291 research outputs found
Constructions of Pure Asymmetric Quantum Alternant Codes Based on Subclasses of Alternant Codes
In this paper, we construct asymmetric quantum error-correcting codes(AQCs)
based on subclasses of Alternant codes. Firstly, We propose a new subclass of
Alternant codes which can attain the classical Gilbert-Varshamov bound to
construct AQCs. It is shown that when , -parts of the AQCs can attain
the classical Gilbert-Varshamov bound. Then we construct AQCs based on a famous
subclass of Alternant codes called Goppa codes. As an illustrative example, we
get three AQCs from the well
known binary Goppa code. At last, we get asymptotically good
binary expansions of asymmetric quantum GRS codes, which are quantum
generalizations of Retter's classical results. All the AQCs constructed in this
paper are pure
New Identities Relating Wild Goppa Codes
For a given support and a polynomial with no roots in , we prove equality
between the -ary Goppa codes where
denotes the norm of , that is In
particular, for , that is, for a quadratic extension, we get
. If has roots in
, then we do not necessarily have equality and we prove that
the difference of the dimensions of the two codes is bounded above by the
number of distinct roots of in . These identities provide
numerous code equivalences and improved designed parameters for some families
of classical Goppa codes.Comment: 14 page
The Dimension of Subcode-Subfields of Shortened Generalized Reed Solomon Codes
Reed-Solomon (RS) codes are among the most ubiquitous codes due to their good
parameters as well as efficient encoding and decoding procedures. However, RS
codes suffer from having a fixed length. In many applications where the length
is static, the appropriate length can be obtained by an RS code by shortening
or puncturing. Generalized Reed-Solomon (GRS) codes are a generalization of RS
codes, whose subfield-subcodes are extensively studied. In this paper we show
that a particular class of GRS codes produces many subfield-subcodes with large
dimension. An algorithm for searching through the codes is presented as well as
a list of new codes obtained from this method
Variations of the McEliece Cryptosystem
Two variations of the McEliece cryptosystem are presented. The first one is
based on a relaxation of the column permutation in the classical McEliece
scrambling process. This is done in such a way that the Hamming weight of the
error, added in the encryption process, can be controlled so that efficient
decryption remains possible. The second variation is based on the use of
spatially coupled moderate-density parity-check codes as secret codes. These
codes are known for their excellent error-correction performance and allow for
a relatively low key size in the cryptosystem. For both variants the security
with respect to known attacks is discussed
On the number of spurious memories in the Hopfield model
The outer-product method for programming the Hopfield model is discussed. The method can result in many spurious stable states-exponential in the number of vectors that are to be stored-even in the case when the vectors are orthogonal
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