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Compute the word length, minimum distance and dimension of some given Hamming codes.

Decode a message of two codewords encoded using Hamming's [7,4] code, with at most one error per codeword.

Correct an error in a received message which was encoded using Hamming's square code

Write down the lexicographic parity check matrix for a Hamming code, and correct two received codewords.

Write down a lexicographic parity check matrix for a Hamming code and correct two received codewords.

Given a set of codewords generating a code, write down a generator matrix, encode three data vectors, and decode one codeword.

Three 3 dim vectors, one with a parameter $\lambda$ in the third coordinate. Find value of $\lambda$ ensuring vectors coplanar. Scalar triple product.

Parametric form of a curve, cartesian points, tangent vector, and speed.

Parametric form of a curve, cartesian points, tangent vector, and speed.

Calculations of the lengths of two 3D vectors, the distance between their terminal points, their sum, difference, and dot and cross products.

Given vectors $\boldsymbol{v}$ and $\boldsymbol{w}$, find their inner product.

(Green’s theorem). $\Gamma$ a rectangle, find: $\displaystyle \oint_{\Gamma} \left(ax^2-by \right)\;dx+\left(cy^2+px\right)\;dy$.

Given vectors $\boldsymbol{A,\;B}$, find $\boldsymbol{A\times B}$

Given a pair of 3D position vectors, find the vector equation of the line through both.  Find two such lines and their point of intersection.

Given two 3 dim vectors, find vector equation of line through one vector in the direction of another. Find two such lines and their point of intersection.

Find all words with given Hamming distance from a given codeword.

Two double integrals with numerical limits

Calculate a repeated integral of the form $\displaystyle I=\int_0^1\;dx\;\int_0^{x^{m-1}}mf(x^m+a)dy$

The $y$ integral is trivial, and the $x$ integral is of the form $g'(x)f'(g(x))$, so it straightforwardly integrates to $f(g(x))$.

Find the coordinates of the stationary point for $f: D \rightarrow \mathbb{R}$: $f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$, $D$ is a disk centre $(c,d)$.

Given a set of codewords generating a code, give a generating matrix, encode three data vectors, and decode one codeword.

Given a set of codewords generating a code, give a generating matrix, encode three data vectors, and decode one codeword.

Given vectors  $\boldsymbol{v,\;w}$, find the angle between them.

3 Repeated integrals of the form $\int_a^b\;dx\;\int_c^{f(x)}g(x,y)\;dy$ where $g(x,y)$ is a polynomial in $x,\;y$ and $f(x)$ is a degree 0, 1 or 2 polynomial in $x$.

Determine if various combinations of vectors are defined or not.

Given a matrix in the field $\mathbb{Z}_n$. By reducing it to row-echelon form (or otherwise), find a basis for the row space of the matrix, as a list of vectors.

Compute the minimum distance between codewords of a code, given a parity check matrix.

Given a set of vectors, find a basis which generates their span as a subspace of $\mathbb{Z}_n$.

Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

Double integrals (2) with numerical limits

Determine if various combinations of vectors are defined or not.