Abstract
We investigate the structure of codes over \(\mathbb{F}_q[u]/(u^s)\) rings with respect to the Rosenbloom-Tsfasman (RT) metric. We define a standard form generator matrix and show how we can determine the minimum distance of a code by taking advantage of its standard form. We define MDR (maximum distance rank) codes with respect to this metric and give the weights of the codewords of an MDR code. We explore the structure of cyclic codes over \(\mathbb{F}_q[u]/(u^s)\) and show that all cyclic codes over \(\mathbb{F}_q[u]/(u^s)\) rings are MDR. We propose a decoding algorithm for linear codes over these rings with respect to the RT metric.
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Mehmet Ozen, Irfan Siap and Fethi Callialp, The structure of linear codes with respect to a non-Hamming (Rosenbloom-Tsfasman) metric (Turkish) to appear in Anadolu Journal of Science.
Mehmet Ozen, Irfan Siap and Fethi Callialp, The structure of quaternary codes with respect to a Rosenbloom-Tsfasman metric, submitted.
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J. D. Key
AMS Classification: 94B05, 94B60
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Ozen, M., Siap, I. Linear Codes over \(\mathbb{F}_{q}[u]/(u^s)\) with Respect to the Rosenbloom–Tsfasman Metric. Des Codes Crypt 38, 17–29 (2006). https://doi.org/10.1007/s10623-004-5658-5
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DOI: https://doi.org/10.1007/s10623-004-5658-5