We suggest a new approach to obtain bounds on locally correctable and some locally testable binary linear codes, by arguing that these codes (or their subcodes) have coset leader graphs with high discrete Ricci curvature. The bounds we obtain for locally correctable codes are worse than the best known bounds obtained using quantum information theory, but are better than those obtained using other methods, such as the "usual" information theory. (We remark that our methods are completely elementary.) The bounds we obtain for a family of locally testable codes improve the best known bounds.