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10.12.2024 | Original Paper
New extremal Type II \({\mathbb {Z}}_4\)-codes of length 64
Erschienen in: Applicable Algebra in Engineering, Communication and Computing
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Abstract
Type II \({\mathbb {Z}}_4\)-codes are a class of self-dual \({\mathbb {Z}}_4\)-codes with Euclidean weights divisible by eight. A Type II \({\mathbb {Z}}_4\)-code of length n is extremal if its minimum Euclidean weight is equal to \(8\left\lfloor \frac{n}{24}\right\rfloor +8.\) A small number of such codes is known for lengths greater than or equal to 48. Based on the doubling method, in this paper we develop a method to construct new extremal Type II \({\mathbb {Z}}_4\)-codes starting from a free extremal Type II \({\mathbb {Z}}_4\)-code of length 48, 56 or 64. Using this method, we construct extremal Type II \({\mathbb {Z}}_4\)-codes of length 64 and type \(4^{31}2^2\). Extremal Type II \({\mathbb {Z}}_4\)-codes of length 64 of this type were not known before. Moreover, the residue codes of the constructed extremal \({\mathbb {Z}}_4\)-codes are best known [64, 31] binary codes and the supports of the minimum weight codewords of the residue code and the torsion code of one of these codes yields self-orthogonal 1-design.
