Planar near-rings form a special class of algebraic systems. Here is a short account on the basics of near-rings in general and planar near-rings in particular.

The set of all real functions, together with pointwise addition
and composition, does not form a ring. Observe that
holds by
definition, while
holds for all
if and only if is *linear*, i.e., if
for all
. Structures of the type like
are
called *near-rings*:

Of course, every ring is a near-ring; hence near-rings are generalized rings. Two ring axioms are missing: the commutativity of addition and (much more important) the other distributive law.

Actually, the first near-rings considered were
*near-fields*,
near-rings
in which
forms a group. In 1905,
L. E. Dickson constructed the first *proper* near-fields by
``distorting'' the multiplication in a field. These types of
near-fields are now called *Dickson near-fields*. Two years later,
Veblen and Wedderburn used near-fields to coordinatize geometric
planes. In a monumental paper, H. Zassenhaus showed in 1936 that all
finite near-fields are Dickson ones, with the exception of 7
(well-known) cases. 51 years later, Zassenhaus showed that there do
exist non-Dickson infinite near-fields of every prime characteristic.

For near-rings, we now have a sophisticated structure theory. Quite often, the results look similar to the ring case, but the proofs are completely different. For instance, matrix rings (``the stuff rings are made of'') have to be replaced by centralizer near-rings , where is a fixed-point free automorphism group.

These types of automorphisms turn up in another context as well; this one has nice applications to real-world problems.

This condition is motivated by geometry. It implies that two ``non-parallel lines and have exactly one point of intersection''. One example is , where . There are various methods to construct planar near-rings (see [Cla92,Pil83,Pil96]). Most of them use fixed-point-free automorphism groups; we just present the easiest way.

Field | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |

Fertilizer | ||||||||||||||

x | x | x | x | x | x | |||||||||

x | x | x | x | x | x | |||||||||

x | x | x | x | x | x | |||||||||

x | x | x | x | x | x | |||||||||

x | x | x | x | x | x | |||||||||

x | x | x | x | x | x | |||||||||

x | x | x | x | x | x | |||||||||

Yields: | 2.8 | 6.1 | 1.2 | 8.5 | 8.0 | 3.9 | 7.7 | 5.5 | 4.0 | 10.7 | 3.2 | 4.9 | 3.8 | 5.5 |

The statistical analysis uses the incidence matrix
of the
design:
compute
.
Then Linear Algebra tells us that
are the best
estimates for the effects of the ingredients
.
In our case, we get the best estimates for the effects of
as
.
If we take
, we can expect a
yield of
, which is considerably better than the
yields which we got in our experiment. A more detailed
statistical analysis gives confidence intervals for the effects of , as well as information concerning their *synergy effects*.

Even more can be done with incidence matrices coming from BIB-Designs. The rows in the matrix are -sequences, so they can be taken as a binary code . This is a nonlinear, constant weight and constant distance code of length with codewords. These codes have the largest number of codewords among all codes with equal weight , length , and fixed minimal distance . So they also solve discrete sphere packing problems!