Galois geometries and coding theory

Etzion, Tuvi; Storme, Leo

doi:10.1007/s10623-015-0156-5

Cited by 73 publications

(57 citation statements)

References 130 publications

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“…The inductive step follows since P j+1 is a subspace partition of H j+1 ∼ = V (n − j − 1, q) of the type given in (13), which satisfies the conditions in (14) and (12).…”

Section: Proof Of Theoremmentioning

confidence: 99%

The maximum size of a partial spread in a finite projective space

Nastase

Sissokho

2017

Journal of Combinatorial Theory, Series A

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Abstract. Let n and t be positive integers with t < n, and let q be a prime power. A partial (t − 1)-spread of PG(n − 1, q) is a set of (t − 1)-dimensional subspaces of PG(n − 1, q) that are pairwise disjoint. Let r = n mod t and 0 ≤ r < t. We prove that if t > (q r − 1)/(q − 1), then the maximum size, i.e., cardinality, of a partial (t − 1)-spread of PG(n − 1, q) is (q n − q t+r )/(q t − 1) + 1. This essentially settles a main open problem in this area. Prior to this result, this maximum size was only known for r ∈ {0, 1} and for r = q = 2.

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“…The inductive step follows since P j+1 is a subspace partition of H j+1 ∼ = V (n − j − 1, q) of the type given in (13), which satisfies the conditions in (14) and (12).…”

Section: Proof Of Theoremmentioning

confidence: 99%

The maximum size of a partial spread in a finite projective space

Nastase

Sissokho

2017

Journal of Combinatorial Theory, Series A

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“…To this end, the quantity A q (n, 2d, k) was defined as the maximum size of a code in G q (n, k) with minimum Grassmannian distance d. There has been extensive work on Grassmannian codes in the last ten years, e.g. [13], [14], [15], [16], [36] and references therein. A related concept is a subspace design or a block design t-(n, k, λ) q which is a collection S of k-subspaces from G q (n, k) (called blocks) such that each subspace of G q (n, t) is contained in exactly λ blocks of S, where t is called the strength of the design.…”

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confidence: 99%

Grassmannian Codes with New Distance Measures for Network Coding

Etzion

Zhang

2018

2018 IEEE International Symposium on Information Theory (ISIT)

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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.

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“…all ℓ-dimensional subspaces of G q (2ℓ, ℓ). The size of G q (2ℓ, ℓ) is 2ℓ ℓ q which is of the order q ℓ 2 [6]. For the vector solution, we can use an MRD[ℓt×ℓt, t] q code whose size is q ℓ(ℓ−1)t 2 +ℓt .…”

Section: E Arbitrary Number Of Messagesmentioning

confidence: 99%

Vector Network Coding Based on Subspace Codes Outperforms Scalar Linear Network Coding

Etzion

Wachter-Zeh

2018

IEEE Trans. Inform. Theory

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Abstract-This paper considers vector network coding solutions based on rank-metric codes and subspace codes. The main result of this paper is that vector solutions can significantly reduce the required alphabet size compared to the optimal scalar linear solution for the same multicast network. The multicast networks considered in this paper have one source with h messages, and the vector solution is over a field of size q with vectors of length t. For a given network, let the smallest field size for which the network has a scalar linear solution be qs, then the gap in the alphabet size between the vector solution and the scalar linear solution is defined to be qs −q t . In this contribution, the achieved gap is q (h−2)t 2 /h+o(t) for any q ≥ 2 and any even h ≥ 4. If h ≥ 5 is odd, then the achieved gap of the alphabet size is q (h−3)t 2 /(h−1)+o(t) . Previously, only a gap of size size one had been shown for networks with a very large number of messages. These results imply the same gap of the alphabet size between the optimal scalar linear and some scalar nonlinear network coding solution for multicast networks. For three messages, we also show an advantage of vector network coding, while for two messages the problem remains open. Several networks are considered, all of them are generalizations and modifications of the well-known combination networks. The vector network codes that are used as solutions for those networks are based on subspace codes, particularly subspace codes obtained from rank-metric codes. Some of these codes form a new family of subspace codes, which poses a new research problem.

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Galois geometries and coding theory

Cited by 73 publications

References 130 publications

The maximum size of a partial spread in a finite projective space

The maximum size of a partial spread in a finite projective space

Grassmannian Codes with New Distance Measures for Network Coding

Vector Network Coding Based on Subspace Codes Outperforms Scalar Linear Network Coding

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