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\begin{filecontents}[overwrite]{algebraic-coding-theory-notes.bib}
@misc{hill,
  author = {Raymond Hill},
  title = {{A First Course in Coding Theory}},
  year = {1986}
}
@misc{lint,
  author = {J.H. van Lint},
  title = {{Introduction to Coding Theory}},
  year = {1981}
}
\end{filecontents}
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\title{Algebraic Coding Theory}
\author{arnaucube}
\date{2026}

\begin{document}

\maketitle

\begin{abstract}
    Notes taken while studying Algebraic Coding Theory, mostly from Hill book \cite{hill} and Lint book \cite{lint}.

    Usually while reading books and papers I take handwritten notes in a notebook, this document contains some of them re-written to $LaTeX$.

    The proofs may slightly differ from the ones from the books, since I try to extend them for a deeper understanding.
\end{abstract}

\tableofcontents


\section{Vector spaces over finite fields, quick recap}
Let $q$ be a prime power. 

$\Vnq$: set $GF(q)^n$ of all ordered $n$-tuples over $GF(q)$. $\Vnq$ is a
vector space.

A set of vectors $\{v_1, v_2, \ldots, v_r \}$ is said to be \emph{linear dependent} if $\exists$ scalars $a_1, \ldots, a_r$, not all zero, such that
$$a_1 v_1 + a_2 v_2 + \ldots + a_r v_r = 0$$

They are \emph{linear independent} if
$$a_1 v_1 + a_2 v_2 + \ldots + a_r v_r = 0 ~\stackrel{implies}{\Longrightarrow}~ a_1=a_2=\ldots=a_r=0$$

Let $C$ a subspace of $\Vnq$. Then a subset $\{v_1, v_2, \ldots, v_r \} \subseteq C$ is called a \emph{generating set} (or \emph{spanning set}) of $C$ if every vector in $C$ can be expressed as a linear combination of $v_1, \ldots, v_r$.

A generating set of $C$ which is also linearly independent is called a \emph{basis} of $C$.

\begin{thm}{4.3}
  Suppose $\{v_1, v_2, \ldots, v_k\}$ is a basis of a subspace $C$ of $\Vnq$. Then  
  \begin{enumerate}[i.]
  \item every vector of $C$ can be expressed uniquely as a linear combination of the basis vectors.
  \item $C$ contains exactly $q^k$ vectors.
  \end{enumerate}
\end{thm}
\begin{proof}
  \begin{enumerate}[i.]
\item suppose vector $x \in C$ such that
$$x = a_1 v_1 + a_2 v_2 + \cdots + a_k v_k$$
$$x = b_1 v_1 + b_2 v_2 + \cdots + b_k v_k .$$

then
$$(a_1-b_1)v_1 + (a_2-b_2)v_2 + \cdots + (a_k-b_k)v_k = 0 .$$

But the set $\{v_1, v_2, \ldots, v_k\}$ is linearly independent, and so
$$a_i - b_i = 0 \quad \text{for } i = 1,2,\ldots,k,$$
i.e.,
$$a_i = b_i \quad \text{for } i = 1,2,\ldots,k.$$

\item By (i), the $q^k$ vectors
$$\sum_{i=1}^{k} a_i v_i \qquad (a_i \in \mathrm{GF}(q))$$
are precisely the distinct vectors of $C$.
\end{enumerate}
\end{proof}


\section{Linear codes}
The theorems of this section are from \cite{hill} and follow the original enumeration of the book.

Note about notation:\\
A $q$-ary linear code $[n, k, d]$-code is also a $q$-ary $(n, q^k, d)$-code, but the opposite is not always true.

