Abstract
Codes over commutative Frobenius rings are studied with a focus on local Frobenius rings of order 16 for illustration. The main purpose of this work is to present a method for constructing a generating character for any commutative Frobenius ring. Given such a character, the MacWilliams identities for the complete and symmetrized weight enumerators can be easily found. As examples, generating characters for all commutative local Frobenius rings of order 16 are given. In addition, a canonical generator matrix for codes over local non-chain rings is discussed. The purpose is to show that when working over local non-chain rings, a canonical generator matrix exists but is less than useful which emphases the difficulties in working over such rings.
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Acknowledgements
Esengül Saltürk would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their support while writing this paper. Steve Szabo would like to thank the University of Scranton for their hospitality during his visit when some of this work was completed.
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Dougherty, S.T., Saltürk, E. & Szabo, S. On codes over Frobenius rings: generating characters, MacWilliams identities and generator matrices. AAECC 30, 193–206 (2019). https://doi.org/10.1007/s00200-019-00384-0
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DOI: https://doi.org/10.1007/s00200-019-00384-0
Keywords
- Codes over rings
- MacWilliams relations
- Local rings
- Frobenius rings
- Generating character
- Weight enumerator