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For density $\rho$ and Lamé parameters $\lambda ,\mu$, the displacement satisfiesIts longitudinal and transverse wave speeds are

In a homogeneous isotropic elastic body without body forces, static equilibrium of the displacement field is

This is the static, homogeneous-isotropic displacement form of linear elasticity.

For the two-dimensional convention $\nabla \mathbf{u}=({\partial }_i{u}_j)$, the energy densityproduces the static isotropic elastic equations by variation.

The two Lamé parameters $\lambda$ and $\mu$ determine the linear isotropic relation between stress and strain; $\mu$ is the shear modulus.

For the one-dimensional reduction of isotropic linear elasticity, nondegeneracy means that the effective longitudinal modulus $\lambda +2\mu$ is nonzero. The static Navier-Cauchy equation then forces the displacement to be affine.

An elastic plate is a thin elastic body whose leading deformation is bending of its middle surface.

The equation $ax+by=c$ has an integer solution exactly when $\gcd (a,b)$ divides $c$. Bézout's identity proves sufficiency, while every integer linear combination of $a,b$ is divisible by their greatest common divisor.

A code of minimum Hamming distance $d$ detects every pattern of at most $d-1$ errors and uniquely corrects every pattern of at most $\lfloor (d-1)/2\rfloor$ errors. The correction claim follows because Hamming balls of that radius around distinct codewords are disjoint.

The binary Hamming code of redundancy $r$ has parameters $[2^r-1,2^r-r-1,3]$. Its parity-check matrix has every nonzero vector of ${\mathbb{F}}_2^r$ as a column.

The binary Reed-Muller code $\mathrm{RM}(d,r)$ evaluates Boolean polynomials of degree at most $r$ on ${\mathbb{F}}_2^d$. It has parametersThe dimension counts square-free monomials, and the minimum weight follows inductively from the decomposition $(u,u+v)$.

The dual code $C^{\perp }$ consists of the words orthogonal to every word of $C$ under the standard dot product. A parity-check matrix for $C$ is a generator matrix for $C^{\perp }$.

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.

A parity-check matrix $H$ for a linear code $C$ satisfies

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.