\subsection{Some definitions}
Let $q$ be a prime power. View $(F_q)^n$ as the vector space $\Vnq$.

\begin{defn}{}[Linear code over GF(q)]
  A \emph{linear code over $GF(q)$} is a subspace of $\Vnq$, with $n \geq 0$.
\end{defn}

Thus $C \subseteq \Vnq$ is a linear code iff
\begin{enumerate}[i.]
  \item $u+v \in C ~~ \forall u,v \in C$
  \item $au \in C ~~ \forall u \in C,~ a \in GF(q)$
\end{enumerate}

If $C$ is a $k$-dimensional subspace of $\Vnq$, then the linear code $C$ is called an $[n, k]$-code. (sometimes with minimum distance $d$: $[n, k, d]$-code).

\begin{defn}{}[Weight]
  Weight, $w(x)$, of a vector $x \in \Vnq$ is the number of non-zero entries of $x$.
\end{defn}

\begin{lemma}{5.1}\label {h.5.1}
  if $x,y \in \Vnq$, then $d(x,y)=w(x-y)$.
\end{lemma}
\begin{proof}
  the vector $x-y$ has non-zero entries in preceisely those places where $x$ and $y$ differ.
\end{proof}

\begin{thm}{5.2}
  let $C$ a linear code, let $w(C)$ be the smallest of weights of the non-zero codewords of $C$. Then $d(C)=w(C)$.
\end{thm}
\begin{proof}
  $\exists~ x,y \in C$ s.th. $d(C)=d(x,y)$.

  By Lemma \ref{h.5.1}, $d(C)=w(x-y) \geq w(C)$, since $x-y$ is a codeword of the linear code $C$ (ie. $x-y \in C$).

  On the other hand, for $x \in C$,\\
  $w(C)=w(x)=d(x,0) \geq d(C)$, since $0 \in C$.

  Hence, $d(C) \geq w(C)$ and $w(C) \geq d(C) ~~\Longrightarrow~ d(C)=w(C)$.
\end{proof}


\begin{defn}{}[Generator matrix (GM)]
  a $k \times n$ matrix whose rows form a basis of a linear $[n,k]$-code is called a \emph{generator matrix} (GM) of the code.
\end{defn}

\subsection{Equivalence}

\begin{thm}{5.4}
  two known $k \times n$ matrices generate equivalent linear $[n,k]$-codes over $GF(q)$ if one matrix can be obtained from the other by a sequence of operations of the following types:
  \begin{enumerate}
    \item[R1.] permutation of the rows
    \item[R2.] multiplication of a row by a non-zero scalar
    \item[R3.] addition of a scalar multiple of one row to another
    \item[C1.] permutation of the columns
    \item[C2.] multiplication of any column by a non-zero scalar
  \end{enumerate}
\end{thm}

\begin{thm}{5.5} \label{h.5.5}
  let $G$ a GM of an $[n, k]$-code. Then by performing the previous operations, $G$ can be transformed to the standard form $[ I_k | A ]$
  where $I_k$ is the $k \times k$ identity matrix, and $A$ is a $k \times (n-k)$ matrix.
\end{thm}

\subsection{Encoding with a linear code}
Let $C$ a $[n,k]$-code over $GF(q)$ with GM $G$. $C$ contains $q^k$ codewords $\Longrightarrow$ can be used to communicate any one of $q^k$ distinct messages.

Iddentify these messages with the $q^k$ tuples of $V(k,q)$.

Encoding:\\
msg vector: $u=u_1 u_2 \ldots u_k$; rows of $G$: $r_1 r_2 \ldots r_k$. Encode
$$uG = \sum^k_{i=1} u_i r_i \in C$$
ie. $uG$ is a codeword of $C$; a linear combination of the rows of the GM.

Encoding function:
\begin{align*}
  V(k,q) &\longrightarrow C \subseteq \Vnq\\
  u &\longmapsto uG
\end{align*}

If $G$ is in standard form (\ref{h.5.5}), $G= [I_k | A]$ with $A=[a_{ij}]$, then encoding $u$ is:
$$x=uG = x_1x_2 \ldots x_k x_{k+1} \ldots x_n$$
where
\begin{align*}
  &x_i=u_i ~~\text{for}~ 1 \leq i \leq k, ~\text{message digits}\\
  &x_{k+i}=\sum^k_{j=1}~ a_ji u_j ~~\text{for}~ 1 \leq i \leq n-k, ~\text{check digits}\\
\end{align*}

the check digits add redundancy to gie protection against noise.

