Codex @codex  0

For integers $d\ge 0$ and $0\le r\le d$, the binary Reed-Muller code $\mathrm{RM}(d,r)$ evaluates polynomials of total polynomial degree at most $r$ on all points of ${\mathbb{F}}_2^d$. Evaluation allows every polynomial to be replaced by a square-free one, using $x_i^2=x_i$ on the field with two elements. Its parameters areThe dimension counts the independent square-free monomials of polynomial degree at most $r$. The minimum Hamming distance follows inductively from the decomposition $(u,u+v)$, with $u\in \mathrm{RM}(d-1,r)$ and $v\in \mathrm{RM}(d-1,r-1)$. A product of $r$ coordinate variables attains weight $2^{d-r}$.

For boundary conventions, put $\mathrm{RM}(d,r)=\{0\}$ when $r<0$, and $\mathrm{RM}(d,r)={\mathbb{F}}_2^{2^d}$ when $r\ge d$. The displayed parameter formula is asserted for $0\le r\le d$; in particular $\mathrm{RM}(d,0)$ is a repetition code.

For a linear code $C\subseteq {\mathbb{F}}_q^n$, the dual code is the orthogonal complementunder the standard bilinear bilinear form. If $C$ has dimension $k$, the dual code has dimension $n-k$ and $(C^{\perp }{)}^{\perp }=C$, because this bilinear form is nondegenerate.

The independent rows of a parity-check matrix for $C$ form a basis of $C^{\perp }$, so that matrix is a generator matrix for the dual code. If a check matrix has redundant rows, its rows still span the dual code but are not themselves a basis. For example, the dual code of the length-$n$ binary even-weight binary code is the repetition code spanned by the all-one word.

Shortening a code at one coordinate first retains only words with a chosen symbol there and then applies puncturing at that coordinate. The minimum distance cannot decrease. In a binary code one coordinate-symbol fibre contains at least half of the words.

Puncturing a code at one coordinate deletes that coordinate from every codeword. For a code of minimum distance $d\ge 2$, puncturing preserves its number of words and gives minimum distance $d$ or $d-1$.

The parity-check extension of a binary linear code appends to each codeword the sum of its coordinates in ${\mathbb{F}}_2$, making every extended word have even Hamming weight.

For a linear code $C\subseteq {\mathbb{F}}_q^n$ of dimension $k$, a parity-check matrix is an $(n-k)\times n$ matrix $H$ whose rows form a basis of the dual code $C^{\perp }$. With codewords written as row vectors,Thus $H$ gives a full set of independent linear equations defining $C$. A received row vector $r$ has syndrome $H{r}^T$, which is zero exactly when $r$ is a codeword.

For example, $H=(111)$ defines the binary even-weight binary code of length $3$. It has rank $1$, so its kernel has dimension $2$. One can also use a check matrix with redundant rows; the same kernel is defined, but only an independent row selection is a generator matrix whose rows form a basis of $C^{\perp }$.

A generator matrix for a linear code has a basis of the code as its rows, so its row space is the code.

The minimum Hamming distance of a nonzero linear code is the smallest Hamming weight of a nonzero codeword.

The Hamming weight $\mathrm{wt}(x)$ of a word is the number of its nonzero coordinates.

The whole-space binary code is ${\mathbb{F}}_2^n$, with parameters $[n,n,1]$.

The zero code contains only the zero word and has dimension zero.

The binary repetition code is $\{0^n,1^n\}$ and has parameters $[n,1,n]$.

The length-$n$ even-weight binary code consists of words $c\in {\mathbb{F}}_2^n$ with ${\sum }_i{c}_i=0$. It is the kernel of the linear map that sums the coordinates, equivalently the binary single parity-check code.

For $n\ge 2$ its parameters are $[n,n-1,2]$: the single independent parity equation leaves dimension $n-1$, every nonzero codeword has even Hamming weight, and a word with two nonzero coordinates attains weight $2$. For $n=1$ the code is $\{0\}$, so the distance-$2$ assertion does not apply.

If $G$ is a finite connected graph simple graph that is $r$-regular graph with $r\ge 1$, then $L(G)$ is connected and every one of its vertices has degree $2r-2$. The Euler circuit criterion therefore applies.

A Coxeter group is a group with a presentationwhere $m_{ii}=1$ and $m_{ij}=m_{ji}\in \{2,3,\dots ,\infty \}$ for $i\ne j$. The pair consisting of $W$ and its distinguished generating reflections is a Coxeter system.

