arXiv:1907.11324v1 [math.AC] 25 Jul 2019 RELATIVE GENERALIZED MINIMUM DISTANCE FUNCTION MANUEL GONZ ´ ALEZ–SARABIA, M. EDUARDO URIBE–PACZKA, ELISEO SARMIENTO, AND CARLOS RENTER ´ IA Abstract. In this paper we introduce the relative generalized minimum distance function (RGMDF for short) and it allows us to give an algebraic approach to the relative generalized Hamming weights of the projective Reed–Muller–type codes. Also we introduce the relative generalized footprint function and it gives a tight lower bound for the RGMDF which is much easier to compute. 1. Introduction This work is a non–trivial generalization of [24], where the case of an algebraic approach to the minimum distance of a Reed–Muller–type code is treated, and [16], where a similar approach is given for the case of the generalized Hamming weights of these codes. The main goal here is the study of the relative generalized Hamming weights (Definition 2.4) of the Reed–Muller–type codes from an algebraic point of view. In order to do this, we introduce the relative generalized minimum distance function (Definition 2.1) and the relative footprint function (Definition 2.3). The Reed–Muller–type codes and their parameters have been studied extensively. If X is a subset of a projective space P s´1 over a finite field K “ F q , and C X pdq is the correspond- ing Reed–Muller–type code (Definition 2.5), several cases have been described [1], [3], [7], [8],[9],[10],[11],[12],[14],[15],[19], [21], [25], [26], [27], [28], [29], [30], [31]: ‚ Projective Reed–Muller codes: X “ P s´1 . ‚ Generalized Reed–Muller codes: X “ ϕpA s´1 q, where A s´1 is an affine space and ϕ : A s´1 Ñ P s´1 , ϕpa 1 ,...,a s´1 q“r1: a 1 : ¨¨¨ : a s´1 s. ‚ Reed–Muller–type codes arising from the Segre variety or the Veronese variety: X is the set of K–rational points of the variety. ‚ Reed–Muller–type codes arising from a complete intersection: X is such that its defining ideal is a set–theoretic complete intersection. ‚ Codes parameterized by a set of monomials: X is the toric set associated to these mono- mials. ‚ Codes parameterized by the edges of a graph: X is the toric set associated to the edges of a simple graph. ‚ Affine cartesian codes: X is the image of a cartesian product of subsets of K under the map K s´1 Ñ P s´1 , x Ñrx :1s. ‚ Projective cartesian codes: X is the image of the cartesian product A 1 ˆ¨¨¨ˆ A s zt  0u under the map K s zt  0uÑ P s´1 , x Ñrxs, 2010 Mathematics Subject Classification. Primary 13P25; Secondary 14G50, 94B27, 11T71. Key words and phrases. Relative generalized minimum distance function, Relative generalized footprint func- tion, Relative generalized Hamming weights. The first and fourth authors are partially supported by COFAA–IPN and SNI, M´ exico. The second author is partially supported by a scholarship from CONACyT, M´ exico. The third author is partially supported by SNI, M´ exico. 1