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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Aydin, N., Karadeniz, S., Yildiz, B.: Some new binary quasi-cyclic codes from codes over the ring \({\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2\). Appl. Algebra Eng. Commun. Comput. 24(5), 355–367 (2013). https://doi.org/10.1007/s00200-013-0207-y
Dougherty, S.T., Karadeniz, S., Yildiz, B.: Constructing formally self-dual codes over \(R_k\). Discrete Appl. Math. 167, 188–196 (2014). https://doi.org/10.1016/j.dam.2013.11.017
Dougherty, S.T., Liu, H.: Independence of vectors in codes over rings. Des. Codes Cryptogr. 51(1), 55–68 (2009). https://doi.org/10.1007/s10623-008-9243-1
Dougherty, S.T., Saltürk, E., Szabo, S.: Codes over local rings of order 16 and binary codes. Adv. Math. Commun. 10(2), 379–391 (2016). https://doi.org/10.3934/amc.2016012
Martínez-Moro, E., Szabo, S.: On codes over local Frobenius non-chain rings of order 16. In: Dougherty, S., Facchini, A., Leroy, A., Puczyłowski, E., Solé, P. (eds.) Noncommutative rings and their applications. Contemporary Mathematics, vol. 634, pp. 227–241. American Mathematical Society, Providence (2015). https://doi.org/10.1090/conm/634/12702
Martínez-Moro, E., Szabo, S., Yildiz, B.: Linear codes over \(\frac{{\mathbb{Z}}_4[x]}{\langle x^2+2x\rangle }\). Int. J. Inf. Coding Theory 3(1), 78–96 (2015)
McDonald, B.R.: Finite Rings with Identity. Marcel Dekker Inc, New York (1974)
Wood, J.A.: Duality for modules over finite rings and applications to coding theory. Am. J. Math. 121(3), 555–575 (1999)
Yildiz, B., Karadeniz, S.: Linear codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2\). Des. Codes Cryptogr. 54(1), 61–81 (2010). https://doi.org/10.1007/s10623-009-9309-8
Yildiz, B., Karadeniz, S.: Self-dual codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2\). J. Frankl. Inst. 347(10), 1888–1894 (2010). https://doi.org/10.1016/j.jfranklin.2010.10.007
Yildiz, B., Karadeniz, S.: Linear codes over \({\mathbb{Z}}_4+u{\mathbb{Z}}_4\): MacWilliams identities, projections, and formally self-dual codes. Finite Fields Appl. 27, 24–40 (2014). https://doi.org/10.1016/j.ffa.2013.12.007
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