On systematicity of linear function-correcting codes

What the math gives to ML

The paper turns task-aware robustness into a distance-design problem: messages that differ in the prescribed function value must be mapped to codewords that are far apart, while messages with the same function value need not be separated. The transferable object is the relative Hamming distance of a linear encoder with respect to the kernel of a task map, which gives a principled way to spend redundancy only on task-relevant directions rather than protecting every latent bit equally. A practical neural adaptation is a quantized latent or output head with systematic task-dependent parity/repetition coordinates, trained under bit noise and evaluated by task accuracy at a fixed redundancy budget.