# Algebra of error models and error correcting codes?

In coding theory we typically consider the situation where we have a channel that connects a sender and receiver. The messages flowing from the sender to the receiver are corrupted by an error source ES. We have an error detection and correction mechanism that is typically fine-tuned to ES. Wikipedia (https://en.wikipedia.org/wiki/Information_theory) uses this image:

But what if we have more than one error source, say $$ES1$$ and $$ES2$$. We can have a error detection and correction mechanism for each but they can then be combined in different ways. Moreover, even if the error detection and correction mechanism for $$ES1$$ is fine-tuned to $$ES1$$, and likewise for $$ES2$$, it is not a priori clear that any combined error detection and correction mechanism has the required properties for the combined error model. So we can imagine an algebra of error sources and error detection and correction mechanisms. I'm sure this has been investigated before.

Where can I find related work in this direction?

• Where did you see that the error control is fine-tuned to the error source ?
– user16034
Commented Jul 9, 2022 at 15:16
• @YvesDaoust For example in micro-processors and dRAMs, where the additional costs of error correction (transistors cost power and "area") weighted against their benefits. Another example is mobile phone data transmission. Complex error correction schemes cost a lot of compute which increases compute, hence energy consumption, which mobile phones are sensitive to. Commented Jul 10, 2022 at 8:00
• Any reference about "fine-tuning" of error codes in these contexts ? (I mean specifically, not about general usage of error control codes.)
– user16034
Commented Jul 10, 2022 at 9:45
• Not to the top of my head, I'd have to dig through papers. But Shannon's original work on channel capacity tunes the codes that achieve the Shannon limit to the distribution of the noise source. Commented Jul 10, 2022 at 13:13
• For example the Wikipedia entries en.wikipedia.org/wiki/ECC_memory and en.wikipedia.org/wiki/Coding_theory both state this. It's kind of obvious that, e.g., you dont need a (7,19)-Hamming code if your error model says at most one bit gets flipped. Commented Jul 10, 2022 at 13:28