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.
A code is said to be a r-local locally repairable code (LRC) if each of its coordinates can be repaired by accessing at most r other coordinates. When some of the r coordinates are also erased, the r-local LRC can not accomplish the local repair, which leads to the concept of (r, δ)-locality. A q-ary [n, k] linear code C is said to have (r, δ)-locality (δ ≥ 2) if for each coordinate i, there exists a punctured subcode of C with support containing i, whose length is at most r + δ − 1, and whose minimum distance is at least δ. The (r, δ)-LRC can tolerate δ − 1 erasures in total, which degenerates to a r-local LRC when δ = 2. A q-ary (r, δ) LRC is called optimal if it meets the Singleton-like bound for (r, δ)-LRCs. A class of optimal q-ary cyclic r-local LRCs with lengths n | q − 1 were constructed by Tamo, Barg, Goparaju and Calderbank based on the q-ary Reed-Solomon codes. In this paper, we construct a class of optimal q-ary cyclic (r, δ)-LRCs (δ ≥ 2) with length n | q − 1, which generalizes the results of Tamo et al. Moreover, we construct a new class of optimal q-ary cyclic r-local LRCs with lengths n | q + 1 and a new class of optimal q-ary cyclic (r, δ)-LRCs (δ ≥ 2) with lengths n | q + 1. The constructed optimal LRCs with length n = q + 1 have the best-known length q + 1 for the given finite field with size q when the minimum distance is larger than 4.
In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, which we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. In particular, all possible parameters for the q-ary MDS EAQECCs of length n ≤ q are completely determined and several new classes of q-ary MDS EAQECCs of length n > q are also obtained.
Abstract-Recently, network error correction coding (NEC) has been studied extensively. Several bounds in classical coding theory have been extended to network error correction coding, especially the Singleton bound. In this paper, following the research line using the extended global encoding kernels proposed in [12], the refined Singleton bound of NEC can be proved more explicitly. Moreover, we give a constructive proof of the attainability of this bound and indicate that the required field size for the existence of network maximum distance separable (MDS) codes can become smaller further. By this proof, an algorithm is proposed to construct general linear network error correction codes including the linear network error correction MDS codes. Finally, we study the error correction capability of random linear network error correction coding. Motivated partly by the performance analysis of random linear network coding [6], we evaluate the different failure probabilities defined in this paper in order to analyze the performance of random linear network error correction coding. Several upper bounds on these probabilities are obtained and they show that these probabilities will approach to zero as the size of the base field goes to infinity. Using these upper bounds, we slightly improve on the probability mass function of the minimum distance of random linear network error correction codes in [7], as well as the upper bound on the field size required for the existence of linear network error correction codes with degradation at most d.Index Terms-Network coding, network error correction coding, the refined Singleton bound, maximum distance separable (MDS) code, random linear network error correction coding, the extended global encoding kernels, network error correction code construction.
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