Grassmannian Codes with New Distance Measures for Network Coding

Abstract

Grassmannian codes are known to be useful in error-correction for random network coding. Recently, they were used to prove that vector network codes outperform scalar linear network codes, on multicast networks, with respect to the alphabet size. The multicast networks which were used for this purpose are generalized combination networks. In both the scalar and the vector network coding solutions, the subspace distance is used as the distance measure for the codes which solve the network coding problem in the generalized combination networks. In this work we show that the subspace distance can be replaced with two other possible distance measures which generalize the subspace distance. These two distance measures are shown to be equivalent under an orthogonal transformation. It is proved that the Grassmannian codes with the new distance measures generalize the Grassmannian codes with the subspace distance and the subspace designs with the strength of the design. Furthermore, optimal Grassmannian codes with the new distance measures have minimal requirements for network coding solutions of some generalized combination networks. The coding problems related to these two distance measures, especially with respect to network coding, are discussed. Finally, by using these new concepts it is proved that codes in the Hamming scheme form a subfamily of the Grassmannian codes.

On the Gap between Scalar and Vector Solutions of Generalized Combination Networks

“…upper and lower bounds on the maximal number of nodes in the middle layer of such networks depending on the other parameters of the network. Our new upper bounds are better than a previous bound from [8] in some parameter range and the lower bounds cover a wide range of network parameters. We then convert these bounds to bounds on the minimal alphabet size for a linear solution for many networks.…”

“…Definition 1 (Covering Grassmannian Codes [8]). An α-(n, k, δ) c q covering Grassmannian code C is a subset of G(n, k) such that each subset with α codewords of C spans a subspace whose dimension is at least δ + k in F n q .…”

“…An α-(n, k, δ) c q covering Grassmannian code C is a subset of G(n, k) such that each subset with α codewords of C spans a subspace whose dimension is at least δ + k in F n q . The following theorem from [8] shows the connection between covering Grassmannian codes and linear network coding solutions.…”

Subspace packings: constructions and bounds

“…Various upper bounds for A q (n, k, t; λ) are considered in Section 3. The classic bounds which were obtained in [33] will be revisited as well as other generalizations of the bounds for λ = 1 and also some new upper bounds. In Section 4 some more constructions to obtain lower bounds on A q (n, k, t; λ) will be considered.…”

2. q-analogs of group divisible designs

“…Quite surprisingly, it turns out that q-analogs of group divisible designs have interesting connections to scattered subspaces which are central objects in finite geometry, as well as to coding theory via q r -divisible projective sets. We will also discuss the connection to q-Steiner systems [BEÖ + 16] and to packing designs [EZ18].…”