Equidistant subspace codes

In this paper we study equidistant subspace codes, i.e. subspace codes with the property that each two distinct codewords have the same distance. We provide an almost complete classification of such codes under the assumption that the cardinality of the ground field is large enough. More precisely, we prove that for most values of the parameters, an equidistant code of maximum cardinality is either a sunflower or the orthogonal of a sunflower. We also study equidistant codes with extremal parameters, and establish general properties of equidistant codes that are not sunflowers. Finally, we propose a systematic construction of equidistant codes based on our previous construction of partial spread codes, and provide an efficient decoding algorithm. Introduction Network coding is a branch of information theory concerned with data transmission over noisy and lossy networks. A network is modeled by a directed acyclic multigraph, and information travels from one or multiple sources to multiple receivers through intermediate nodes. Network coding has several applications, e.g. peer-to-peer networking, distributed storage and patches distribution. In [1] it was proved that the information rate of a network communication may be improved employing coding at the nodes of the network, instead of simply routing the received inputs. In [16] it was shown that maximal information rate can be achieved in the multicast situation by allowing the intermediate nodes to perform linear combination of the inputs they receive, provided that the cardinality of the ground field is sufficiently large. Random linear network coding was introduced in [13], and a mathematical approach was proposed in [14] and [15], together with the definition of subspace code. In this paper we study equidistant subspace codes, i.e., subspace codes with the property that the intersection of any pair of codewords has the same dimension. Equidistant subspace codes were shown to have relevant applications in distributed storage in [20]. In the same 2010 Mathematics subject classification: 11T71, 14G50, 94B60, 51E23, 15A21.