🤖 AI Summary
This work investigates the construction and characterization of additive complementary dual (ACD) codes over the finite field $mathbb{F}_{q^2}$. Addressing both trace-Euclidean and trace-Hermitian inner products, we establish— for the first time—necessary and sufficient conditions on generator matrices for ACD codes. We propose a novel systematic framework for constructing high-dimensional additive ACD codes from linear LCD codes over $mathbb{F}_q$, uncovering intrinsic connections between their dimensions and duality properties. Furthermore, we provide explicit constructions and parameter-enhancement strategies that improve code length, dimension, or minimum distance. Our results extend the theory of ACD codes to general quadratic extension fields, unifying additive code structures with the trace-based inner product framework. This advancement furnishes new theoretical tools for quantum error-correcting codes and secret sharing schemes.
📝 Abstract
Shi et al. [Additive complementary dual codes over F4. Designs, Codes and Cryptography, 2022.] studied additive codes over the finite field F4 with respect to trace Hermitian and trace Euclidean inner products. In this article, we define additive codes of length n over finite field Fq2 as additive subgroups of Fn q2 where q is a prime power. We associate an additive code with a matrix called a generator matrix. We characterize trace Euclidean ACD and trace Hermitian ACD codes in terms of generator matrices over the finite field Fq2 . Also, we construct these codes over Fq2 from linear LCD codes over Fq.
