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\begin{document}

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\section*{binary linear code}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definitions_and_framework}{Definitions and framework}\dotfill \pageref*{definitions_and_framework} \linebreak
\noindent\hyperlink{linear_codes}{Linear codes}\dotfill \pageref*{linear_codes} \linebreak
\noindent\hyperlink{binary_linear_codes}{Binary linear codes}\dotfill \pageref*{binary_linear_codes} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definitions_and_framework}{}\subsection*{{Definitions and framework}}\label{definitions_and_framework}

\hypertarget{linear_codes}{}\subsubsection*{{Linear codes}}\label{linear_codes}

A ($q$-ary) \textbf{linear code} of length $n$ and rank $k$ is a $k$-dimensional subspace of the $n$-dimensional vector space $\mathbb{F}_q^n$ over a [[finite set|finite]] [[field]] $\mathbb{F}_q$ with $q$ elements, or of any $n$-dimensional space over $\mathbb{F}_q$ that comes equipped with a ``standard'' basis. A principal application is in the construction of [[error-correcting code]]s.

There is a [[species|Joyal species]] $Code_{q, k}: Core(FinSet) \to Set$ of $q$-ary linear codes of rank $k$, which assigns to a finite set $\Omega$ the set of $k$-dimensional subspaces of the free $\mathbb{F}_q$-vector space generated by $\Omega$ as basis. The corresponding [[groupoid]] of $q$-linear codes is the [[category of elements]] of $Code_{q, k}$. This gives one notion of [[isomorphism]] of linear codes, where an isomorphism $(S, V \subseteq \mathbb{F}_q S) \to (T, W \subseteq \mathbb{F}_q T)$ is a bijection $\phi: S \to T$ such that the linear map $\mathbb{F}_q \phi$ carries $V$ onto $W$. Thus, [[isomorphisms]] in this sense are given by [[permutation matrices]].

An alternative notion that recurs in coding theory is that an isomorphism (often called an \emph{equivalence}) of codes consists of a [[monomial matrix]] which, when interpreted as a linear isomorphism using the assigned linear bases, carries $V$ onto $W$.

Each assigned basis $\{e_1, \ldots, e_n\}$ of a vector space $U$ can be construed as a self-dual basis with respect to a bilinear form, so that $\langle e_i, e_j \rangle = \delta_{i j}$. In terms of coordinates with respect to the chosen basis, this bilinear form is the usual dot product

\begin{displaymath}
(\vec{x}, \vec{y}) \mapsto x_1 y_1 + \ldots + x_n y_n
\end{displaymath}
and this bilinear form is used to define the notion of \emph{dual code} (see below).

\hypertarget{binary_linear_codes}{}\subsubsection*{{Binary linear codes}}\label{binary_linear_codes}

The following is largely adapted from \hyperlink{FLM}{Frankel, Lepowsky, Meurman}.

A \textbf{(binary linear) code} is a $q$-ary code with $q = 2$. In this case, isomorphisms defined by permutation matrices are the same as isomorphisms defined by monomial matrices.

As a combinatorial structure on a finite set $\Omega$, a binary linear code is typically construed as a subspace of a finite power set $P(\Omega)$, considered as an $\mathbb{F}_2$-space whose addition is given by taking the symmetric difference of sets, and with standard basis given by the [[atoms]].

In that case, the standard dot product can be identified with the nondegenerate symmetric $\mathbb{F}_2$-bilinear pairing on the Boolean ring $P(\Omega)$ given by

\begin{displaymath}
P(\Omega) \times P(\Omega) \stackrel{\langle -, - \rangle}{\to} \mathbb{F}_2
\end{displaymath}
\begin{displaymath}
\,
\end{displaymath}
\begin{displaymath}
(S, T) \mapsto {|S \cap T|} \pmod 2
\end{displaymath}
where ${|C|}$ is the [[cardinality]] of a set $C$. In the sequel, we let $n = {|\Omega|} = \dim(P(\Omega))$.