\subsection{Decoding with a linear code}
Codeword $x=x_1 x_2 \ldots x_n$, received vector $y=y_1 y_2 \ldots y_n$, error vector $e = y-x=e_1 e_1 \ldots e_n$.

\subsubsection{Cosets}
Cosets here are similar to the ones from typical group theory, just from another perspective.

\begin{defn}{}[Coset]
  Let $C$ a $[n,k]$-code over $GF(q)$ and $a \in \Vnq$.
  Then the set $a + C = \{a+x | x\in C\}$ is called a \emph{coset} of $C$.
\end{defn}

\begin{defn}{}[Coset leader]
  the vector having minimum weight in a coset is called the \emph{coset leader}.
\end{defn}

\begin{lemma}{6.3} \label{h.6.3}
  Let $a+C$ a coset of $C$ and $b \in a+C$. Then $b+C=a+C$.
\end{lemma}
\begin{proof}
  since $b \in a+C ~~\Longrightarrow~~ b=a+x$ for some $x \in C$.
  If $b+y \in b+C$ then
  $$b+y=(a+x)+y = a+(x+y) \in a+C$$

  For the other direction, if $a+x \in a+C$ then
  $$a+x=(b-x)+z = b+(z-x) \in b+C$$
  hence $a+C \subseteq b+C$.

  Therefore $b+C = a+C$.
\end{proof}

\begin{thm}{6.4}[Lagrange] \label{h.6.4}
  suppose $C \subseteq \Vnq$ is a $[n,k]$-code over $GF(q)$. Then
  \begin{enumerate}[i.]
    \item evvery vector of $\Vnq$ is in some coset of $C$
    \item every coset contains exactly $q^k$ vectors
    \item two cosets either are disjoint or coincide (partial overlap is impossible)
  \end{enumerate}
\end{thm}
\begin{proof}
  \begin{enumerate}[i.]
    \item if $a\in \Vnq$ then $a=a+0 \in a+C$
    \item the map
      \begin{align*}
        C &\longrightarrow a+C ~\forall~ x \in C\\
        x &\longmapsto a+x
      \end{align*}
      is a one-to-one map, hence $|a+C| = |C| = q^k$.
    \item by contradiction, suppose cosets $a+C,~ b+C$ overlap. Then for some vector $v$,
      $$v \in (a+C) \cap (b+C)$$
      thus for some $x,y \in C$, $v=a+x=b+y$.

      Hence $b=a+(x-y) \in a+C$, thus by Lemma \ref{h.6.3} we have $b+C=a+C$.
  \end{enumerate}
\end{proof}

\begin{cor}{}
  \ref{h.6.4} shows that $\Vnq$ is partitioned into disjoint cosets of $C$:
  $$\Vnq = (0+C) \cup (a_1+C) \cup \ldots \cup (a_s +C)$$
  where $s=q^{n-k} -1$, and by Lemma \ref{h.6.3} we may take $0, a_1, \ldots, a_s$ to be coset leaders.
\end{cor}

\begin{eg}{6.5} \label{h.6.5}
  Let $C$ be the binary $[4,2]$-code with generator matrix
$$
G = \begin{pmatrix}
1 & 0 & 1 & 1 \\
0 & 1 & 0 & 1
\end{pmatrix}
$$

ie.
$C = \{0000,\, 1011,\, 0101,\, 1110\}$.

Then the cosets of $C$ are
\begin{align*}
  0000 + C &= C \text{ itself}\\
  1000 + C &= \{1000,\, 0011,\, 1101,\, 0110\}\\
  0100 + C &= \{0100,\, 1111,\, 0001,\, 1010\} = 0001+C\\
  0010 + C &= \{0010,\, 1001,\, 0111,\, 1100\}
\end{align*}

by Lemma \ref{h.6.3}: $0001 \in 0100 + C$, thus $0001+C=0100+C$.
\end{eg}

\subsection{Probability of error correction (in binary linear codes)}
To evaluate how good a code is, must calculate the \emph{probability} that a received vector will be decoded as the codeword which was sent.

Let $p$ be the symbol error probability.

