2,496 research outputs found
Strong Singleton type upper bounds for linear insertion-deletion codes
The insertion-deletion codes was motivated to correct the synchronization
errors. In this paper we prove several Singleton type upper bounds on the
insdel distances of linear insertion-deletion codes, based on the generalized
Hamming weights and the formation of minimum Hamming weight codewords. Our
bound are stronger than some previous known bounds. These upper bounds are
valid for any fixed ordering of coordinate positions. We apply these upper
bounds to some binary cyclic codes and binary Reed-Muller codes with any
coordinate ordering, and some binary Reed-Muller codes and one
algebraic-geometric code with certain special coordinate ordering.Comment: 22 pages, references update
An MDS-PIR Capacity-Achieving Protocol for Distributed Storage Using Non-MDS Linear Codes
We propose a private information retrieval (PIR) protocol for distributed
storage systems with noncolluding nodes where data is stored using an arbitrary
linear code. An expression for the PIR rate, i.e., the ratio of the amount of
retrieved data per unit of downloaded data, is derived, and a necessary and a
sufficient condition for codes to achieve the maximum distance separable (MDS)
PIR capacity are given. The necessary condition is based on the generalized
Hamming weights of the storage code, while the sufficient condition is based on
code automorphisms. We show that cyclic codes and Reed-Muller codes satisfy the
sufficient condition and are thus MDS-PIR capacity-achieving.Comment: To be presented at 2018 IEEE International Symposium on Information
Theory (ISIT). arXiv admin note: substantial text overlap with
arXiv:1712.0389
Weight hierarchies of a family of linear codes associated with degenerate quadratic forms
We restrict a degenerate quadratic form over a finite field of odd
characteristic to subspaces. Thus, a quotient space related to is
introduced. Then we get a non-degenerate quadratic form induced by over the
quotient space. Some related results on the subspaces and quotient space are
obtained. Based on this, we solve the weight hierarchies of a family of linear
codes related to Comment: 12 page
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