To each code $\mathbf{C} \subset P(\Omega)$ there is a \textbf{dual code} $\mathbf{C}^\perp$:

\begin{displaymath}
\mathbf{C}^\perp = \{S \in P(\Omega): \forall T \in \mathbf{C} \; \langle S, T \rangle = 0\}.
\end{displaymath}
Notice that $\dim(\mathbf{C}^\perp) = n - \dim(\mathbf{C})$. A code $\mathbf{C}$ is \textbf{self-dual} if $\mathbf{C} = \mathbf{C}^\perp$. In this case $n$ is even and $\dim(\mathbf{C}) = n/2$.

Now suppose $\Omega$ is a set with an even number $n$ of elements. The collection $E(\Omega)$ of subsets of even cardinality is a vector subspace of $P(\Omega)$ of dimension $n-1$, and we have

\begin{displaymath}
E(\Omega) = 1^\perp,
\end{displaymath}
where $1$ is the multiplicative unit of $P(\Omega)$ (the subset $\Omega$), under the assumption $n$ is even. On $E(\Omega)$ we may define a quadratic form

\begin{displaymath}
q: E(\Omega) \to \mathbb{F}_2
\end{displaymath}
taking $S \in E(\Omega)$ to $\frac{{|S|}}{2} \pmod 2$; this $q$ gives rise to the bilinear form $\langle -, - \rangle$ on $E(\Omega)$ via

\begin{displaymath}
\langle S, T \rangle = q(S + T) - q(S) - q(T).
\end{displaymath}
A code $\mathbf{C} \subset P(\Omega)$ is of \textbf{type II} if

\begin{itemize}%
\item $4$ divides ${|C|}$ for every $C \in \mathbf{C}$ and $\Omega \in \mathbf{C}$,

\end{itemize}
and of \textbf{type I} if

\begin{itemize}%
\item $\mathbf{C} \subset E(\Omega)$ and $\Omega \in \mathbf{C}$, but $\mathbf{C}$ is not of type II.

\end{itemize}
Notice that the quadratic form (hence the bilinear form) vanishes on a type II code $\mathbf{C}$, so that we have $\mathbf{C} \subset \mathbf{C}^\perp$ (i.e., $\mathbf{C}$ is isotropic). If $\mathbf{C}$ is maximal isotropic -- has half the dimension of $P(\Omega)$ -- then it follows that $\mathbf{C}$ is self-dual.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
\textbf{(Hamming $(8, 4)$ code)} Up to isomorphism, there is a unique type II self-dual code $\mathbf{C} \subset P(\Omega)$ where $\Omega$ is an 8-element set. Such a code may be constructed as follows: let $\Omega$ be the projective line $\mathbb{P}^1 \mathbb{F}_7$, so as a set $\Omega = \{0, 1, 2, 3, 4, 5, 6, \infty\}$. Let $Q$ be the set of squares $0, 1, 2, 4$ and let $N$ be its complement in $\Omega$. The span of the set of elements $\{N + i: i \in \mathbb{F}_7\}$ (where $\infty + i = \infty$ for each $i$) is a 4-dimensional space consisting of 16 elements:

\begin{displaymath}
\itexarray{
\emptyset, & \{3, 5, 6, \infty\}, & \{0, 4, 6, \infty\}, & \{0, 1, 5, \infty\} \\  
\{1, 2, 6, \infty\}, & \{0, 2, 3, \infty\}, & \{1, 3, 4, \infty\}, & \{2, 4, 5, \infty\} \\ 
\{0, 3, 4, 5\}, & \{1, 4, 5, 6\}, & \{0, 2, 5, 6\}, & \{0, 1, 3, 6\} \\  
\{0, 1, 2, 4\}, & \{1, 2, 3, 5\}, & \{2, 3, 4, 6\}, & \Omega.
}
\end{displaymath}
For this subspace $\mathbf{C} \subset P(\Omega)$, every element is a set that is of cardinality $0$, $4$, or $8$. It is clearly of type II, and self-dual because it is maximal isotropic.

