0
votes
1answer
17 views

Polynomial Arithmetic Modulo 2 (CRC Error Correcting Codes)

I'm trying to understand how to calculate CRC (Cyclic Redundancy Codes) of a message using polynomial division modulo 2. The textbook Computer Networks: A Systems Approach gives the following rules ...
1
vote
1answer
13 views

If gen matrix has even weigth rows, do codewords have even weigth for non binary code?

Is that true that in a non binary code C every codeword has even weight if and only if every row of G has even weight?
0
votes
1answer
57 views

Is there a binary [10,6,4] code?

Using the sphere padding packing bound formula I can conclude that 1 + 12 + 66 $\ge$ $2^{6}$ which indicates that there MAY be a binary [10,6,4] code, however I cannot prove that there is. How can I ...
0
votes
1answer
28 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 $ ...
0
votes
0answers
64 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 ...
2
votes
2answers
50 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
3answers
101 views

Finding element in binary representation of $GF(2^6)$

I got the following task: Let $F = GF(2^6 )$ be K[x] modulo the primitive polynomial $h(x) = 1 +x ^2 +x ^3 +x ^5 +x ^6$ , and let $\alpha$ be the class of x. I have a table with the binary ...
1
vote
2answers
441 views

Why should we append zeros during CRC calculation?

Say we have M as message bits , why do we need to append r-zeros to M message bits before performing the division to obtain r-bit checksum. Why don't we directly perform the division on the M message ...
1
vote
1answer
258 views

What is the average Levenshtein distance between two random binary strings of length $L$?

For example, for length $L=7$, two random binary strings might be: 0100101 1010011 The Levenshtein distance here would be 5, as it would require 5 bit-flips to ...
3
votes
2answers
79 views

Recognize a valid binary Golay codeword

Are there any properties of a binary [24,12,8] Golay code which would allow me to say, for example, that a given 24-bit word is or is not a Golay codeword for some generator matrix? That is to say, is ...
1
vote
1answer
141 views

Problem about BCH code

I have another homework to do, please give me some hints in order to solve this problem: "Determine whether the dual of an arbitrary BCH code is a BCH code."
1
vote
1answer
200 views

Problem about narrow sense BCH-Code

I have an homework to do and i have no idea where to start. The question is: "Show that a narrow-sense binary BCH-code of length $ n=2^m-1 $ and designed distance $ 2t+1 $ has minimum distance $ 2t+1 ...
4
votes
1answer
272 views

A question dealing with residual codes.

I've been reading about residual codes, and have come across several statements that I am having trouble showing. (1.) By using the residual code, one can show that a $[16, 5, 8]$ binary code ...
3
votes
0answers
216 views

An Upper Bound for an $[n,k,d]$ Linear Binary Code.

I've been reading about the various upper bounds for different types of codes. Recently, I came across a statement that is similar to the Singleton Upper Bound that I am having trouble proving. The ...
4
votes
1answer
77 views

Proving that $A_2(13,7) = 8$

It is not too difficult to find a binary code consisting of $8$ words, each $13$ bits long, keeping the distance between every pair of words at least $7$. I know it is not possible to find $9$ words ...
3
votes
1answer
66 views

q-ary code/Latin squares

For any value of $q$ the largest number of elements in any q-ary code $C$ of length $4$, distance $3$ is $q^2$. How can we prove that this is attainable iff there are a pair of mutually orthogonal ...
3
votes
1answer
7k views

Calculating CRC by long division: How to decide the top number of long division?

Ok, I know how to use long division by using regular numbers, but when comes to binary numbers I'm getting confused. In following calculation I can see the equation solved but I don't understand ...