Codes and Anticodes in the Grassman Graph

Schwartz, Moshe; Etzion, Tuvi

doi:10.1006/jcta.2001.3188

Cited by 83 publications

(79 citation statements)

References 5 publications

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“…Some bounds on A q [n, 2δ, l], e.g., the Hamming type upper bound, the Gilbert type lower bound, and the Singleton type upper bound, were derived in [1]. It is known that the Hamming type bound is not very good [1] and there exist no non-trivial perfect codes meeting the Hamming type bound [5,6]. The Singleton type bound developed in [1] is the following:…”

Section: Introductionmentioning

confidence: 98%

Johnson type bounds on constant dimension codes

Xia

2008

Des. Codes Cryptogr.

121

100

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Very recently, an operator channel was defined by Koetter and Kschischang when they studied random network coding. They also introduced constant dimension codes and demonstrated that these codes can be employed to correct errors and/or erasures over the operator channel. Constant dimension codes are equivalent to the so-called linear authentication codes introduced by Wang, Xing and Safavi-Naini when constructing distributed authentication systems in 2003. In this paper, we study constant dimension codes. It is shown that Steiner structures are optimal constant dimension codes achieving the WangXing-Safavi-Naini bound. Furthermore, we show that constant dimension codes achieve the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures. Then, we derive two Johnson type upper bounds, say I and II, on constant dimension codes. The Johnson type bound II slightly improves on the Wang-Xing-Safavi-Naini bound. Finally, we point out that a family of known Steiner structures is actually a family of optimal constant dimension codes achieving both the Johnson type bounds I and II.

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Section: Introductionmentioning

confidence: 98%

Johnson type bounds on constant dimension codes

Xia

2008

Des. Codes Cryptogr.

121

100

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“…The q-analog Steiner structure [19] could be used in the construction that was presented in Section 4.3, which does not require the size of n. We will state it in detail in the following.…”

Section: Exactly Attained Codesmentioning

confidence: 99%

Subspace Coding for Networks with Different Level Messages

Cai

Guo

2015

Entropy

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Abstract:We study the asymptotically-achievable rate region of subspace codes for wireless network coding, where receivers have different link capacities due to the access ways or the faults of the intermediate links in the network. Firstly, an outer bound of the achievable rate region in a two-receiver network is derived from a combinatorial method. Subsequently, the achievability of the outer bound is proven by code construction, which is based on superposition coding. We show that the outer bound can be achieved asymptotically by using the code presented by Koetter and Kschischang, and the outer bound can be exactly attained in some points by using a q-analog Steiner structure. Finally, the asymptotically-achievable rate region is extended to the general case when the network has m receivers with different levels.

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“…Note that no nontrivial Steiner structures, except for the case s = 0 when we have a partition of GF (q) m by k-spaces, are known. Properties of Steiner structures in P q (n), introduced in [1] are studied in [19].…”

Section: New Constructionmentioning

confidence: 99%