Construction and Classification of Generalized Hadamard Codes over Eisenstein Local Rings Z_(2^s )[ω]

The research paper examines the design principles and structural features of Generalized Hadamard (GH) codes that operate within Eisenstein local rings Z_(2^s )[ω] utilizing a primitive cube root of unity ω that fulfills ω^2+ω+1=0. The paper first introduces an algebraic Eisenstein integer framework before developing appropriate Gray mapping to examine binary-domain representations of these codes. We establish the essential criteria and necessary checks for determining the linear properties of GH codes based on Z_(2^s ) [ω] structures. This research work defines the kernel structure of these codes together with their rank specification and structural properties evaluation. A classification system for Z_(2^s )[ω]-linear Hadamard codes presents itself in the last part according to their algebraic and combinatorial characteristics. Future studies about coding techniques within algebraic integer rings can begin from our current work because our research expands the understanding of code theory on non-traditional rings.