0
votes
0answers
22 views

Finding a parity check matrix of a binary code

I'm supposed to find a parity check matrix of a binary [6,3,3] code. Given a generator matrix G I can find a parity check matrix by row reducing until I get the identity matrix, then take $-A^{\top} ...
0
votes
1answer
22 views

Finding a standard generator matrix given a binary code

My question is how do I find the standard generator matrix of a binary [7,6,2] code? From what I understand a generator matrix for $C$ is any $ k \times n$ matrix $ G$ with entries in $ ...
2
votes
2answers
29 views

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. ...
1
vote
0answers
63 views

Equivalence of Hadamard matrix

This question is from The Theory of Error-Correcting Codes by MacWilliams and Sloane, Problem 2.(3). If n = $2^m$, let $u_1$, $u_2$,...,$u_n$ denote the distinct $m$-tuples. Show that the matrix $H = ...
1
vote
0answers
34 views

Generator Matrix

I have a C in $F_2^6$ $(x_1,x_2,x_3,x_4) \to (x_1,x_2,x_3,x_4,x_1+x_2,x_3+x_4)$ for $x = (1,0,1,1)$ i get $c = (1,0,1,1,1,0)$ we know that $$c = G . x$$ G is the Generator Matrix in the solution ...
2
votes
1answer
77 views

linear binary code problem

Let $\mathcal C$ be a $[n,k,d]$ linear binary code such that $\mathcal C$ has a systematic generator matrix $G=[I_k\mid A]$. (i) Prove that $u\in (\mathbb F_2)^k$ is coded by $c=(u\mid uA)\in ...
1
vote
0answers
70 views

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} ...
1
vote
0answers
93 views

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 $n$ ...
0
votes
1answer
17 views

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 ...
1
vote
0answers
148 views

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 ...
1
vote
1answer
2k views

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 ...
2
votes
1answer
294 views

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 ...
3
votes
1answer
487 views

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. ...
1
vote
1answer
659 views

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 ...
0
votes
0answers
347 views

increase the number of linearly independent rows/columns in a matrix

I have the following problem which is a part of a larger problem. I would like to hear your comments or point me to the correct direction. I am making use of a randomly generated boolean matrix of ...
4
votes
3answers
312 views

Codes: Distance, number of codewords, etc

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 ...