The Weights in MDS Codes

The weights in MDS codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n-k+1 to n. The proof uses the covering radius of the dual codeComment: 5 pages, submitted to IEEE Trans. IT. This version 2 is the revised version after the refereeing process. Accepted for publicatio

Generalized weights and bounds for error probability over erasure channels

New upper and lower bounds for the error probability over an erasure channel are provided, making use of Wei's generalized weights, hierarchy and spectra. In many situations the upper and lower bounds coincide and this allows improvement of existing bounds. Results concerning MDS and AMDS codes are deduced from those bounds

Integer sequences that are generalized weights of a linear code

Which integer sequences are sequences of generalized weights of a linear code? In this paper, we answer this question for linear block codes, rank-metric codes, and more generally for sum-rank metric codes. We do so under an existence assumption for MDS and MSRD codes. We also prove that the same integer sequences appear as sequences of greedy weights of linear block codes, rank-metric codes, and sum-rank metric codes. Finally, we characterize the integer sequences which appear as sequences of relative generalized weights (respectively, relative greedy weights) of linear block codes.Comment: 19 page

Improved Private Information Retrieval for Coded Storage From Code Decomposition

We consider private information retrieval (PIR) for distributed storage systems with noncolluding nodes where data is stored using a non maximum distance separable (MDS) linear code. Recently, it was shown that when data is stored using certain non-MDS codes, the MDS-PIR capacity can be achieved, and is indeed the capacity of the system. In this paper, for storage codes not belonging to this class, we present a heuristic algorithm for their decomposition into punctured subcodes and a PIR protocol based on these punctured subcodes. The code decomposition is guided by the generalized Hamming weights of the storage code. We show that the proposed PIR protocol can achieve a larger PIR rate than that of all existing PIR protocols

The Partition Weight Enumerator of MDS Codes and its Applications

A closed form formula of the partition weight enumerator of maximum distance separable (MDS) codes is derived for an arbitrary number of partitions. Using this result, some properties of MDS codes are discussed. The results are extended for the average binary image of MDS codes in finite fields of characteristic two. As an application, we study the multiuser error probability of Reed Solomon codes.Comment: This is a five page conference version of the paper which was accepted by ISIT 2005. For more information, please contact the author