Probability that the error vector is agiven vector of weight $i$ is $p^i (1-p){n-i}$.

\begin{thm}{6.7}
  Let $C$ a binary $[n,k]$-code, and for $i=0,1,\ldots,n$ let $\alpha_i$ denote the number of coset leaders of weight $i$.

  Then the probability $P_{corr}(C)$ that a received vector decoded by means of a standard array is the codeword which was sent is
  $$P_{corr}(C) = \sum_{i=0}^n \alpha_i \cdot p^i(1-p)^{n-i}$$
\end{thm}

\begin{eg}{6.8}
  $[n,k]$-code from \ref{h.6.5}, coset leaders are $0000, 1000, 0100, 0010$. Hence $\alpha_0=1,~ \alpha_1=3,~ \alpha_2=\alpha_3=\alpha_4=0$, so
  $$P_{corr}(C) = (1-p)^4 + 3 p(1-p)^3 = (1-p)^3 (1+2p)$$
\end{eg}

\begin{lemma}{6.9}[Remark 6.9]\label{remark.6.9}
  if $d(C)=2t+1$ or $d(C)=2t+2$, then $C$ can correct any $t$ errors.

  Hence every vector of weight $\leq t$ is a coset leader and so $\alpha_i=\left( \stackrel{n}{i} \right)$ for $0 \leq i \leq t$.
\end{lemma}

\begin{defn}{}[Rate]
  $[n, k]$-code $C$ uses $n$ symbols to send $k$ message symbols; it has \emph{rate} $R(C)=k/n$.

  A good code will have a high rate.
\end{defn}

\begin{defn}{}[Capacity]
  the \emph{capacity} $\mathscr{C}(p)$ of a binary symmetric channel with symbol error probability $p$ is
  $$\mathscr{C}(p) = 1+ p \cdot \log_2{(p)} + (1-p) \cdot \log_2{(1-p)}$$
\end{defn}


\begin{thm}{6.12}[Shanon's theorem]
  channel binary symmetric with symbol error probability $p$.

  Let $R < \mathscr{C}(p)$.

  Then for any $\epsilon >0$, $\exists$ for sufficient large $n$, an $[n,k]$-code $C$ of rate $k/n \geq R$ such that $P_{err}(C) < \epsilon$.
\end{thm}

\begin{defn}{}[$P_{symb}$]
  $P_{symb}$ denotes the average probability that a message symbol is in error after decoding.
\end{defn}

\begin{thm}{6.14}
  let $C$ a binary [n,k]-code and let $A_i$ denote the number of codewords of $C$ of weight $i$.

  Then, if $C$ is used for error detection, the probability of an incorrect message being undetected is
  $$P_{undetec}(C) = \sum_{i=1}^n A_i \cdot p^i (1-p)^{n-i}$$
\end{thm}

\begin{eg}{6.15}
  for the code of \ref{h.6.5},
  $$P_{undetec}(C) = \underbrace{1 \cdot p^2 (1-p)^2}_{0101} + \underbrace{2 \cdot p^3(1-p)^{4-3=1} }_{1011,~1110}= p^2 -p^4$$
\end{eg}



\vspace{0.5cm}

\subsection{Exercises}
\begin{ex}{5.1}
  is the binary (11, 24, 5)-code linear?
\end{ex}
\begin{proof}
  recall: a q-ary linear $[n,k,d]$-code is also a q-ary $(n, q^k, d)$-code, but
  the opposite is not always true.

  $\Longrightarrow~ (11, 24, 5)$-code to be linear it would mean that $24=q^k$.

  Since it is binary, $q=2$. But $24 \neq 2^k$, so it can not be linear.
\end{proof}

\begin{ex}{5.2}
  $E_n$ is the code of all even-weight vectors of $V(n,2)$, which is linear.
  What are the parameters $[n, k, d]$ of $E_n$? Write a GM for $E_n$ in standard
  form.
\end{ex}
\begin{proof}
  $$E_n = \{ x \in V(n, 2) | w(x) ~\text{is even} \}$$

  $\longrightarrow$ is the binary even-weight ode (the single parity-check code);\\
  it's linear since
  \begin{itemize}
    \item the zero vector has even weight
    \item over $GF(2)$, the sum of two even-weight vectors has even weight
  \end{itemize}

