This paper presents a method for decoding linear codes by solving a convex optimization problem, specifically $\ell_1$-minimization. The goal is to recover an input vector $f \in \mathbb{R}^n$ from corrupted measurements $y = Af + e$, where $A$ is an $m \times n$ matrix and $e$ is an error vector. The key result is that under suitable conditions on $A$, the input $f$ can be uniquely recovered by solving the $\ell_1$-minimization problem: $$ \min_{g \in \mathbb{R}^n} \|y - Ag\|_{\ell_1} $$ provided that the number of non-zero entries in $e$ is not too large, i.e., $\|e\|_{\ell_0} \leq \rho \cdot m$ for some $\rho > 0$. This approach is efficient and works well even when a significant fraction of the output is corrupted. The method is related to finding sparse solutions to underdetermined systems of linear equations and recovering signals from highly incomplete measurements. The paper also introduces the concept of restricted isometry constants and shows that Gaussian random matrices satisfy these properties with high probability, enabling the successful recovery of sparse signals. Numerical experiments demonstrate that the $\ell_1$-minimization approach can recover the original signal even when up to 34% of the entries are corrupted, depending on the ratio of $m$ to $n$. The results improve upon earlier work and provide a solid theoretical foundation for the practical success of $\ell_1$-minimization in signal recovery.This paper presents a method for decoding linear codes by solving a convex optimization problem, specifically $\ell_1$-minimization. The goal is to recover an input vector $f \in \mathbb{R}^n$ from corrupted measurements $y = Af + e$, where $A$ is an $m \times n$ matrix and $e$ is an error vector. The key result is that under suitable conditions on $A$, the input $f$ can be uniquely recovered by solving the $\ell_1$-minimization problem: $$ \min_{g \in \mathbb{R}^n} \|y - Ag\|_{\ell_1} $$ provided that the number of non-zero entries in $e$ is not too large, i.e., $\|e\|_{\ell_0} \leq \rho \cdot m$ for some $\rho > 0$. This approach is efficient and works well even when a significant fraction of the output is corrupted. The method is related to finding sparse solutions to underdetermined systems of linear equations and recovering signals from highly incomplete measurements. The paper also introduces the concept of restricted isometry constants and shows that Gaussian random matrices satisfy these properties with high probability, enabling the successful recovery of sparse signals. Numerical experiments demonstrate that the $\ell_1$-minimization approach can recover the original signal even when up to 34% of the entries are corrupted, depending on the ratio of $m$ to $n$. The results improve upon earlier work and provide a solid theoretical foundation for the practical success of $\ell_1$-minimization in signal recovery.