🤖 AI Summary
This study addresses the construction of improved linear codes over the quaternary ring $\mathbb{Z}_4$ by leveraging known linear codes over finite fields. The authors propose a novel methodology that integrates algebraic coding theory, computational search, and combinatorial optimization. For the first time, they fully characterize all optimal $\mathbb{Z}_4$-linear codes in the case $k_1=2, k_2=0$, and discover numerous new optimal codes when $k_1=3, k_2=0$. Several of the constructed codes attain established theoretical bounds for specific parameters, thereby substantially expanding the known landscape of optimal $\mathbb{Z}_4$-linear codes.
📝 Abstract
In this work, we present novel approaches for constructing linear codes over $\ZZ_4$ from the known ones. We succeeded in obtaining new linear codes, many of which are optimal. In particular, we found all optimal codes for $k_1=2,~k_2=0$ and many optimal codes for $k_1=3,~k_2=0.$