Consider the $[4,2]$ ternary Hamming code with check matrix $ \left( \begin{array}{ccc} 1 & 1 & 2 &0\\ 0 & 1 & 1 & 1\end{array} \right). $ Clearly, the code has $4$-tuple minimum-weight error vectors $\{\vec e_{m1}=\vec0,…,\vec e_{m9} \}$ corresponding to all the syndromes $\{\vec s_1=\vec0,…,\vec s_9 \}$. My textbook says to list $\vec0$ and […]

I have a (seemingly) simple question about coding theory. I feel like the answer to this should be very obvious, but I’m having some troubles with it. I’m trying to create binary code which is $N$ bits in length and consists of $M < 2^N$ codewords, $\{m_1, m_2, … , m_M\}$. For a given $N$ […]

Consider the polynomial $p(x) = 1+x^5+x^{10}$ with binary coefficients. Consider the multiplicative group of $\mathbb{F}_{16}$, and let $p(x)$ be evaluated at each of these $15$ elements. The only possible evaluations are $0$ and $1$. I am looking for more such polynomials which have binary coefficients, which when evaluated on the elements of an extension field […]

Wikipedia says that CRC algorithm is based on cyclic codes, but it doesn’t say that it is a cyclic code. If I understood correctly, a linear code of length $n$ called cyclic if and only if its generator polynomial divides $x^n-1$. So I think that in general CRC codes are not cyclic. Is it true? […]

I’m working on trying to understand how to use Reed-Solomon decoding to make Shamir secret sharing robust to cheaters as mentioned here. I’m following the setup shown at the bottom of page 40 in this paper. So, here is my simple example. I’m working in $\mathbb{Z}_{11}$. Let $p(x)=8+3x$ be the polynomial I’m using for sharing […]

Let $P:\mathbb{Z}_2^n\to\mathbb{Z}_2$ be a polynomial of degree $k$ over the boolean cube. An affine subcube inside $\mathbb{Z}_2^n$ is defined by a basis of $k+1$ linearly independent vectors and an offset in $\mathbb{Z}_2^n$. [See “Testing Low-Degree Polynomials over GF(2)” by Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, Dana Ron – http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.9.1235 – for more details] […]

I know about the antipodal mapping. What I want to know is what the most significant differences between the sphere and projective space are, and how to think of each of them and their relationship to one another. I come at this from a coding theory/vector quantization perspective; I’m trying to understand the difference between […]

Here is a combinatorial problem: let $\Sigma=\{\alpha_1,\ldots,\alpha_n\}$ be an alphabet and we consider any words over $\Sigma$ of length $n$. We also define over the set of such words the Hamming distance: $$d(\omega,\omega’)=\#\{i\in \{1,\ldots,n\} \vert a_i\neq b_i\}$$ where $\omega=a_1\ldots a_{n}$ and $\omega’=b_1\ldots b_n$ are two words. The questions are the following: Let $\omega_0$ be the […]

My field is Coding Theory and my background is Algebraic, there are many applications of Genetic Algorithm in Coding Theory, I would to know the easiest and the most elementary and introductory note about “Genetic Algorithm in Coding Theory”, also is this algorithm using in Crypto too?

Are there good lower bounds on the size of a collection of $k$-element subsets of an $n$-element set such that each pair of subsets has at most $m$ elements in common?

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