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On generator matrices and parity check matrices of generalized integer codes

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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.

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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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  1. Toyota Technological Institute, Hisakata 2-12-1, Tenpaku, Nagoya, Aichi , 468–8511, Japan

    Hajime Matsui

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  1. Hajime Matsui
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Correspondence to Hajime Matsui.

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Communicated by C. Carlet.

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Matsui, H. On generator matrices and parity check matrices of generalized integer codes. Des. Codes Cryptogr. 74, 681–701 (2015). https://doi.org/10.1007/s10623-013-9883-7

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