Mass Formula for Self-Orthogonal and Self-Dual Codes over Non-Unital Rings of Order Four

Adel Alahmadi (), Altaf Alshuhail, Rowena Alma Betty, Lucky Galvez and Patrick Solé
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Adel Alahmadi: Research Group of Algebraic Structures and Applications, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Altaf Alshuhail: Research Group of Algebraic Structures and Applications, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Rowena Alma Betty: Institute of Mathematics, University of the Philippines Diliman, Quezon City 1101, Philippines
Lucky Galvez: Institute of Mathematics, University of the Philippines Diliman, Quezon City 1101, Philippines
Patrick Solé: I2M, (CNRS, University of Aix-Marseille, Centrale Marseille), 13009 Marseilles, France

Mathematics, 2023, vol. 11, issue 23, 1-17

Abstract: We study the structure of self-orthogonal and self-dual codes over two non-unital rings of order four, namely, the commutative ring I = a , b | 2 a = 2 b = 0 , a 2 = b , ab = 0 and the noncommutative ring E = a , b | 2 a = 2 b = 0 , a 2 = a , b 2 = b , ab = a , ba = b . We use these structures to give mass formulas for self-orthogonal and self-dual codes over these two rings, that is, we give the formulas for the number of inequivalent self-orthogonal and self-dual codes, of a given type, over the said rings. Finally, using the mass formulas, we classify self-orthogonal and self-dual codes over each ring, for small lengths and types.

Keywords: code over ring; non-unital ring; self-orthogonal code; quasi self-dual; self-dual code; mass formulas (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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