0
votes
0answers
34 views

Proving that number of codes with even weight is the same as number of codes with odd weight for a specific code book

Consider the $[n,n]$ code-book $C_0=\{0,1\}^n$ with $n$ being odd and the codes $c_i \in C_0=[c_1,c_2,...,c_{2^n}]$ being sorted in the ascending order of hamming weight (from $0$ to $n$). Now let's ...
1
vote
2answers
44 views

Linear Code (9,5): Is my Parity Check correct?

I have an exercise about designing parity checks for the Hamming (9,5) group code with minimum distance $3$. I use the following notation for the generator matrix: $$ ...
2
votes
1answer
103 views

Why do XOR and other operators on binary variables qualify as linear?

I never fully understood why is the operation $\oplus: \{0,1\}^n \times \{0,1\}^n \mapsto \{0,1\}^n$ considered linear ? I am well aware of the definition of linearity on real numbers, and I ...
0
votes
0answers
39 views

Probability, linear independence and study of variant of Lights Out

Using Arduino, some leds and pushbuttons I've created a simple variant of the mathematically popular game "Lights Out". In my variant, the starting configuration is always all lights on; what changes ...
6
votes
2answers
312 views

Complement of all-one vector in binary vector space

Let $V$ be a k-dimensional subspace of $(\mathbb{F}_2)^n$, such that vector $\vec{j}=(1,1,...,1) \in V$. Standard linear algebra shows that it is possible to find a $(k-1)$-dimensional space $W$ such ...
0
votes
3answers
139 views

Mathematical alternative to bit manipulation to set all bits to $0$ except the two most significant (highest order) set bits.

The following describes a function which I want to solve mathematically rather than resorting to binary bit manipulation is possible: $y = f(x)$ where $x$ is an arbitrary integer equal to or ...
4
votes
1answer
415 views

Fun ideas for project on Encoding

My university is organizing a project for college students (age around 17) around the subject of encoding (not encypting! See also http://danielmiessler.com/study/encoding_vs_encryption/). As a ...