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Type II codes over \(\mathbb{Z}_4\). (English) Zbl 0898.94009

Type II codes over \(\mathbb{Z}_4\) are introduced as self-dual codes over the integers modulo 4 containing the all-one vector and with all Euclidean weights divisible by 8. Their weight enumerators are characterized using invariant theory. An upper bound on their minimum Euclidean weight is given which leads to a notion of extremality for the Euclidean weight.
Their binary images under the Gray map are formally self-dual with even weights. The authors give three constructions of families of type II of which the extended quadratic residue \(\mathbb{Z}_4\) codes are the main example. They are obtained by Hensel lifting of the binary quadratic residue codes. Their binary images have good parameters. The authors also show relations between type II \(\mathbb{Z}_4\)-codes and even modular lattices.

MSC:

94B05 Linear codes (general theory)
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11E12 Quadratic forms over global rings and fields
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