Codes from Planar nearrings

On Constructing Codes from Planar Nearrings

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

a~ b: iff forall x in N:ax=bx

for a,b in N and specify the set

N

^*:={ x in N: not (x ~ 0)}

Furthermore let

B^* := { N*a+b:a,b in N, not(a = 0)}.

Then D^*:= (N, B^*, I ), where I is the inclusion relation, is a BIB-design or (v,k,λ)-design with parameters

v = #N,

k = #(N^*/~),

λ = k-1,

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

b = {v(v-1)/k}, and

r = v-1.

In connection with a BIB-design one often uses the so-called incidence matrix

A=( 1-δ_{B_j,B_j\p_i})_ij,

where p_i (for i=1,...,v) and B_j (for j=1,...,b) are the points and blocks of the design, respectively. So we arrive at a matrix consisting of 0's and 1's.

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

n = b,

M = v, and

d = 2(r-\lambda ),

and for C_A we have

n = v,

M = b, and

d = 2(k-\mu ),

where

\mu =max { # (B_{i} \cap B_{j}): B_{i},B_{j} in B*,B_{i} not= B_{j}}

There n denotes the length of a code, M the number of codewords, and d the minimal distance. The maximum number of codewords of a given weight is a question of interest to coding theorists. This question is trivial to answer if we do not require anything but a fixed weight. So let A(n,d,w) denote the mentioned bound also for a fixed minimal distance. Then we have the following:

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

B:={ Na+b:a,b in N, not (a= 0)},

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