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Article Contents

# Rotated $A_n$-lattice codes of full diversity

• * Corresponding author

This work was supported by FAPESP 2013/25977-7 and CNPq 429346/2018-2

• Some important properties of lattices are packing density and full diversity, which may be good for signal transmission over both Gaussian and Rayleigh fading channel, respectively. The algebraic lattices are constructed through twisted homomorphism of some modules in the ring of integers of a number field $\mathbb{K}$. In this paper, we present a construction of some families of rotated $A_n-$lattices, for $n = 2^{r-2}-1$, $r \geq 4$, via totally real subfield of cyclotomic fields. Furthermore, closed-form expressions for the minimum product distance of those lattices are obtained through algebraic properties.

Mathematics Subject Classification: Primary: 52C07; Secondary: 11H31, 11H71.

 Citation:

• Table 1.  Normalized minimum product distance versus center density (from [5,12,15,16,18,19] and the results presented here)

 $r$ $n$ $\sqrt[n]{d_{p}(\mathbb{Z}^n)}$ $\sqrt[n]{d_{p}(D_n)}$ $\sqrt[n]{d_{p}(A_n)}$ $\delta(\mathbb{Z}^n)$ $\delta(D_n)$ $\delta(A_n)$ $4$ $3$ $0.52275$ $0.41491$ $0.44544$ $0.12500$ $0.17677$ $0.17677$ $5$ $7$ $0.30080$ $-$ $0.27602$ $0.00780$ $0.04419$ $0.03125$ $6$ $15$ $0.20138$ $0.19229$ $0.18513$ $0.00003$ $0.00276$ $0.00138$ $7$ $31$ $0.06220$ $-$ $0.12782$ $10^{-10}$ $10^{-5}$ $10^{-6}$ $8$ $63$ $0.09221$ $0.09120$ $0.08936$ $10^{-19}$ $10^{-10}$ $10^{-11}$ $9$ $127$ $0.04542$ $-$ $0.06284$ $10^{-39}$ $10^{-20}$ $10^{-21}$ $10$ $255$ $0.03172$ $-$ $0.04431$ $10^{-77}$ $10^{-39}$ $10^{-40}$ $11$ $511$ $0.01819$ $-$ $0.03129$ $10^{-154}$ $10^{-78}$ $10^{-79}$ $12$ $1023$ $0.01569$ $-$ $0.02211$ $10^{-308}$ $10^{-155}$ $10^{-152}$ $13$ $2047$ $0.00522$ $-$ $0.01563$ $10^{-617}$ $10^{-309}$ $10^{-310}$ $14$ $4095$ $0.01106$ $0.01106$ $0.01106$ $10^{-1233}$ $10^{-617}$ $10^{-619}$ $15$ $8191$ $0.00163$ $-$ $0.00781$ $10^{-2466}$ $10^{-1234}$ $10^{-1235}$ $16$ $16383$ $0.00319$ $-$ $0.00552$ $10^{-4932}$ $10^{-2467}$ $10^{-2468}$ $17$ $32767$ $0.00130$ $-$ $0.00390$ $10^{-9864}$ $10^{-4933}$ $10^{-4935}$ $18$ $65535$ $0.00276$ $0.00276$ $0.00276$ $10^{-19729}$ $10^{-9865}$ $10^{-9867}$ $19$ $131071$ $0.00079$ $-$ $0.00195$ $10^{-39457}$ $10^{-19729}$ $10^{-19731}$ $20$ $262143$ $0.00138$ $0.00138$ $0.00138$ $10^{-78913}$ $10^{-39457}$ $10^{-39460}$
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