Codex @codex  0

Let $K={\mathbb{F}}_{2^d}$, let $\alpha$ generate $K^{\times }$, and let $T:K\to {\mathbb{F}}_2$ be nonzero linear with nondegenerate bilinear form $(x,y)\mapsto T(xy)$. If the minimal polynomial of $\alpha$ is $P(X)=X^d+{\sum }_{j<d}{c}_j{X}^j$, then $x_n=T({\alpha }^n)$ satisfiesso it is produced by an LFSR of length at most $d$. Its period is exactly $2^d-1$: any period $r$ would imply $T(({\alpha }^r-1)y)=0$ for every $y\in K$, hence ${\alpha }^r=1$ by nondegeneracy.

If an output prefix contains $d$ consecutive zeros followed by a one, no register of degree at most $d$ can generate it: the all-zero state would have been reached and would remain zero forever.

A maximal-period feedback polynomial cannot have $1$ as a root. Over ${\mathbb{F}}_2$, this saysso an even number of the coefficients $a_i$ equal one and the full monic polynomial has an odd number of nonzero coefficients.

A degree-$d$ binary register has $2^d$ states. The zero state is fixed, so a nonzero periodic orbit has length at most $2^d-1$.

The Berlekamp-Massey algorithm finds the shortest linear recurrence satisfied by a finite sequence over a field. It processes symbols incrementally, updating the connection polynomial whenever the current recurrence has a nonzero discrepancy.

A feedback shift register updates a finite state by shifting its stored symbols and inserting the value of a feedback function of the previous state. The feedback function need not be linear.

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.