Our starting point is a finite planar (left) nearring (N,+,°). We define an equivalence relation on N via

k = #(N^{*}/~),

λ = k-1,

with #S denoting the cardinality of a set S. In addition we have

r = v-1.

Now, why not considering the rows and the columns of A, respectively,
as codewords. Doing this we receive ** C**^{A}, the row code of A,
and **C**_A, the column code of A, two non-linear equal-weight codes.
The parameters of these codes depend on those of the underlying BIB-design,
namely for **C**^A we have

M = v, and

d = 2(r-\lambda ),

M = b, and

d = 2(k-\mu ),

* C^A is a maximal code, i.e., M=A(n,d,w) for
n,M,d as above and w=r being the weight
of each codeword of C^A.
*

Although the definition of the column code is dual to the one of the row
code, we do not have maximality for the **C**_A's in general.
But there are situations, where both codes behave
in the same way with respect
to maximality. To show these situations, we have to mention some more facts
about planar nearrings.
There exists a construction procedure which
takes some group N (finite for our purpose)
and Φ a non-trivial
group of fixed point free automorphisms of N as input,
and produces numerous planar nearrings
as output. But all of them yield the same (N,**B**^*, I) and
therefore the same codes. Now one possible choice for N is the additive
group of a field or a nearfield. In this case
Φ can be simply identified
with a subgroup of the multiplicative group of one of these structures.
A class of interesting codes is the following:

*
Let N=GF(q^2), where q is a prime power, and let Φ
have order q+1. Then C_A is maximal.
*

But not only the (N,**B**^*,I ) can be used for constructing codes.
There exists a further incidence structure,
namely **D**:=(N,**B**,I)
with

which is often a BIB-design, for instance in a similar situation as above:
*
*

*
Let N=GF(q^2), where q is a prime power, and let Φ
have order q-1. Then the corresponding (N, B,I) is a
BIBD and C_A is maximal.
*

Furthermore these designs are affine planes, where λ =1. In the same spirit there is the more general result:

*
Let (N, B,I) be a BIBD with λ =1 and incidence matrix
A. Then the corresponding C_A is maximal.
*

Also some other geometric structures can be mentioned in this connection:

Let (N,**B**,I) be a Möbius plane with v=m^2+1 and incidence
matrix A. Then the corresponding **C**_A is maximal.

Much attention has been also paid to circular planar nearrings due to their abundance of geometric properties. For these we can at least determine the minimal distance of the column code, namely:

*
Let (N,+,°) be a circular planar nearring. Then the minimal
distance of the column code is 2(k-2).
*

The latest developments also concentrate on linear codes, which are defined by taking the linear hull, and decoding methods for codes from planar nearrings.

Dipl.Ing. Roland A. Eggetsberger

Univ. of Linz, Dep. of Mathematics

Tel. ++43-732-2468-9141