Tagged Questions

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Binary Hamming code - number of words weight $i$

Define $A_i$ as the number of words in binary Hamming code of weight $i$. Prove: $$A_1=0, A_0=1$$ $$(i+1)A_{i+1}+A_i+(n-i+1)A_{i-1}= {n \choose i}$$ I am a tad clueless as to how to proceed. ...
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Shortened Generator Matrix

goodmorning, could someone tell me if the following code has been handled correctly? I have this generator matrix (which I should modify in order to have it correct): G=\begin {bmatrix} ...
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Distributing partially known data between n parties

Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ ...
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u | u + v Coding

G1 is generator of C1 and G2 is generator of C2. If C is c1 || c1 + c2 then how do you find the generator and parity-check of C? I have tried two examples and I see a pattern in the G and H of C. I ...
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How to find the parity check matrix for 101101101101101 in Hamming Codes (15,11) in graphic way?

I am trying to find hamming matrix for safe coded word: 101101101101101 My questions are: 1) What matrix check I should use? I mean there are two types of 15,11 => one starting with 1111 and one ...
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Get code words from generator matrix

I have some issue regarding the generator matrix. Please can some body can explain me "How to get Codebook from Generator matrix?" Following is my issue Generator matrix has 3 code words. Then ...
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Properties of diagonal and permutation matrices.

I've been reading about equivalent codes, and the topic of monomial automorphisms came up. These are the set of monomial matrices (square matrices with exactly one nonzero entry in each row and ...
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Finding Null Space Basis over a Finite Field

I have more a systems background, but I have a math-y type question so I figured I'd give it a shot here...This is more of an implementation question, I don't need to prove anything at the moment. ...
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Probability of full rank of a random matrix.

Suppose, $G$ is a $k \times n$ binary matrix with $\operatorname{rank}(G) = k$. The first $k$ columns of $G$ are linearly independent and the next $n-k$ columns are linear combinations of the first ...
Let $p\geq 3$ be any prime and consider the code $C = N(H)\subseteq\mathbb{Z}_p^2$, where \$H = \begin{pmatrix} 1 & 1 & 1 & \dots & 1 & 1 \\ 0 & 1 & 2 & \dots & p-2 ...