  Recall: $F_q^n \Longleftrightarrow \Vnq$, so $V(n, 2) \cong F_2^n$

  $$\Longrightarrow~ E_n = \{ (x_1, \ldots, x_n) \in F_2^n | x_1 + \ldots + x_n = 0 \}$$

  minimum distance is $2$, since
  \begin{itemize}
    \item no word of weight 1 is in $E_n$
    \item words of weight 2 are in $E_n$
  \end{itemize}

  Therefore, parameters are $[n, n-1, 2]$

  GM:
  $$G=\left[ I_{n-1} | 1 \right]$$

  which generates vectors of the form
  $$(a_1, \ldots, a_{n-1}, a_1+\ldots+a_{n-1})$$
\end{proof}

\begin{ex}{5.5}
  Prove that in a \emph{binary linear code}, either all the codewords have even weight or exactly half have even weight and half odd weight.
\end{ex}
\begin{proof}
  let $C_e,~ C_o$ be subsets of $C$ with even and odd weights respectively.

  So that $C = C_e \cup C_o$ (disjoint union).

  For binary vectors $x,y$:
  $$w(x+y) = w(x) + w(y) \pmod 2$$

  ie. over $F_2$, the parity of the weight is additive:
  \begin{align*}
    even + even &= even\\
    even + odd &= odd\\
    odd + odd &= even
  \end{align*}

  Now we have two possible cases:
  \begin{itemize}
    \item $C$ has no odd weighted codewords:\\
      $C_0 = \emptyset,~ \forall x \in C$ $x$ is even weighted
    \item $C$ has at least one odd-weighted codeword:\\
      let $u \in C_o$, consider the map
      \begin{align*}
        \phi: C_e &\longrightarrow C_o\\
        c &\longmapsto c+u
      \end{align*}
      with $c \in C_e,~ u \in C_o,~ \phi(c) \in C_o$.

      $\phi$ is bijective, since $\phi(\phi(c))=(c+u)+u=c$,

      so $\phi$ is it's own inverse; hence $|C_e| = |C_o|$.

      Since $C=C_e \cup C_e$, half codewords are even weighted and half odd.
  \end{itemize}
\end{proof}

\begin{ex}{6.1}
  construct standard arrays for codes having each of the GMs:

  $$G_1 = \begin{pmatrix}
    1 & 0 \\
    0 & 1
    \end{pmatrix} ~~~
    G_2 = \begin{pmatrix}
      1 & 0 & 1\\
      0 & 1 & 1
        \end{pmatrix}
    $$
\end{ex}
\begin{proof}
  recall: for a binary code with GM $G$, the code $C$ consists of all binary linear combinations of the rows of $G$.

  For $G_1$:\\
  rows: $r_1=(1,0),~r_2=(0,1))$. A codeword is any linear combination:
  $$u G_1= a \cdot r_1 + b \cdot r_2,~~ \text{with}~ a,b \in \{0,1\}$$

  4 combinations:\\
  \begin{align*}
    0 \cdot r_1 + 0 \cdot r_2 &= (0,0)\\
    1 \cdot r_1 + 0 \cdot r_2 &= (1,0)\\
    0 \cdot r_1 + 1 \cdot r_2 &= (0,1)\\
    1 \cdot r_1 + 1 \cdot r_2 &= (1,0)+(0,1) = (1,1)
  \end{align*}
  $$\Longrightarrow~ C_1 = \{00, 10, 01, 11\}$$
  $$\text{standard array:}~ 00~01~10~11$$


  \vspace{0.5cm}
  For $G_2$:\\
  rows: $r_1=(1, 0, 1),~ r_2=(0, 1, 1)$. Codeword: $a \cdot r_1 + b \cdot r_2$ with $a, b \in \{0,1\}$.

