Towards the Complete Determination of Next-to-Minimal Weights of Projective Reed-Muller Codes
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Towards the Complete Determination of Next-to-Minimal Weights of Projective Reed-Muller Codes Cícero Carvalho1
· Victor G. L. Neumann1
Received: 19 June 2020 / Revised: 14 October 2020 / Accepted: 4 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Projective Reed-Muller codes are obtained by evaluating homogeneous polynomials of degree d in Fq [X 0 , . . . , X n ] on the points of a projective space of dimension n defined over a finite field Fq . They were introduced by Lachaud, in 1986, and their minimum distance was determined by Serre and Sørensen. As for the higher Hamming weights, contributions were made by Rodier, Sboui, Ballet and Rolland, mostly for the case where d < q. In 2016 we succeeded in determining all next-to-minimal weights when q = 2, and in 2018 we determined all next-to-minimal weights for q = 3, and almost all of these weights for the case where q ≥ 4. In the present paper we determine some of the missing next-to-minimal weights of projective Reed-Muller codes when q ≥ 4. Our proofs combine results of geometric nature with techniques from Gröbner basis theory. Keywords Projective Reed–Muller codes · Next-to-minimal weights · Higher Hamming weights · Evaluation codes Mathematics Subject Classification 94B05 · 94B27 · 11T71 · 14G50
1 Introduction Let Fq be a finite field with q elements, and denote by Pn (Fq ) the projective space of dimension n defined over Fq . It is known that the (homogeneous) ideal J of all polynomials in q q Fq [X] = Fq [X 0 , X 1 , . . . , X n ] which vanish on Pn (Fq ) is generated by {X j X i − X i X j | 0 ≤ i < j ≤ n} (see e.g. [12] or [14]). We write Fq [X] and J with the usual grading as
Communicated by G. Korchmaros. Cícero Carvalho and Victor Neumann were partially supported by Grants from CNPq and FAPEMIG.
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Cícero Carvalho [email protected] Victor G. L. Neumann [email protected]
1
Faculdade de Matemática, Universidade Federal de Uberlândia, Av. J. N. Ávila 2121, Uberlândia, MG 38.408-902, Brazil
123
C. Carvalho, V.G.L. Neumann (2)
Table 1 Values for WPRM (n, d) after [5] (2)
(2)
n
k
WGRM (n, d − 1)
WPRM (n, d)
n=2
k=0
=1
q2
q2
=1
q n−k
q n−k − q n−k−2 Unknown (q − 1)(q − + 1)q n−k−2
n≥3
k WPRM (n, d) (2) then | f | ≥ WGRM (n, d − 1). (1) (2) Proposition 2.5 If 1 + q1 WPRM (n, d) ≤ WGRM (n, d − 1), then 1+
1 q
(1)
(2)
(2)
WPRM (n, d) ≤ WPRM (n, d) ≤ WGRM (n, d − 1) ,
(2) (1) whereas if WGRM (n, d − 1) < 1 + q1 WPRM (n, d), then (2)
(2)
WPRM (n, d) = WGRM (n, d − 1) . Proof Let f ∈ Fq [X 0 , . . . , X n ]d be a nonzero polynomial such that 1 (1) WPRM (n, d). |f| < 1+ q Then, from Lemma 2.3, the support of f satisfies the hypotheses of Proposition 2.1. Hence there exists a hyperplane H ⊂ Pn (Fq ) which does not intersect the support of (1) (2) f , and from Lemma 2.4 we get | f | = WPRM (n, d) or | f | ≥ WGRM (n, d − 1). Thus if (1)
(2)
(1)
WPRM (n, d) ≤ WGRM (n, d − 1) we get | f | = WPRM (n, d), and recalling that (2) (2) (1) WPRM (n, d) ≤ WGRM (n, d −1)
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