diff --git a/README.md b/README.md
index 3c38032..34e57eb 100644
--- a/README.md
+++ b/README.md
@@ -143,8 +143,8 @@ points at $g^i$ where $i < n$ like so:
 
 <p align="center">
 <img
-    alt="P(x) = \left(x-1\right)\left(x-g\right)\cdots\left(x-g^{n-1}\right) = \prod_{i=0}^{n-1} \left(x - g^i\right)"
-    src="https://latex.codecogs.com/svg.image?P%28x%29%20%3d%20%5cleft%28x%2d%31%5cright%29%5cleft%28x%2dg%5cright%29%5ccdots%5cleft%28x%2dg%5e%7bn%2d%31%7d%5cright%29%20%3d%20%5cprod_%7bi%3d%30%7d%5e%7bn%2d%31%7d%20%5cleft%28x%20%2d%20g%5ei%5cright%29"
+    alt="P(x) = \prod_{i=0}^{n-1} \left(x - g^i\right) = \left(x-1\right)\left(x-g\right)\cdots\left(x-g^{n-1}\right)"
+    src="https://latex.codecogs.com/svg.image?P%28x%29%20%3d%20%5cprod_%7bi%3d%30%7d%5e%7bn%2d%31%7d%20%5cleft%28x%20%2d%20g%5ei%5cright%29%20%3d%20%5cleft%28x%2d%31%5cright%29%5cleft%28x%2dg%5cright%29%5ccdots%5cleft%28x%2dg%5e%7bn%2d%31%7d%5cright%29"
 >
 </p>
 
@@ -210,8 +210,8 @@ to our original codeword, where $E(x)$ contains up to $e$ non-zero terms:
 
 <p align="center">
 <img
-    alt="E(x) = E_{j_0} x^{j_0} + E_{j_1} x^{j_1} + \cdots + E_{j_{e-1}} x^{j_{e-1}} = \sum_{j \in E} E_j x^j"
-    src="https://latex.codecogs.com/svg.image?E%28x%29%20%3d%20E_%7bj_%30%7d%20x%5e%7bj_%30%7d%20%2b%20E_%7bj_%31%7d%20x%5e%7bj_%31%7d%20%2b%20%5ccdots%20%2b%20E_%7bj_%7be%2d%31%7d%7d%20x%5e%7bj_%7be%2d%31%7d%7d%20%3d%20%5csum_%7bj%20%5cin%20E%7d%20E_j%20x%5ej"
+    alt="E(x) = \sum_{j \in E} E_j x^j = E_{j_0} x^{j_0} + E_{j_1} x^{j_1} + \cdots + E_{j_{e-1}} x^{j_{e-1}}"
+    src="https://latex.codecogs.com/svg.image?E%28x%29%20%3d%20%5csum_%7bj%20%5cin%20E%7d%20E_j%20x%5ej%20%3d%20E_%7bj_%30%7d%20x%5e%7bj_%30%7d%20%2b%20E_%7bj_%31%7d%20x%5e%7bj_%31%7d%20%2b%20%5ccdots%20%2b%20E_%7bj_%7be%2d%31%7d%7d%20x%5e%7bj_%7be%2d%31%7d%7d"
 >
 </p>
 
@@ -281,8 +281,8 @@ To help with this, we introduce a very special polynomial, the
 