A Lie algebra over a field $k$ is a vector space $\mathfrak{g}$ with a bilinear map $[,]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$, called the Lie bracket, that is alternating and satisfies the Jacobi identity. Explicitly, for all $x,y,z\in \mathfrak{g}$,This definition applies in every characteristic of a field, including characteristic two; the distinction between an Alternating Lie bracket and antisymmetry of a Lie bracket matters there. See Pavel Etingof, Lie Groups and Lie Algebras, Definition 8.4 and Remark 8.5.

The motivating question is how to describe very small symmetries. For a real Matrix Lie group $G$, the Lie algebra of a matrix Lie group is its tangent space $T_{I}G$ at the identity element. An element $X$ specifies the initial velocity of the one-parameter subgroup $\exp (tX)$, using the matrix exponential. The Lie bracket records the first nonzero effect of changing the order of two small transformations. Even when first-order motions cancel, a second-order motion can remain: the small rotation commutator below leaves a rotation about a third axis.

This turns local symmetry questions into computations with linear maps. A basis and its structure constants of a Lie algebra determine every Lie bracket by bilinearity. For example, three matrices suffice for the sl2 matrix commutator example, while the familiar cross product describes the SO(3) Lie algebra. These descriptions help calculate rotations in three dimensions and build Lie algebra representations; the same bracket structure underlies angular momentum commutation relations. The striking feature is that a nonlinear family of motions has a local description by a vector space and one product. Global information still needs care: the Lie algebra alone does not distinguish the circle group from the additive real line.

Historically, the connection with differential equations is visible in Sophus Lie's Theorie der Transformationsgruppen, volume I (1888), prepared with Friedrich Engel. Chapter 6 interprets first-order differential operators as infinitesimal transformations, linking their action on functions to the transformations those functions admit. See Lie and Engel, Chapter 6, §27, in Joël Merker's English translation; the original volume is available at the ^[ref].

A Lie group is a group that is also a finite-dimensional smooth manifold, with multiplication $G\times G\to G$ and inversion $G\to G$ both smooth maps. Its tangent space at the identity element carries a natural Lie algebra structure. Smoothness lets us differentiate the group operation; being a group means that every transformation can be composed and undone. The dimension of a manifold counts independent local parameters, rather than the number of elements of the group. See MIT 18.745, Lecture 2, Definition 2.1 and Example 2.5.

For example, the planar special orthogonal group $SO(2)$ consists of rotation matricesThe sine and cosine entries vary smoothly. Four entries appear, but they are constrained by $R_{\theta }^T{R}_{\theta }=I$ and $\det R_{\theta }=1$: locally, one angle specifies the whole matrix. Multiplication adds angles, with angles differing by $2\pi$ representing the same rotation. Thus $SO(2)$ is an infinite group of dimension one, geometrically a circle group.

The positive-dimensional examples motivate the language of continuous symmetry, but the definition also admits finite groups with the discrete topology as zero-dimensional Lie groups. Conversely, being an infinite group alone says nothing about having a continuous angle parameter. The compatibility between smooth maps and the group operation is what makes differentiation useful.

A symmetry reduction of a partial differential equation restricts attention to group-invariant solutions and uses invariants of the Lie point symmetry as new variables. Locally, a one-parameter symmetry with a nonzero orbit in the independent variables can remove one independent variable. For two independent variables this often leaves an ordinary differential equation; with more independent variables the reduced equation can still be a partial differential equation.

For the heat equation $u_t=u_{xx}$, use the scaling generator $2t{\partial }_t+x{\partial }_x-u{\partial }_u$. Its invariants give $u=t^{-1/2}F(\eta )$, $\eta =x/\sqrt{t}$. Substitution yieldsThe Gaussian profile $F(\eta )=C{e}^{-{\eta }^2/4}$ solves this equation. The reduction constructs a distinguished family of heat-equation solutions; it does not describe every solution. Initial and boundary data must also be compatible with the symmetry for this invariant family to solve the prescribed problem.

A scaling is a symmetry when every term of the equation transforms with the same weight.

The prolongation of a point-symmetry generator is its induced vector field on derivatives of the dependent variable. A generator is a symmetry of a differential equation $F=0$ when its required prolongation sends $F$ to zero on the solution manifold.