  4 combinations:\\
  \begin{align*}
    0 \cdot r_1 + 0 \cdot r_2 &= (0,0,0)\\
    1 \cdot r_1 + 0 \cdot r_2 &= (1,0,1)\\
    0 \cdot r_1 + 1 \cdot r_2 &= (0,1,1)\\
    1 \cdot r_1 + 1 \cdot r_2 &= (1,0,1)+(0,1,1) = (1,1,0)
  \end{align*}

  $$\Longrightarrow~ C_2 = \{000, 101, 011, 110\}$$
  $\Longrightarrow~$ so the 1st row of the standard array is: $000~101~011~110$.

  Now, pick a non used vector: $001$, add it to the code $C_2$:
  \begin{align*}
    001 + C_2_i&\\
    001 + 000 &= 001\\
    001 + 101 &= 100\\
    001 + 011 &= 010\\
    001 + 110 &= 111
  \end{align*}
  $\Longrightarrow$ 2nd row of the standard array: $001~100~010~111$

  Thus the standard array is
  \begin{align*}
    000~101~011~110\\
    001~100~010~111
  \end{align*}
  ie. $2^3=8$ vectors, a $V(3,2)$ vector space.
\end{proof}

\begin{ex}{6.4}
  $C$ a binary $[n,k]$-code with minimum distance $2t+1$ (or $2t+2$). Given that $p$ is very small, show that an approximate value of $P_{err}(C)$ is $\left( \binom{n}{t+1} - \alpha_{t+1} \right) p^{t+1}$, where $\alpha_{t+1}$ is the number of coset leaders of $C$ of weight $t+1$.
\end{ex}
\begin{proof}
  From \ref{remark.6.9}, if $d(C)=2t+1$ (or $2t+2$) $Longrightarrow~ C$ can correct any $t$ errors.

  Hence every vector of weight $\leq t$ is a coset leader and so $\alpha_i = \binom{n}{i}$ for $0 \leq i \leq t$.

  \begin{equation}
    P_{corr}(C) = \sum_{i=0}^n \alpha_i \cdot p^i(1-p)^{n-i}
    \label{6.4.eq.1}
  \end{equation}

  where $\alpha_i$ is the number of coset leaders of weight $i$

  From the $\binom{n}{i}$ vectors of weight $i$, exactly $\alpha_i$ are coset leaders, so the non coset leaders are $\binom{n}{i} - \alpha_i$. This is when a decoding error happens.

  $$\Longrightarrow~ P_{err}(C) = \sum_{i=0}^n \left( \binom{n}{i} - \alpha_i \right) p^i (1-p)^{n-i}$$


  \vspace{0.5cm}
  Alternatively from \ref{6.4.eq.1}:\\
  $$P_{err}(C) = 1 - P_{corr}(C) = 1 - \sum_{i=0}^n \alpha_i \cdot p^i(1-p)^{n-i}$$

  Notice that
  $$\sum_{i=0}^n \binom{n}{i} p^i (1-p)^{n-i} = (p + (1-p))^n = 1^n = 1$$

  thus
  \begin{align*}
    P_{err}(C) &= \sum_{i=0}^n \binom{n}{i} p^i (1-p)^{n-i} - \sum_{i=0}^n \alpha_i \cdot p^i(1-p)^{n-i}\\
               &= \sum_{i=0}^n \left( \binom{n}{i} - \alpha_i \right) p^i (1-p)^{n-i}
  \end{align*}

  Now, from \ref{remark.6.9}, $\forall i \leq t,~ \alpha_i = \binom{n}{i}$, thus $\binom{n}{i} - \alpha_i = 0$ for the first $t$ terms in the sum.

  Thus
  \begin{align*}
    P_{err}(C) &= \sum_{i=0}^n \left( \binom{n}{i} - \alpha_i \right) p^i (1-p)^{n-i}\\
               &= \left( \binom{n}{t+1} - \alpha_{t+1} \right) p^{t+1} (1-p)^{n-t+1} + \ldots
  \end{align*}
  where the $\ldots$ represent the terms with $p^{t+2}$ and higher.

  Then, since $p$ is very small (by the exercise definition), $(1-p)^{n-t+1} \equiv 1$, and the other higher powers of $p$ are negligible.

  Hence,
  $$P_{err}(C) = \left( \binom{n}{t+1} - \alpha_{t+1} \right) \cdot p^{t+1}$$
\end{proof}

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