Code word

Abstract: 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 capU 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.

Abstract: Techniques for detecting and correcting burst errors in data bytes formed in a two-level block code structure. A second level decoder uses block level check bytes to detect columns in a two-level block code structure that contain error bytes. The second level decoder generates erasure pointers that identify columns in the two-level block structure effected by burst errors. A first level decoder then uses codeword check bytes to correct all of the bytes in the columns identified by the erasure pointers. The first level decoder is freed to use all of the codeword check bytes only for error byte value calculations. The first level decoder does not need to use any of the codeword check bytes for error location calculations, because the erasure pointers generated by the second level decoder provide all of the necessary error locations. This techniques doubles the error correction capability of the first level decoder.

Abstract: Presents a novel approach to soft decision decoding for binary linear block codes. The basic idea is to achieve a desired error performance progressively in a number of stages. For each decoding stage, the error performance is tightly bounded and the decoding is terminated at the stage where either near-optimum error performance or a desired level of error performance is achieved. As a result, more flexibility in the tradeoff between performance and decoding complexity is provided. The decoding is based on the reordering of the received symbols according to their reliability measure. The statistics of the noise after ordering are evaluated. Based on these statistics, two monotonic properties which dictate the reprocessing strategy are derived. Each codeword is decoded in two steps: (1) hard-decision decoding based on reliability information and (2) reprocessing of the hard-decision-decoded codeword in successive stages until the desired performance is achieved. The reprocessing is based on the monotonic properties of the ordering and is carried out using a cost function. A new resource test tightly related to the reprocessing strategy is introduced to reduce the number of computations at each reprocessing stage. For short codes of lengths N/spl les/32 or medium codes with 32 >

Abstract: The problem of error-control in a "noncoherent" random network coding channel is considered Information transmission is modelled 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 suitable coding metric on subspaces is defined, under which a minimum distance decoder achieves correct decoding if the dimension of the space V U is large enough When the dimension of each codeword is restricted to a fixed integer, the code forms a subset of the vertices of the Grassmann graph Sphere-packing, sphere-covering bounds and a Singleton bound are provided for such codes A Reed-Solomon-like code construction is provided and decoding algorithm given