Context: roughly speaking the construction of a error correcting code is a problem to choose some subset in a metric space, such that the points of the subset pairwise as far-distant as possible. Because "transmission via noisy channel" is just some random perturbation of a point, and "optimal decoding" is just taking the nearest point from the initial subset. Shannon taught us (in particular) that just random choice of points as a such subset is basically the best what one can achieve. However the problem we cannot decode efficiently such a random encoder, so it is not practical.
Practically used codes are "linear" - that means we just take a vectors space (typically isomorphic to $F_2^{N}$ (for 0,1 codes)) and the "subset" is LINEAR subspace.
What is more interesting that the codes which were the most practical for the long time in the past - convolutional codes have very simple idea - consider as an ambient vector space several copies of polynomial ring $R = F[x] \oplus ... \oplus F[x]$ and take as "encoder" a map from $F[x] \to R: p(x) \mapsto (g_1 p, g_2 p, ... , g_n p) $, i.e. just multiply input polynomial $p(x)$ by a sequence of fixed polynomials $(g_1(x), g_2(x), ... g_n(x))$.
That simple ! (I would like I would know it when I was a student).
(Such codes, are not the best ones, but quite good, and Viterbi famously proposed efficient algorithm to decode such codes. ) See simple examples of such codes discussed on MO: MO103001, MO101374
Question: So what would we get if we consider ideals in some other rings (and not necessarily commutative - so one-sided ideals) ? For example take a group algebra of your favorite (may be finite) group over $F_2$ (or other field) and consider ideals there - we know their structure, but how good they would be as error-correcting codes ? That means what will be the minimal (=worst) (or 90%-percentile minimal) distance between pairs of points in such linear subspaces ? Are they as good as random subspaces ?
As for the "distance" I mean the Hamming distance on the group algebra, but the question make sense for ideals in any ring with a metric.
I wanted to ask that question for 10 years - really ! It is appeared several days ago I learned that such topics seems to be discussed in literature for many years, but seems not very actively. So I guess some answers are known, at least partially.
PS
In particular it is discussed in https://arxiv.org/abs/2005.08283 and one of the authors
kindly agreed to make an online seminar on his work tomorrow for small group of us,
and everyone is welcome to join: see announce https://t.me/sberlogabig/86
(well, the language is planned not the most common one, sorry about that).
Any way, any opinions, comments, suggestions from MO community would be invaluable (as usually). In particular that would help to better understand the talk, I hope.
PSPS
For some newer codes than convolutional see MO: Hot-topics in error correcting coding related to interesting math. ?
