diff --git a/src/thesis/chapters/2_fundamentals.tex b/src/thesis/chapters/2_fundamentals.tex
index f60f9ae..608dc7f 100644
--- a/src/thesis/chapters/2_fundamentals.tex
+++ b/src/thesis/chapters/2_fundamentals.tex
@@ -6,8 +6,6 @@ communications engineering and quantum information science.
 This chapter provides the relevant theoretical background on both of
 these topics and subsequently introduces the fundamentals of \ac{qec}.
 
-% TODO: Is an explanation of BP with guided decimation needed in this chapter?
-% TODO: Is an explanation of OSD needed chapter?
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Classical Error Correction}
 \label{sec:Classical Error Correction}
@@ -872,8 +870,6 @@ Take for example the two qubits
     \ket{\psi_2} = \alpha_2 \ket{0} + \beta_2 \ket{1}
     .%
 \end{align*}
-% TODO: Fix the fact that \psi is used above for the single-qubit
-% case and below for the multi-qubit case
 We examine the state $\ket{\psi}$ of the composite system.
 Assuming the qubits are independent, this is a \emph{product state}
 $\ket{\psi} = \ket{\psi_1}\otimes\ket{\psi_2}$.
@@ -1374,7 +1370,6 @@ We can describe it using the check matrix
     \right]
     .%
 \end{align}
-% TODO: Check X vs. Z
 The first $n$ columns correspond to $X$ operators acting on the
 corresponding physical qubit, the rest to the $Z$ operators.
 
@@ -1383,7 +1378,7 @@ corresponding physical qubit, the rest to the $Z$ operators.
 
     % tex-fmt: off
     \begin{quantikz}
-        \lstick[2]{$E\ket{\psi}_\text{L}$}           &          & \gate[2]{P_1} & \gate[2]{P_2} & \gate[style={draw=none},2]{\ldots} & \gate[2]{P_{n-k}} &          &          &                 \\
+        \lstick[2]{$E\ket{\psi}_\text{L}$}  &          & \gate[2]{P_1} & \gate[2]{P_2} & \gate[style={draw=none},2]{\ldots} & \gate[2]{P_{n-k}} &          &          &                 \\
                                             &          &               &               &                                    &                   &          &          &                 \\
         \lstick{$\ket{0}_{\text{A}_1}$}     & \gate{H} & \ctrl{-1}     &               &                                    &                   & \gate{H} & \meter{} & \setwiretype{c} \\
         \lstick{$\ket{0}_{\text{A}_2}$}     & \gate{H} &               & \ctrl{-2}     &                                    &                   & \gate{H} & \meter{} & \setwiretype{c} \\
@@ -1409,8 +1404,8 @@ Stabilizer codes are especially practical to work with when they can
 handle $X$ and $Z$ type errors independently.
 As $Z$ errors anti-commute with $X$ operators in the stabilizers and
 vice versa, this property translates into being able to split the
-stabilizers into some being made up of only $X$
-operators and some only of $Z$ operators.
+stabilizers into a subset being made up of only $X$
+operators and the rest only of $Z$ operators.
 We call such codes \ac{css} codes.
 We can see this property in \autoref{eq:steane} in the check matrix
 of the Steane code.
@@ -1428,12 +1423,13 @@ Using the dual code of $\mathcal{C}_2$ \cite[Eq.~3.4]{ryan_channel_2009}
     \bm{x}' \bm{x}^\text{T} = 0 ~\forall \bm{x} \in \mathcal{C}_2 \right\}
     ,%
 \end{align*}
-we can construct the check matrix as
+we define $\bm{H}_X := \bm{H}(\mathcal{C}_2^\perp)$ and $\bm{H}_Z
+:= \bm{H}(\mathcal{C}_1)$, and construct the check matrix as
 \begin{align*}
     \left[
         \begin{array}{c|c}
-            \bm{H}(\mathcal{C}_2^\perp) & \bm{0} \\
-            \bm{0} & \bm{H}(\mathcal{C}_1)
+            \bm{H}_X & \bm{0} \\
+            \bm{0} & \bm{H}_Z
         \end{array}
     \right]
     .%
@@ -1442,7 +1438,7 @@ In order to yield a valid stabilizer code, $\mathcal{C}_1$ and
 $\mathcal{C}_2$ must satisfy the commutativity condition
 \begin{align}
     \label{eq:css_condition}
-    \bm{H}(\mathcal{C}_2^\perp) \bm{H}(\mathcal{C}_1)^\text{T} = \bm{0}
+    \bm{H}_X \bm{H}_Z^\text{T} = \bm{0}
     .%
 \end{align}
 We can ensure this is the case by choosing them such that
@@ -1470,7 +1466,6 @@ code, scaling up of which would be prohibitively expensive
 
 % Bivariate Bicycle codes
 
-% TODO: Introduce H_X and H_Z above
 A recent addition to the class of \ac{qldpc} codes is that of \ac{bb}
 codes \cite[Sec.~3]{bravyi_high-threshold_2024}.
 These are a special type of \ac{css} code, where $\bm{H}_X$ and