In this paper, we study cyclic stabiliser codes over F-p of length dividing p(t) + 1 for some positive integer t. We call these t-Frobenius codes or just Frobenius codes for short. We give methods to construct them an...
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ISBN:
(纸本)9781457705953
In this paper, we study cyclic stabiliser codes over F-p of length dividing p(t) + 1 for some positive integer t. We call these t-Frobenius codes or just Frobenius codes for short. We give methods to construct them and show that they have efficient decoding algorithms. An important subclass of stabiliser codes are the linear stabiliser codes. For linear Frobenius codes we have stronger results: We completely characterise all linear Frobenius codes. As a consequence, we show that for every integer n that divides p(t) + 1 for an odd t, there are no linear cycliccodes of length n. On the other hand for even t, we give an explicit method to construct all of them. This gives us many explicit examples of Frobenius code which include the well studied Laflamme code. We show that the classical notion of BCH distance can be generalised to all the Frobenius codes that we construct, including the non-linear ones, and show that the algorithm of Berlekamp can be generalised to correct quantum errors within the BCH limit. This gives, for the first time, a family of codes that are neither CSS nor linear for which efficient decoding algorithm exits.
Many good quantum error-correcting codes were constructed from cycliccodes. However, it is a difficult problem to determine the true minimum distance of quantum cyclic codes for large length n. In this work, we const...
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Many good quantum error-correcting codes were constructed from cycliccodes. However, it is a difficult problem to determine the true minimum distance of quantum cyclic codes for large length n. In this work, we construct nonbinary quantum cyclic codes and asymmetric quantum cyclic codes that are derived from repeated-root cycliccodes for an arbitrary length p(s), and determine the true minimum distance of all those codes. Some proposed quantum cyclic codes are optimal. Additionally, some proposed asymmetric quantum cyclic codes have better parameters than the ones available in the literature.
In this work we present a new construction of quantum error correcting codes. It is based on the stabilizer formalism but does not use the CSS construction. It takes advantage of the properties of the difference set t...
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ISBN:
(纸本)9781538647813
In this work we present a new construction of quantum error correcting codes. It is based on the stabilizer formalism but does not use the CSS construction. It takes advantage of the properties of the difference set to construct the code. The technique may also be used to generate the quantum LDPC codes as well as the cycliccodes.
There are mainly two efficient constructing methods for quantumcodes, the construction named after Calderbank, Shor and Stean, and the hermitian code construction. We compare two methods in quantum cyclic codes and q...
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ISBN:
(纸本)9781467368506
There are mainly two efficient constructing methods for quantumcodes, the construction named after Calderbank, Shor and Stean, and the hermitian code construction. We compare two methods in quantum cyclic codes and quantum duadic codes separately, and show that the hermitian code construction is better in terms of existence conditions of codes and the code parameters.
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