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This paper mainly study $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes. A Gray map from $ \mathbb{Z}_{2}^{\alpha}\times\mathbb{Z}_{4}^{\beta}[u] $ to $ \mathbb{Z}_{4}^{\alpha+2\beta} $ is defined, and we prove that is a weight preserving and distance preserving map. A MacWilliams-type identity between the Lee weight enumerator of a $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive code and its dual is proved. Some properties of one-weight $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes and two-weight projective $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes are discussed. As main results, some construction methods for one-weight and two-weight $ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $-additive codes are studied, meanwhile several examples are presented to illustrate the methods.
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Table 1.
One-weight
Cases | Weight | Remark |
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Table 2.
Two-weight
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Table 3. Code parameters comparison
Examples | Length of $\Phi(\mathcal{C})$ | Size of $\Phi(\mathcal{C})$ | Lee weight of $\Phi(\mathcal{C})$ | Lee weight in Database in http://www.Z4codes.info/ | Remark |
Ex. 5.3 (i) | 8 | 4 | 8 | 8/10 | As good as in Database |
Ex. 5.3 (ii) | 10 | 2 | 12 | / | New value |
Ex. 5.5 (i) | 32 | 4 | 32 | 32/42 | As good as in Database |
Ex. 5.5 (ii) | 30 | 4 | 32 | 30/40 | Better than Database |
Ex. 5.5 (iii) | 62 | 4 | 64 | 82 | |
Ex. 5.7 | 36 | 8 | 32 | / | New value |
Ex. 6.2 (i) | 9 | 4 | 6 and 12 | 9/12 | Optimal as per Database |
Ex. 6.2 (ii) | 16 | 4 | 8 and 16 | 16/21 | Optimal as per Database |
Ex. 6.4 (i) | 18 | 4 | 16 and 20 | 18/24 | Improves on Database |
Ex. 6.4 (ii) | 11 | 4 | 12 and 13 | 11/14 | Improves on Database |
Ex. 6.6 | 46 | 8 | 36 and 48 | / | New value |
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