Self-Orthogonality of $q$ -Ary Images of $q^{m}$ -Ary Codes and Quantum Code Construction

@article{Sundeep2007SelfOrthogonalityO,
  title={Self-Orthogonality of \$q\$ -Ary Images of \$q^\{m\}\$ -Ary Codes and Quantum Code Construction},
  author={B Sundeep and Andrew Thangaraj},
  journal={IEEE Transactions on Information Theory},
  year={2007},
  volume={53},
  pages={2480-2489}
}
  • B. Sundeep, A. Thangaraj
  • Published 26 June 2006
  • Mathematics, Computer Science
  • IEEE Transactions on Information Theory
A code over GF can be imaged or expanded into a code over GF using a basis for the extension field over the base field. The properties of such an image depend on the original code and the basis chosen for imaging. Problems relating the properties of a code and its image with respect to a basis have been of great interest in the field of coding theory. In this work, a generalized version of the problem of self-orthogonality of the q-ary image of a qm-ary code has been considered. Given an inner… 
Self-orthogonality of Images and Traces of Codes with Applications to Quantum Codes
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    2007 IEEE International Symposium on Information Theory
  • 2007
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References

SHOWING 1-10 OF 15 REFERENCES
Orthogonality of binary codes derived from Reed-Solomon codes
  • C. Retter
  • Mathematics, Computer Science
    IEEE Trans. Inf. Theory
  • 1991
The author provides a simple method for determining the orthogonality of binary codes derived from Reed-Solomon codes and other cyclic codes of length 2/sup m/-1 over GF(2/sup m/) for m bits.
The q-ary image of some qm-ary cyclic codes: Permutation group and soft-decision decoding
Using a particular construction of generator matrices of the q-ary image of q/sup m/-ary cyclic codes, it is proved that some of these codes are invariant under the action of particular permutation
On the minimum distance of a q-ary image of a qm-ary cyclic code
TLDR
A lower bound on the minimum distance of a q-ary image of q/sup m/-ary cyclic code, based on its concatenated structure due to Jensen and Seguin is discussed.
Quantum Error Correction Via Codes Over GF(4)
TLDR
In the present paper the problem of finding quantum-error-correcting codes is transformed into one of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product.
The q-ary image of a qm-ary cyclic code
  • G. Séguin
  • Mathematics, Computer Science
    IEEE Trans. Inf. Theory
  • 1995
TLDR
All of the bases with respect to which the q-ary image of V is cyclic are determined.
Quantum Error Correction Via Codes Over GF
The problem of finding quantum-error-correcting codes is transformed into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner
Theory of Error-correcting Codes
The field of channel coding started with Claude Shannon's 1948 landmark paper. Fifty years of efforts and invention have finally produced coding schemes that closely approach Shannon's channel
Quantum codes from cyclic codes over GF(4m)
We provide a construction for quantum codes (Hermitian-self-orthogonal codes over GF(4)) starting from cyclic codes over GF(4/sup m/). We also provide examples of these codes some of which meet the
Bit-level soft-decision decoding of Reed-Solomon codes
A Reed-Solomon decoder that makes use of bit-level soft-decision information is presented. A Reed-Solomon generator matrix that possesses a certain inherent structure in GF(2) is derived. This
An average weight-distance enumerator for binary expansions of Reed-Solomon codes
  • C. Retter
  • Mathematics, Computer Science
    IEEE Trans. Inf. Theory
  • 2002
An average Hamming weight enumerator is derived for the codewords at each Hamming distance from a received pattern in the set of all possible binary expansions of a Reed-Solomon code. Since these
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