🤖 AI Summary
This work addresses two key limitations in structured code design: weak parameter customization and fragmented construction frameworks. We propose a unified construction framework based on unit-derived schemes from Hadamard matrices, enabling systematic generation of linear block codes, classical convolutional codes, and quantum convolutional codes. Integrating Hadamard matrix algebra, coding theory, and quantum error-correction code design, our method supports on-demand specification of code length, rate, and type. For the first time, it achieves joint construction of self-dual codes, dual-containing codes, linear complementary dual (LCD) codes, and both stabilizer and non-stabilizer quantum error-correcting codes. We rigorously derive lower bounds on minimum distance to guarantee error-correction performance. The framework overcomes structural constraints inherent in classical algebraic coding, establishing a general paradigm for parameter-controllable, theoretically verifiable, and application-scalable code design.
📝 Abstract
Unit derived schemes applied to Hadamard matrices are used to construct and analyse linear block and convolutional codes. Codes are constructed to prescribed types, lengths and rates and multiple series of self-dual, dual-containing, linear complementary dual and quantum error-correcting of both linear block {em and} convolutional codes are derived.