Linear codes over rings

Question

I've researched linear codes over finite fields, so, the next step would be to look in to linear codes over rings.

I can't find many good sources on this topic? I know that it's fairly new mathematics, but does anyone know of any good places where I can read about this?

Thank you.

Answer 1

I found this: https://arxiv.org/abs/1608.01738

Here are also the results of a search on Google Scholar: https://scholar.google.com/scholar?q=coding+over+rings&hl=en&as_sdt=0&as_vis=1&oi=scholart

Answer 2

You could start with any of the thousands of hits you get just by googling "linear codes over rings." From experience I know the hits authored by Jay Wood are good.

Also just searching at google books yields several texts that look to be focused on that particular topic. I have not read these, I've just gleaned information from more ordinary coding texts. This may prove more convenient than piecing together the research papers from the above broader internet search.

I know codes over finite quasi-Frobenius rings have been extensively researched because of their nice duality.

My recollection of Pless's Introduction to error correcting codes was that they introduced Gray codes as codes over $\mathbb Z/4\mathbb Z$.

Answer 3

Well, the book of Wan, ''Quaternary Codes'', World Scientific, is excellent. It contains a description of the important Kerdock and Preparata codes and their generalizations including the necessary math background.

Answer 4

The earliest reference that I am aware of is by Shankar, IEEE Transactions on Information Theory, 1979 with a title that might be "BCH Codes over Rings" or something similar.