The Z2s -additive codes are subgroups of Z n 2 s , and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s -additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z4-linear Hadamard codes. However, when s > 2, the dimension of the kernel of Z2slinear Hadamard codes of length 2 t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is computed for s = 3. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once t ≥ 3 is fixed. In this case, the number of nonequivalent such codes is also established.
The Zps -additive codes of length n are subgroups of Z n p s , and can be seen as a generalization of linear codes over Z2, Z4, or more general over Z2s . In this paper, we show two methods for computing a parity-check matrix of a Zps -additive code from a generator matrix of the code in standard form. We also compare the performance of our results implemented in Magma with the current available function in Magma for codes over finite rings in general. A time complexity analysis is also shown.
The Z 2 s -additive codes are subgroups of Z n 2 s , and can be seen as a generalization of linear codes over Z 2 and Z 4 . A Z 2 s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z 2 s -additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some Z 2 s -linear Hadamard codes of length 2 t are equivalent, once t is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to t = 11, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on s ∈ {2, 3}, the full classification of the Z 2 s -linear Hadamard codes of length 2 t is established by giving the exact number of such codes.
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