Coding for Errors and Erasures in Random Network Coding

• Published in: IEEE transactions on information theory Vol. 54; no. 8; pp. 3579 - 3591
• Main Authors: Koetter, R., Kschischang, F.R.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.08.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Codes; Coding; Coding, codes; Construction; Context modeling; Cryptography; Data encryption; Decoding; Detection, estimation, filtering, equalization, prediction; Error correction; Error correction codes; Errors; Exact sciences and technology; Geometry; Information theory; Information, signal and communications theory; Jamming; Mathematical models; Network coding; network error correction; Networks; Receivers; Signal and communications theory; Signal, noise; Strontium; subspace metric; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); Transmitters; Vector spaces; Vectors; Reed Solomon code; Finite field; Parameter estimation; sub- space metric; Coding errors; Transfer characteristic; Transmitter; Decoding; Geometrical projection; Random coding; Algorithm; Information transmission; network error correction; Linear coding; Vector space; Channel estimation; Network coding; Minimal distance; Metric; Error correction
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2008.926449

The problem of error-control in random linear network coding is considered. A ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic. Motivated by the property that linear network coding is vector-space preserving, information transmission is modeled as the injection into the network of a basis for a vector space V and the collection by the receiver of a basis for a vector space U . A metric on the projective geometry associated with the packet space is introduced, and it is shown that a minimum-distance decoder for this metric achieves correct decoding if the dimension of the space V cap U is sufficiently large. If the dimension of each codeword is restricted to a fixed integer, the code forms a subset of a finite-field Grassmannian, or, equivalently, a subset of the vertices of the corresponding Grassmann graph. Sphere-packing and sphere-covering bounds as well as a generalization of the singleton bound are provided for such codes. Finally, a Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ldquolist-1rdquo minimum-distance decoding algorithm is provided.