 <p align="center">
 <img
-    alt="\Lambda(x) = \left(1 - X_{j_0} x\right)\left(1 - X_{j_1} x\right)\cdots\left(1 - X_{j_{e-1}} x\right) = \prod_{j \in E} \left(1 - X_j x\right)"
-    src="https://latex.codecogs.com/svg.image?%5cLambda%28x%29%20%3d%20%5cleft%28%31%20%2d%20X_%7bj_%30%7d%20x%5cright%29%5cleft%28%31%20%2d%20X_%7bj_%31%7d%20x%5cright%29%5ccdots%5cleft%28%31%20%2d%20X_%7bj_%7be%2d%31%7d%7d%20x%5cright%29%20%3d%20%5cprod_%7bj%20%5cin%20E%7d%20%5cleft%28%31%20%2d%20X_j%20x%5cright%29"
+    alt="\Lambda(x) = \prod_{j \in E} \left(1 - X_j x\right) = \left(1 - X_{j_0} x\right)\left(1 - X_{j_1} x\right)\cdots\left(1 - X_{j_{e-1}} x\right)"
+    src="https://latex.codecogs.com/svg.image?%5cLambda%28x%29%20%3d%20%5cprod_%7bj%20%5cin%20E%7d%20%5cleft%28%31%20%2d%20X_j%20x%5cright%29%20%3d%20%5cleft%28%31%20%2d%20X_%7bj_%30%7d%20x%5cright%29%5cleft%28%31%20%2d%20X_%7bj_%31%7d%20x%5cright%29%5ccdots%5cleft%28%31%20%2d%20X_%7bj_%7be%2d%31%7d%7d%20x%5cright%29"
 >
 </p>
 
@@ -329,8 +329,8 @@ definition of $\Lambda(x)$:
 
 <p align="center">
 <img
-    alt="\Lambda(x) = 1 + \Lambda_1 x + \Lambda_2 x^2 + \cdots + \Lambda_e x^e = 1 + \sum_{k=1}^e \Lambda_k x^k"
-    src="https://latex.codecogs.com/svg.image?%5cLambda%28x%29%20%3d%20%31%20%2b%20%5cLambda_%31%20x%20%2b%20%5cLambda_%32%20x%5e%32%20%2b%20%5ccdots%20%2b%20%5cLambda_e%20x%5ee%20%3d%20%31%20%2b%20%5csum_%7bk%3d%31%7d%5ee%20%5cLambda_k%20x%5ek"
+    alt="\Lambda(x) = 1 + \sum_{k=1}^e \Lambda_k x^k = 1 + \Lambda_1 x + \Lambda_2 x^2 + \cdots + \Lambda_e x^e"
+    src="https://latex.codecogs.com/svg.image?%5cLambda%28x%29%20%3d%20%31%20%2b%20%5csum_%7bk%3d%31%7d%5ee%20%5cLambda_k%20x%5ek%20%3d%20%31%20%2b%20%5cLambda_%31%20x%20%2b%20%5cLambda_%32%20x%5e%32%20%2b%20%5ccdots%20%2b%20%5cLambda_e%20x%5ee"
 >
 </p>
 
@@ -389,8 +389,8 @@ $S_{i-1}, S_{i-2}, \cdots, S_{i-e}$:
 
 <p align="center">
 <img
-    alt="S_i = \Lambda_1 S_{i-1} + \Lambda_2 S_{i-2} + \cdots + \Lambda_e S_{i-e} = \sum_{k=1}^e \Lambda_k S_{i-k}"
-    src="https://latex.codecogs.com/svg.image?S_i%20%3d%20%5cLambda_%31%20S_%7bi%2d%31%7d%20%2b%20%5cLambda_%32%20S_%7bi%2d%32%7d%20%2b%20%5ccdots%20%2b%20%5cLambda_e%20S_%7bi%2de%7d%20%3d%20%5csum_%7bk%3d%31%7d%5ee%20%5cLambda_k%20S_%7bi%2dk%7d"
+    alt="S_i = \sum_{k=1}^e \Lambda_k S_{i-k} = \Lambda_1 S_{i-1} + \Lambda_2 S_{i-2} + \cdots + \Lambda_e S_{i-e}"
+    src="https://latex.codecogs.com/svg.image?S_i%20%3d%20%5csum_%7bk%3d%31%7d%5ee%20%5cLambda_k%20S_%7bi%2dk%7d%20%3d%20%5cLambda_%31%20S_%7bi%2d%31%7d%20%2b%20%5cLambda_%32%20S_%7bi%2d%32%7d%20%2b%20%5ccdots%20%2b%20%5cLambda_e%20S_%7bi%2de%7d"
 >
 </p>