Table 1 The description of the LLR-based TA in ZF criterion

Perform standard LR algorithm such as LLL or ELR on H and generate

the reduced basis \(\bar {\textbf {H}}\) and unimodular matrix T;

Using (14) to get the PEP of each component, and set j=1;

Do

If j=N+1

break;

end

Find the jth largest PEP \(P\left (x_{k}\neq \hat {x}_{k}|\bar {y}_{k}\right)\);

Using (38) to obtain \(\lambda _{k,i}^{*}\) and get \(P_{i}\left (\tilde {x}_{k}\neq \hat {\tilde {x}}_{k}|\tilde {y}_{k}\right),i\neq k\) ;

Calculate \(i^{*}=\arg \min _{i} P_{i}\left (\tilde {x}_{k}\neq \hat {\tilde {x}}_{k}|\tilde {y}_{k}\right)\);

If \(P_{i^{*}}\left (\tilde {x}_{k}\neq \hat {\tilde {x}}_{k}|\tilde {y}_{k}\right)<P\left (x_{k}\neq \hat {x}_{k}|\bar {y}_{k}\right)\)

Set j=1,

\(\ \ \ \ \ \ \ \ \ \ \ \ \ \bar {y}_{k}\leftarrow \bar {y}_{k}+\lambda _{k,i^{*}}\bar {y}_{i^{*}}\),

\(\ \ \ \ \ \ \ \ \ \ \ \ \ \textbf {t}^{i}\leftarrow \textbf {t}^{i^{*}}-\lambda _{k,i^{*}}\textbf {t}^{k}\);

\(\ \ \ \ \ \ \ \ \ \ \ \ \ \textbf {h}^{i}\leftarrow \textbf {h}^{i^{*}}-\lambda _{k,i^{*}}\textbf {h}^{k}\);

Using (28) and (29) to update the covariance matrix C;

Use \(P_{i^{*}}\left (\tilde {x}_{k}\neq \hat {\tilde {x}}_{k}|\tilde {y}_{k}\right)\) to update \(P\left (x_{k}\neq \hat {x}_{k}|\bar {y}_{k}\right)\);

else

j=j+1;

End if

While (true)