Is there any particular application for which only codes over rings can succeed, not codes over fields? Why there are so many codes based over rings not over fields proposed recently? I know rings are more general structure. So it gives more flexibility. But is there any particular application for which only codes over rings can succeed?
 A: There is unlikely to be any particular application for which only codes over rings can succeed.
However there are applications for which codes over rings (and sequences over rings where cyclically equivalent codewords are taken to be a single sequence) perform better under the relevant performance measure.
The rest is long but gives a significant example.
For example, in 1990 P. Sole had already observed that $Z_4$ codes might be better than binary codes (in particular binary Kerdock codes which were the best known for the relevant parameters could be written in a $Z_4$ form as well).
In S. Boztas' PhD Thesis in 1990, $Z_4-$ valued sequences (dubbed Family A) were constructed using Galois Rings and these sequences settled a long problem about the discrepancy between binary and nonbinary performance measures for CDMA (wireless communications) spread spectrum sequence sets. More on that at the end.
A bit later Hammons, Kumar, Calderbank, Sloane and Sole showed that binary nonlinear Kerdock and Preparata codes which obeyed the MacWilliams' identities (normally only obeyed by binary linear codes) could be viewed as linear (using Galois rings) over $Z_4$ symbol alphabet. This led to the intense activity in codes over rings. And some codes over rings have superior parameters. Below, I describe the application where this difference in going to the ring $Z_4$ instead of the field $GF(2)$ or $GF(4)$ is significant.
Binary vs Nonbinary Correlation and Family A
Note: Sequences $s_t$ over $Z_4$ can also be written as $i^{s_t}$ over the 4th roots of unity.
In spread spectrum CDMA, each symbol is modulated by multiplying it with a pseudorandom binary sequence at a much higher rate. Each user is assigned a sequence and different users transmit without frame synchronisation, which means that in using signal detection other users' interference appears as noise in the desired user's receiver via an inner product of a sequence and cyclic shifts of another sequence.
Mathematical abstraction of this problem is that one needs many sequences over some symbol alphabet such that each sequence is nearly orthogonal to its own cyclic shifts as well as all cyclic shifts of other sequences. Define a set of vectors by including all the cyclic shifts of each sequence into an augmented set
$\{a_1,\ldots,a_k\} \in X^d$. Consider $X$ to be real or complex. The goal is to minimize the highest nontrivial innner product magnitude which measures the worst case interference.
Relevant results for this case are due to Welch, Kabatianski, Levenshtein, Sidelnikov. Welch's applies to arbitrary vectors, real or complex. The others apply to vectors constructed from complex roots of unity of some finite order.
Welch's bound states
Let $e\geq 1$ be an integer and let $a_1,\ldots,a_k$ be distinct vectors in $\mathbb{C}^d.$ Then the following inequalities hold
$$
\sum_{i=1}^k \sum_{j=1}^k \left| \langle a_i, a_j \rangle \right|^{2e} \geq \frac{\left(\sum_{i=1}^k \lVert a_i \rVert^{2e}\right)^2}{\binom{d+e-1}{e}},
$$
If the set of vectors you are interested in is of size roughly $d^u,$ the tightest lower bound is obtained by choosing $e=\lfloor u\rfloor.$
Sidelnikov bound applies to roots of unity, and has $\{-1,+1\}$-valued and higher order of roots of unity valued versions.
The $\{-1,+1\}$ valued Sidelnikov bound gives a lower bound on magnitude of roughly $\sqrt{2d}$ while the nonbinary lower bound is only $\sqrt{d}$ if family size is roughly $d^2$ (i.e., $d$ cyclically distinct sequences, supporting $d$ different transmitters).
The construction of Family A, demonstrates that this gap in lower bounds is real and using quaternary signals one can reduce interference by a factor of $\sqrt{2}$ or 3 dB.
