Abstract:This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce q-polymatroids, the q-analogue of polymatroids, and develop their basic properties. We associate a pair of q-polymatroids to a rank-metric codes and show that several invariants and structural properties of the code, such as generalized weights, the property of being MRD or an optimal anticode, and duality, are captured by the associated combinatorial object.We start by establishing the nota… Show more
“…There is a natural connection between duals of Delsarte codes and the duals of (q, m)polymatroids. It is shown by Shiromoto [19] as well as Gorla et al [10], and we record it below.…”
Section: Duality Of Delsarte Rank Metric Codessupporting
confidence: 68%
“…This proves that P = (E, ρ) is a (q, m)demi-polymatroid. The desired formula for the conullity function of P is immediate from (10). This, in turn, shows that Indeed, the inequality d r (P) ≤ min{d r (P(C)), d r (P(C T ))} is clear from the definition and equation (5).…”
Section: Consequently M-fold Wei Duality As In Theorem 17 Holds For mentioning
confidence: 78%
“…The proof is quite short and natural and given in [19,Proposition 11], and also in [10,Theorem 8.1]. An immediate consequence is the following.…”
Section: Duality Of Delsarte Rank Metric Codesmentioning
confidence: 89%
“…It follows that ρ * (X ) = m dim X − max{dim C(X ), dim C T (X )}. (10) This implies that ρ * (X ) ≤ m dim X . Moreover, it also implies that ρ * (X ) ≥ 0, because from (5) and Proposition 6 we see that both m dim X − dim C(X ) and m dim X − dim C T (X ) are nonnegative.…”
Section: Consequently M-fold Wei Duality As In Theorem 17 Holds For mentioning
confidence: 93%
“…Also, considering flags of length 1, we can deduce the results of Ravagnani [18] for the GW's of Delsarte codes and their duals. We remark that q-analogues of matroids, called q-matroids and q-polymatroids, have been considered by Jurrius and Pellikaan [12] and by Gorla et al [10], respectively. However, as far as we can see, Wei-type duality for their generalized weights is not shown in these papers.…”
We consider the notion of a (q, m)-polymatroid, due to Shiromoto, and the more general notion of (q, m)-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martínez-Peñas and Matsumoto for relative generalized rank weights are derived as a consequence.
“…There is a natural connection between duals of Delsarte codes and the duals of (q, m)polymatroids. It is shown by Shiromoto [19] as well as Gorla et al [10], and we record it below.…”
Section: Duality Of Delsarte Rank Metric Codessupporting
confidence: 68%
“…This proves that P = (E, ρ) is a (q, m)demi-polymatroid. The desired formula for the conullity function of P is immediate from (10). This, in turn, shows that Indeed, the inequality d r (P) ≤ min{d r (P(C)), d r (P(C T ))} is clear from the definition and equation (5).…”
Section: Consequently M-fold Wei Duality As In Theorem 17 Holds For mentioning
confidence: 78%
“…The proof is quite short and natural and given in [19,Proposition 11], and also in [10,Theorem 8.1]. An immediate consequence is the following.…”
Section: Duality Of Delsarte Rank Metric Codesmentioning
confidence: 89%
“…It follows that ρ * (X ) = m dim X − max{dim C(X ), dim C T (X )}. (10) This implies that ρ * (X ) ≤ m dim X . Moreover, it also implies that ρ * (X ) ≥ 0, because from (5) and Proposition 6 we see that both m dim X − dim C(X ) and m dim X − dim C T (X ) are nonnegative.…”
Section: Consequently M-fold Wei Duality As In Theorem 17 Holds For mentioning
confidence: 93%
“…Also, considering flags of length 1, we can deduce the results of Ravagnani [18] for the GW's of Delsarte codes and their duals. We remark that q-analogues of matroids, called q-matroids and q-polymatroids, have been considered by Jurrius and Pellikaan [12] and by Gorla et al [10], respectively. However, as far as we can see, Wei-type duality for their generalized weights is not shown in these papers.…”
We consider the notion of a (q, m)-polymatroid, due to Shiromoto, and the more general notion of (q, m)-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martínez-Peñas and Matsumoto for relative generalized rank weights are derived as a consequence.
A perfect matroid design (PMD) is a matroid whose flats of the same rank all have the same size. In this paper we introduce the q-analogue of a PMD and its properties. In order to do so, we first establish a new cryptomorphic definition for q-matroids. We show that q-Steiner systems are examples of q-PMD’s and we use this q-matroid structure to construct subspace designs from q-Steiner systems. We apply this construction to the only known q-Steiner system, which has parameters S(2, 3, 13; 2), and hence establish the existence of a new subspace design with parameters 2-(13, 4, 5115; 2).
It is well-known that the dimension of optimal anticodes in the rank-metric is divisible by the maximum m between the number of rows and columns of the matrices. Moreover, for a fixed k divisible by m, optimal rank-metric anticodes are the codes with least maximum rank, among those of dimension k. In this paper, we study the family of rank-metric codes whose dimension is not divisible by m and whose maximum rank is the least possible for codes of that dimension, according to the Anticode bound. As these are not optimal anticodes, we call them quasi optimal anticodes (qOACs). In addition, we call dually qOAC a qOAC whose dual is also a qOAC. We describe explicitly the structure of dually qOACs and compute their weight distributions, generalized weights, and associated q-polymatroids.
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