The theory of error-correcting codes☆

Software-defined networking (SDN) and network function virtualization (NFV) processed in multi-access edge computing (MEC) cloud systems have been proposed as critical paradigms for achieving the low latency requirements of the tactile Internet. While virtual network functions (VNFs) allow greater flexibility compared to hardware-based solutions, the VNF abstraction also introduces additional packet processing delays. In this paper, we investigate the practical feasibility of NFV with respect to the tactile Internet latency requirements. We develop, implement, and evaluate Chain-based Low latency VNF ImplemeNtation (CALVIN), a low-latency management framework for distributed Service Function Chains (SFCs). CALVIN classifies VNFs into elementary, basic, and advanced VNFs; moreover, CALVIN implements elementary and basic VNFs in the kernel space, while the advanced VNFs are implemented in the user space. Throughout, CALVIN employs a distributed mapping with one VNF per Virtual Machine (VM) in a MEC system. Furthermore, CALVIN avoids the metadata structure processing and batch processing of packets in the conventional Linux networking stack so as to achieve short per-packet latencies. Our rigorous measurements on off-the-shelf conventional networking and computing hardware demonstrate that CALVIN achieves round-trip times from a MEC ingress point via two elementary forwarding VNFs (one in kernel space and one in user space) and a MEC server to a MEC egress point on the order of 0.32 ms. Our measurements also indicate that MEC network coding and encryption are feasible for small 256 byte packets with an MEC latency budget of 0.35 ms; whereas, large 1400 byte packets can complete the network coding, but not the encryption within the 0.35 ms.

Reducing Latency in Virtual Machines: Enabling Tactile Internet for Human-Machine Co-Working

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Algebraic Geometric Codes

The entanglement-assisted formalism generalizes the standard stabilizer formalism, which can transform arbitrary classical linear codes into entanglement-assisted quantum error-correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this work, we construct six classes of q-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical negacyclic MDS codes by exploiting two or more pre-shared maximally entangled states. We show that two of these six classes q-ary EAQMDS have minimum distance more larger than $$q+1$$
 . Most of these q-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature.

Entanglement-assisted quantum MDS codes from negacyclic codes

New entanglement-assisted quantum codes from k-Galois dual codes

On the parameters of r-dimensional toric codes

We consider matrix-product codes $${[C_1\cdots C_s] \cdot A}$$, where $${C_1, \ldots , C_s}$$are nested linear codes and matrix A has full rank. We compute their minimum distance and provide a decoding algorithm when A is a non-singular by columns matrix. The decoding algorithm decodes up to half of the minimum distance.

Construction and decoding of matrix-product codes from nested codes

The order bound for the minimum distance of algebraic geometry codes was originally defined for the duals of one-point codes and later generalized for arbitrary algebraic geometry codes. Another bound of order type for the minimum distance of general linear codes, and for codes from order domains in particular, was given in [1]. Here we investigate in detail the application of that bound to one-point algebraic geometry codes, obtaining a bound d* for the minimum distance of these codes. We establish a connection between d* and the order bound and its generalizations. We also study the improved code constructions based on d*. Finally we extend d* to all generalized Hamming weights.

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On the order bounds for one-point AG codes

We introduce Fulcrum, a network coding framework that achieves three seemingly conflicting objectives: 1) to reduce the coding coefficient overhead down to nearly $n$ bits per packet in a generation of $n$ packets; 2) to conduct the network coding using only Galois field $GF(2)$ operations at intermediate nodes if necessary, dramatically reducing computing complexity in the network; and 3) to deliver an end-to-end performance that is close to that of a high-field network coding system for high-end receivers, while simultaneously catering to low-end receivers that decode in $GF(2)$ . As a consequence of 1) and 3), Fulcrum has a unique trait missing so far in the network coding literature: providing the network with the flexibility to distribute computational complexity over different devices depending on their current load, network conditions, or energy constraints. At the core of our framework lies the idea of precoding at the sources using an expansion field $GF(2^{h}),~h > 1$ , to increase the number of dimensions seen by the network. Fulcrum can use any high-field linear code for precoding, e.g., Reed-Solomon or Random Linear Network Coding (RLNC). Our analysis shows that the number of additional dimensions created during precoding controls the trade-off between delay, overhead, and computing complexity. Our implementation and measurements show that Fulcrum achieves similar decoding probabilities as high field RLNC but with encoders and decoders that are an order of magnitude faster.

/pdf/fulcrum-flexible-network-coding-for-heterogeneous-devices-wz3kt9t8g1.pdf

Fulcrum: Flexible Network Coding for Heterogeneous Devices

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes [26, 24]. In this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes our bound is often tight. Furthermore, for these codes the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights.

Diego Ruano

Papers

On the parameters of r-dimensional toric codes

Construction and decoding of matrix-product codes from nested codes

On the order bounds for one-point AG codes

Fulcrum: Flexible Network Coding for Heterogeneous Devices

Relative generalized Hamming weights of one-point algebraic geometric codes