Faulty share detection in Shamir’s secret sharing

For Shamir's secret key sharing algorithm, we develop the procedure for detection of faulty shares. This procedure consists of the error locator polynomial construction for the data set $ \{ (x_j,y_j)\}_{j=1}^N $ with $ y $ values generated from $ x $ ones by a polynomial interpolant of a degree $ n < N-1 $ with possible occurrence of some errors. The error locator polynomial is sought out in the form of an appropriate Hankel polynomial
$$ \mathcal H_{L}(x;\{ \tau \}) := \left|
\begin{array}{lllll} \tau_0 & \tau_1 & \tau_2 & \ldots & \tau_{L} \\ \tau_1 & \tau_2 & \tau_3 &\ldots & \tau_{L+1} \\ \vdots & \vdots & \vdots & & \vdots \\ \tau_{L-1} & \tau_{L} & \tau_{L+1} & \ldots & \tau_{2L-1} \\ 1 & x & x^2 & \ldots & x^{L} \end{array}
\right| \, , $$
where $ \tau_{\ell} := \displaystyle \sum_{j=1}^{N} y_j \frac{x_j^{\ell}}{W^{\prime}(x_j)} $; $ \displaystyle W(x):=\prod_{j=1}^N (x- x_j) $.