio Jaramillo1 · Maria Vaz Pinto2 · Rafael H. Villarreal1 Received: 27 May 2020 / Revised: 14 August 2020 / Accepted: 3 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The aim of this work is to give degree formulas for the generalized Hamming weights of evaluation codes and to show lower bounds for these weights. In particular, we give degree formulas for the generalized Hamming weights of Reed–Muller-type codes, and we determine the minimum distance of toric codes over hypersimplices, and the 1st and 2nd generalized Hamming weights of squarefree evaluation codes. Keywords Evaluation codes · Toric codes · Minimum distance · Affine torus · Footprint · Degree · Reed–Muller codes · Generalized Hamming weights · Affine variety · Finite field · Gröbner bases Mathematics Subject Classification Primary 13P25 · Secondary 14G50 · 94B27 · 11T71
1 Introduction ∞ Let S = K [t1 , . . . , ts ] = d=0 Sd be a polynomial ring over a finite field K = Fq with the standard grading and let X = {P1 , . . . , Pm } be a set of distinct points in the affine space
Communicated by G. Korchmaros. Delio Jaramillo was supported by a scholarship from CONACYT, Mexico. Maria Vaz Pinto was partially supported by the Center for Mathematical Analysis, Geometry and Dynamical Systems of Instituto Superior Técnico, Universidade de Lisboa. Rafael H. Villarreal was supported by SNI, Mexico.
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Rafael H. Villarreal [email protected] Delio Jaramillo [email protected] Maria Vaz Pinto [email protected]
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Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14–740, 07000 Mexico City, Mexico
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Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1, 1049-001 Lisbon, Portugal
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As := K s . The evaluation map, denoted ev, is the K -linear map given by ev : S → K m ,
f → ( f (P1 ), . . . , f (Pm )) .
The kernel of ev, denoted I (X ), is the vanishing ideal of X consisting of the polynomials of S that vanish at all points of X . This map induces an isomorphism of K -linear spaces between S/I (X ) and K m . Let L be a linear subspace of S of finite dimension. The image of L under the evaluation map, denoted L X , is called an evaluation code on X [53,55]. Let ≺ be a monomial order on S [11, p. 54] and let I = I (X ) be the vanishing ideal of X . The monomials of S are denoted t a := t1a1 · · · tsas , a = (a1 , . . . , as ) in Ns , where N = {0, 1, . . .}. We denote the initial monomial of a non-zero polynomial f ∈ S by in≺ ( f ) and the initial ideal of I by in≺ (I ). A monomial t a is called a standard monomial of S/I , with respect to ≺, if t a ∈ / in≺ (I ). The footprint of S/I , denoted ≺ (I ), is the finite set of all standard monomials of S/I . The footprint has been used in connection with many kinds of codes [13–15,25]. The linear code L X is called a standard evaluation code on X relative to ≺ if L is a
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