There is a second Hamming code $\mathbf{C}'$ (isomorphic to $\mathbf{C}$) that is spanned by the elements $\{-N - i: i \in \mathbb{F}_7\}$. This is ``complementary'' to the first in an appropriate sense: we have

\begin{displaymath}
\mathbf{C} + \mathbf{C}' = E(\Omega), \qquad \mathbf{C} \cap \mathbf{C}' = \mathbb{F}_2 \Omega,
\end{displaymath}
so that $\mathbf{C}/\mathbb{F}_2 \Omega$ is complementary to $\mathbf{C}'/\mathbb{F}_2 \Omega$ in the 6-dimensional space $E(\Omega)/\mathbb{F}_2 \Omega$.

\end{example}
In the theory of error-correcting codes, this is related to the Hamming $(7, 4)$ code, but with an extra parity bit. The Hamming $(7, 4)$ code is one of an infinite family of Hamming codes of type $(2^r - 1, 2^r - r - 1)$.

\begin{example}
\label{}\hypertarget{}{}
(The [[binary Golay code]].) Up to isomorphism, there is a unique self-dual code of type II on a 24-element set with no elements of weight (= cardinality) 4. Indeed, let $3 \Omega$ denote the disjoint union of three copies of $\Omega$. Construct a code $\mathbf{D} \subset P(3\Omega)$ that is spanned by elements of the form

\begin{displaymath}
S \sqcup S \sqcup \emptyset, \qquad S \sqcup \emptyset \sqcup S, \qquad T \sqcup T \sqcup T
\end{displaymath}
where $S$ ranges over elements of the Hamming code $\mathbf{C}$ and $T$ ranges over elements of the Hamming code $\mathbf{C}'$. Any three elements $S \sqcup S \sqcup \emptyset$, $S' \sqcup \emptyset \sqcup S'$, and $T \sqcup T \sqcup T$ are mutually orthogonal in $P(\Omega)$, so evidently $\mathbf{D}$ is an orthogonal direct sum of three 4-dimensional isotropic spaces, making $\mathbf{D}$ an isotropic subspace of dimension 12, hence maximal isotropic. Each element of $\mathbf{D}$ has cardinality a multiple of $4$, making it self-dual type II.

To see that $\mathbf{D}$ has no elements of weight 4, argue as follows. Note that combinations $(S \sqcup S \sqcup \emptyset) + (S' \sqcup \emptyset \sqcup S')$ are of the form $S_1 \sqcup S_2 \sqcup S_3$ such that $S_1 + S_2 + S_3 = 0$. Now suppose we had

\begin{displaymath}
{|S_1 + T|} + {|S_2 + T|} + {|S_3 + T|} = 4.
\end{displaymath}
Since all the summands are even, one must be zero, so say $S_3 + T = 0$. Since $\mathbf{C} \cap \mathbf{C}' = \mathbb{F}_2 \Omega$, we must have $T \in \mathbb{F}_2 \Omega$. But then ${|S_1 + T|}$ and ${|S_2 + T|}$ are multiples of $4$, and therefore one is zero, so say $S_2 + T = 0$. But now we have

\begin{displaymath}
S_1 = S_1 - 2 T = S_1 + S_2 + S_3 = 0
\end{displaymath}
whence we infer that ${|S_1 + T|}$ is a multiple of $8$, and we have reached a contradiction.

\end{example}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[coding theory]]


\item [[binary Golay code]]


\item [[combinatorial design]]


\item [[Leech lattice]]


\item [[adinkra]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Igor Frankel, James Lepowsky, Arne Meurman, \emph{Vertex Operator Algebras and the Monster}, Academic Press 1988.

\end{itemize}
\begin{itemize}%
\item Patrick J. Morandi, \emph{Error Correcting Codes and Algebraic Curves} , lecture notes New Mexico State University 2001. (\href{http://www.math.nmsu.edu/~pmorandi/math601f01/LectureNotes.pdf}{pdf})


\item Jay A. Wood, \emph{Spinor groups and algebraic coding theory} , J.Combinatorial Th. Series A \textbf{51} (1989) pp.277-313. (available online)



\end{itemize}
category: combinatorics

[[!redirects binary linear codes]] [[!redirects binary code]]



\end{document}