Extremal sizes of subspace partitions

ABSTRACT. Following the approach by R. Kötter and F. R. Kschischang, we study network codes as families of k-dimensional linear subspaces of a vector space F n q , q being a prime power and F q the finite field with q elements. In particular, following an idea in finite projective geometry, we introduce a class of network codes which we call partial spread codes. Partial spread codes naturally generalize spread codes. In this paper we provide an easy description of such codes in terms of matrices, discuss their maximality, and provide an efficient decoding algorithm. INTRODUCTIONThe topology of a network is well-modeled by a directed multigraph. Vertices without incoming edges play the role of sources and vertices without outgoing edges play the role of sinks. Vertices which are neither sources nor sinks are called nodes. The interest in network modeling is due to its several applications in technology (distributed storage, peer-to-peer networking and, in particular, wireless communications).In [1] Ahlswede, Cai, Li, and Yeung discovered that the information rate may be improved by employing coding at the nodes of a network (instead of simply routing). Moreover, Li, Cai and Yeung proved in [14] that, in a multicasting situation, maximal information rate can be achieved by allowing the nodes to transmit linear combinations of the inputs they receive, provided that the size of the base field is large enough.A turning point in the study of linear network codes was the paper [12] by R. Kötter and F. R. Kschischang. The authors suggested an algebraic approach to the topic, developing a clear and rigorous mathematical setup. Interesting connections with classical projective geometry also emerged. Several other interesting papers followed the same approach, e.g., [6], [7], and [13].In this paper, we propose and study a class of network codes, which fit within the same framework. In Section 1 the algebraic approach by Kötter and Kschischang is briefly recalled. In Section 2 we introduce a family of network codes which we call partial spread codes, and which generalize spread codes (see [16]). Our codes have the same cardinality and distance distribution as the codes proposed in [8]. The elements of our codes however are given as rowspaces of appropriate matrices in block form. The structure of this family of matrices allow us to derive properties of the code, which we discuss in Section 3. In particular, we establish the maximality of partial spread codes with respect to containment. Based on the same block matrix structure, in Section 4 we are able to give an efficient decoding algorithm.

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“…An alternative proof of Lemma 8 is given in [8], Theorem 11. For interesting discussions on the sharpness of the bound see [5] and [10].…”

Section: | ≤mentioning

confidence: 99%

Partial spreads in random network coding

Gorla

Ravagnani

2014

Finite Fields and Their Applications

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“…For any integer d such that d 1 < d ≤ d m , the d-supertail of P is the set of subspaces in P of dimension less than d, and it is denoted by ST . The size of a subspace partition P is the number of subspaces in P. For 1 ≤ t < n, let σ q (n, t) denote the minimum size of any subspace partition of V in which the largest subspace has dimension t. The exact value of σ q (n, t) is given by the following theorem (see André [1] and Beutelspacher [3] for n (mod t) ≡ 0, and see [14,20] for n (mod t) ≡ 0). Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

The structure of the minimum size supertail of a subspace partition

NăźStase

Sissokho

2016

Des. Codes Cryptogr.

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Abstract. Let V = V (n, q) denote the vector space of dimension n over the finite field with q elements. A subspace partition P of V is a collection of nontrivial subspaces of V such that each nonzero vector of V is in exactly one subspace of P. For any integer d, the d-supertail of P is the set of subspaces in P of dimension less than d, and it is denoted by ST . Let σ q (n, t) denote the minimum number of subspaces in any subspace partition of V in which the largest subspace has dimension t. It was shown by Heden et al. that |ST | ≥ σ q (d, t), where t is the largest dimension of a subspace in ST . In this paper, we show that if |ST | = σ q (d, t), then the union of all the subspaces in ST constitutes a subspace under certain conditions.

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