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On the Classification of ℤpℤp2-Linear Generalized Hadamard Codes | IEEE Conference Publication | IEEE Xplore

On the Classification of ℤpp2-Linear Generalized Hadamard Codes


Abstract:

The {\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - additive}} codes are subgroups of \mathbb{Z}_p^{{\alpha _1}} \times \mathbb{Z}_{{p^2}}^{{\alpha _2}}. A${\mathbb{Z}_p}...Show More

Abstract:

The {\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - additive}} codes are subgroups of \mathbb{Z}_p^{{\alpha _1}} \times \mathbb{Z}_{{p^2}}^{{\alpha _2}}. A{\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - linear}} generalized Hadamard (GH) code is a GH code over {\mathbb{Z}_p} which is the Gray map image of a {\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}-additive code. A recursive construction of {\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - additive}} GH codes of type (α1, α2; t1, t2) with t1, t2 ≥ 1 is known, and for which types the corresponding {\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}-linear GH codes are nonlinear over {\mathbb{Z}_p} is also known. In this paper, we generalize some known results for {\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}-linear GH codes with p = 2 to any p≥3 prime when {\alpha _1} \ne 0. First, we present new recursive constructions of {\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - linear}} GH codes having the same type, and show that we obtained equivalent codes. Then, we compute the rank of some families of {\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - Linear }} GH codes. Finally, we show that, unlike {\mathbb{Z}_4}{\text{ - linear}} Hadamard codes, the {\mathbb{Z}_{{p^2}}}{\text{ - Linear }} GH codes are not included in the family of {\mathbb{Z}_p}{\mathbb{Z}_{{p^2}}}{\text{ - Linear }} GH codes with {\alpha _1} \ne 0 when p ≥ 3 prime.
Date of Conference: 01-09 November 2022
Date Added to IEEE Xplore: 07 December 2022
ISBN Information:
Conference Location: Mumbai, India

Funding Agency:


References

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