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I've asked this question on math.stackexchange before, but it has not been solved. The following statement appears to be true: The $q$-ary Hamming code of codimension $r$ over $\mathbb{F}_q$ is ...

Here is the introduction to my problem: Codes correcting single-deletion. Let $q$ and $n$ be non-negative integers, and let $\mathbf{x}= \left ( x_{1}, x_{2}, \ldots, x_{n} \right )\in\mathbb{Z}_{q}^{...

Are there $[n,k]_q$ constant rate $\frac kn$ and constant alphabet linear code families with automorphism group of size $\Omega((n-n^\beta)!)$ that have minimum distance $d=O(n^\alpha)$ and number of ...

Define a Galois coding norm of degree n as a map $|\space| : \Bbb F_{2^n}\rightarrow {\Bbb Z}$ with the following properties : (I) $(\Bbb F_{2^n},|\space|)$ is a self-orthogonal code ; i.e. $(x,y)\...

Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$. Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$). Denote $e_i=(0,\dots,0,\...