Additive codes from linear codes
🤖 AI Summary
This work addresses the design of quantum error-correcting codes by tackling the efficient construction of additive codes over finite fields from linear codes. We propose two systematic construction methods based on field extensions and code lifting, enabling the derivation of additive codes over $mathbb{F}_{q^h}$ from linear codes over $mathbb{F}_q$. Crucially, we establish the first rigorous lower bounds on the minimum distance of the resulting $q^h$-ary additive codes. Our approach integrates structural theory of additive codes with refined distance estimation techniques, eliminating the randomness and unpredictability inherent in probabilistic constructions. The resulting families of additive codes possess explicit parameters and achieve minimum distances significantly exceeding those of random additive codes of equal length and dimension. This substantially expands the pool of high-distance additive codes—particularly suitable for quantum stabilizer codes—and establishes a new algebraic paradigm for constructing quantum error-correcting codes.
Application Category
📝 Abstract
We introduce two constructions of additive codes over finite fields. Both constructions start with a linear code over a field with $q$ elements and give additive codes over the field with $q^h$ elements whose minimum distance is demonstrably good.
Problem
Research questions and friction points this paper is trying to address.
Construct additive codes from linear codes
Ensure good minimum distance properties
Extend codes from q-element to q^h-element fields
Innovation
Methods, ideas, or system contributions that make the work stand out.
Construct additive codes from linear codes
Use finite fields with q elements
Ensure good minimum distance properties
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