Two “Dual” Families of Nearly-Linear Codes over ℤ
p
, p odd |
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Authors: | Bram van Asch Henk C A van Tilborg |
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Affiliation: | (1) Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (e-mail: {A.G.v.Asch, H.C.A.v.Tilborg}@tue.nl), NL |
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Abstract: | Since the paper by Hammons e.a. 1], various authors have shown an enormous interest in linear codes over the ring ℤ4. A special weight function on ℤ4 was introduced and by means of the so called Gray map ϕ : ℤ4→ℤ2
2 a relation was established between linear codes over ℤ4 and certain interesting non-linear binary codes of even length.
Here, we shall generalize these notions to codes over ℤ
p2
where p is an arbitrary prime. To this end, a new weight function will be proposed for ℤ
p2
. Further, properties of linear codes over ℤ
p2
will be discussed and the mapping ϕ will be generalized to an isometry between ℤ
p2
and ℤ
p
p
, resp. between ℤ
p2
n
and ℤ
p
pn
. Some properties of Galois rings over ℤ
q
will be described and two dual families of linear codes of length n = p
m
− 1, gcd(m, p) = 1, over ℤ
q
will be constructed. Taking q = p
2, their images under the new mapping can be viewed as a generalization of the binary Kerdock and the Preparata code, although
they miss some of their nice combinatorial properties.
Received: June 19, 2000; revised version: November 6, 2000 |
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Keywords: | :p2-ary codes Gray map Galois rings Kerdock code Preparata code |
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