Created the Abstract Algebra theorems and definitions cheat sheet

part-5/chapters/chapter-31/chapter-31.tex

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+\chapter{Introduction to Algebraic Coding Theory}
+\subimport{./}{linear-codes.tex}
+\subimport{./}{parity-check-matrix-decoding.tex}
+\subimport{./}{coset-decoding.tex}

part-5/chapters/chapter-31/coset-decoding.tex

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+\section{Coset Decoding}
+
+\begin{theorem}[Coset Decoding Is Nearest-Neighbor Decoding]
+	In coset decoding, a received word $w$ is decoded as a code word $c$ such that $d(w,c)$ is a minimum.
+\end{theorem}
+
+\begin{definition}[Syndrome]
+	If an $(n,k)$ linear code over $\F$ has parity-check matrix $H$, then, for any vector $u$ in $\F^n$, the vector $uH$ is called the \textit{syndrome} of $u$.
+\end{definition}
+
+\begin{theorem}[Same Coset-Same Syndrome]
+	Let $C$ be an $(n,k)$ linear code over $\F$ with a parity-check matrix $H$. Then, two vectors of $\F^n$ are in the same coset of $C$ if and only if they have the same syndrome.
+\end{theorem}

part-5/chapters/chapter-31/linear-codes.tex

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+\section{Linear Codes}
+
+\begin{definition}[Linear Code]
+	An $(n,k)$ \textit{linear code} of a finite field $\F$ is a $k$-dimensional subspace $V$ of the vector space
+	\[ \F^n = \underbrace{\F \oplus \F \oplus \dots \oplus \F}_\text{$n$ copies} \]
+	over $\F$. The members of $V$ are called the \textit{code words}. When $\F$ is $\Z_2$, the code is called \textit{binary}.
+\end{definition}
+
+\begin{definition}[Hamming Distance, Hamming Weight]
+	The \textit{Hamming distance} between two vectors in $\F^n$ is the number of components in which they differ. The \textit{Hamming weight} of a vector is the number of nonzero components of the vector. The \textit{Hamming weight} of a linear code is the minimum weight of any nonzero vector in the code.
+\end{definition}
+
+\begin{theorem}[Properties of Hamming Distance and Hamming Weight]
+	For any vectors $u$, $v$ and $w$, $d(u,v) \leq d(u,w) + d(w,v)$ and $d(u,v) = \text{wt}(u-v)$.
+\end{theorem}
+
+\begin{theorem}[Correcting Capability of a Linear Code]
+	If the Hamming weight of a linear code is at least $2t + 1$, then the code can correct any $t$ or fewer errors. Alternatively, the same code can detect any $2t$ or fewer errors.
+\end{theorem}

part-5/chapters/chapter-31/parity-check-matrix-decoding.tex

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+\section{Parity-Check Matrix Decoding}
+
+\begin{lemma}[Orthogonality Relation]
+	Let $C$ be a systematic $(n,k)$ linear code over $\F$ with a standard generator matrix $G$ and parity-check matrix $H$. Then, for any vector $v$ in $\F^n$, we have $vH=0$ (the zero vector) if and only if $v$ belongs to $C$.
+\end{lemma}
+
+\begin{theorem}[Parity-Check Matrix Decoding]
+	Parity-check matrix decoding will correct any single error if and only if the rows of the parity-check matrix are nonzero and no one row is a scalar multiple of any other row.
+\end{theorem}