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Bull. Korean Math. Soc. 2018; 55(3): 971-997

Online first article March 9, 2018      Printed May 31, 2018

https://doi.org/10.4134/BKMS.b170443

Copyright © The Korean Mathematical Society.

Primitive idempotents in the ring $F_{4}[x]/\langle x^{p^{n}}-1\rangle$ and cyclotomic $Q$ codes

Sudhir Batra, Rekha Mathur

DCR University of Science and Technology, DCR University of Science and Technology

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The parity of cyclotomic numbers of order 2, 4 and 6 associated with 4-cyclotomic cosets modulo an odd prime $p$ are obtained. Hence the explicit expressions of primitive idempotents of minimal cyclic codes of length $p^{n}$, $n\ge 1$ over the quaternary field $F_{4}$ are obtained. These codes are observed to be subcodes of $Q$ codes of length $p^{n}$. Some orthogonal properties of these subcodes are discussed. The minimal cyclic codes of length 17 and 43 are also discussed and it is observed that the minimal cyclic codes of length 17 are two weight codes. Further, it is shown that a $Q$ code of prime length is always cyclotomic like a binary duadic code and it seems that there are infinitely many prime lengths for which cyclotomic $Q$ codes of order 6 exist.

Keywords: cyclic codes, duadic codes, $Q$ codes, primitive idempotents, cyclotomic numbers

MSC numbers: 12E20, 94B15, 11T71

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