## Z4-Goethals Codes, Decoding and Designs

Kalle Ranto, Z4-Goethals Codes, Decoding and Designs. TUCS Dissertations 42. Turku Centre for Computer Science, 2002.

### Abstract:

In 1990's some classical binary nonlinear error-correcting codes were
presented as linear codes over an alphabet Z4, i.e. the integers
modulo 4, via the Gray map. These codes include Kerdock, Preparata,
Goethals, Delsarte-Goethals, and Nordstrom-Robinson codes. This
invention triggered off the research on Z4-codes and Galois rings
in coding theory.

Helleseth, Kumar, and Shanbhag introduced a generalized family of
Z4-Goethals codes in 1996. In this thesis we consider two aspects
of the codes in this family: their decoding and combinatorial designs
related to them. In both tasks we end up with Dickson polynomials and
equations involving them. This is something that has not appeared in
the literature before.

Chapters 1 and 2 are introductory. We recall some preliminary results
from finite fields and Galois rings. The basic notions in the theory of
error-correcting codes and combinatorial designs are given.

In Chapter 3 we introduce a uniform algebraic decoding algorithm for
all Z4-Goethals codes. This algorithm can correct all errors up to
error-correcting capability, that is, all errors with Lee weight at
most 3.

In Chapters 4 and 5 we construct several new infinite families of
3-designs from some of the Z4-Goethals codes. These designs have
parameters 3-(2^m,8,\lambda) with odd m >= 5 and the smallest
designs have \lambda=14(2^m-8)/3. Also a special connection between
two designs with blocks sizes 7 and 8 is described with the affine
geometry.

### Files:

Full publication in PDF-format

### BibTeX entry:

@PHDTHESIS{phdRanto02a,  title = {Z4-Goethals Codes, Decoding and Designs},  author = {Ranto, Kalle},  number = {42},  series = {TUCS Dissertations},  school = {Turku Centre for Computer Science},  year = {2002},  keywords = {Goethals code, decoding, combinatorial design, Dickson polynomial},  ISBN = {951-29-2314-9},}

Belongs to TUCS Research Unit(s): FUNDIM, Fundamentals of Computing and Discrete Mathematics

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