Duality of Codes over Non-unital Rings of Order Four
Adel Alahmadi, Asmaa Melaibari, Patrick Solé
Abstract
In this paper, we present a basic theory of the duality of linear codes over three of the non-unital rings of order four; namely <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I, E</i> , and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> as denoted in (Fine, 1993). A new notion of duality is introduced in the case of the non-commutative ring <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> . The notion of self-dual codes with respect to this duality coincides with that of quasi self-dual codes over <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> as introduced in (Alahmadi <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> , 2022). We characterize self-dual codes and LCD codes over the three rings, and investigate the properties of their corresponding additive codes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> . We study the connection between the dual of any linear code over these rings and the dual of its associated binary codes. A MacWilliams formula is established for linear codes over <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> .