Designs, Codes and Cryptography

, Volume 74, Issue 3, pp 681–701 | Cite as

On generator matrices and parity check matrices of generalized integer codes

Article
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Abstract

Generalized integer codes are defined as codes over rings of integers modulo \(n\) in which individual code symbols generally have different moduli. In this paper, we use a certain type of matrix identities to derive a necessary and sufficient condition for integer matrices to be equal to the generator matrices of generalized integer codes. Moreover, it is shown that the parity check matrix is generated from this matrix identity of the generator matrix. We also show the close connection between the listing of a certain type of integer codes and Hecke rings. Finally, an efficient algorithm that enumerates theoretically all of the generator matrices of generalized integer codes is provided.

Keywords

Codes over rings Elementary divisors Extended Euclidean algorithm Bézout’s identity Integer lattices  Hecke rings 

Mathematics Subject Classification

94B05 94B25 94B40 94B60 11T71 
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Notes

Acknowledgments

This work was partly supported by KAKENHI, Grant-in-Aid for Scientific Research C, 23560478. The author would like to thank the anonymous referees for their helpful comments which improved the final presentation of the paper.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Toyota Technological InstituteNagoyaJapan

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