A New Method for Constructing Self-Dual Codes over Finite Commutative Rings with Characteristic 2

Yongsheng Ma, Jizhu Nan () and Yuanbo Liu
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Yongsheng Ma: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Jizhu Nan: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Yuanbo Liu: School of Mathematics and Information, China West Normal University, Nanchong 637001, China

Mathematics, 2024, vol. 12, issue 17, 1-28

Abstract: In this work, we present a new method for constructing self-dual codes over finite commutative rings R with characteristic 2. Our method involves searching for k × 2 k matrices M over R satisfying the conditions that its rows are linearly independent over R and M M ⊤ = α ⊤ α for an R -linearly independent vector α ∈ R k . Let C be a linear code generated by such a matrix M . We prove that the dual code C ⊥ of C is also a free linear code with dimension k , as well as C / H u l l ( C ) and C ⊥ / H u l l ( C ) are one-dimensional free R -modules, where H u l l ( C ) represents the hull of C . Based on these facts, an isometry from R x + R y onto R 2 is established, assuming that x + H u l l ( C ) and y + H u l l ( C ) are bases for C / H u l l ( C ) and C ⊥ / H u l l ( C ) over R , respectively. By utilizing this isometry, we introduce a new method for constructing self-dual codes from self-dual codes of length 2 over finite commutative rings with characteristic 2. To determine whether the matrix M M ⊤ takes the form of α ⊤ α with α being a linearly independent vector in R k , a necessary and sufficient condition is provided. Our method differs from the conventional approach, which requires the matrix M to satisfy M M ⊤ = 0 . The main advantage of our method is the ability to construct nonfree self-dual codes over finite commutative rings, a task that is typically unachievable using the conventional approach. Therefore, by combining our method with the conventional approach and selecting an appropriate matrix construction, it is possible to produce more self-dual codes, in contrast to using solely the conventional approach.

Keywords: finite commutative rings; self-dual codes; free linear codes; isometry; hull (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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