1
vote
3answers
94 views

Proof verification: A $(16,5,8)$ binary code does exist.

Well I have used spheres in coding with radius, $r=\left\lfloor\frac{\delta -1}{2} \right\rfloor=\left\lfloor\frac{8 -1}{2} \right\rfloor = 3$ and that means we have $\sum \limits_{i=0}^3 {16 \choose ...
0
votes
1answer
47 views

Modulation and translation properties of DFT

Consider the discrete fourier transform over a finite field $GF(q)$. Let also $\omega$$\in$$GF(q)$ be an element of order $n$ and which is an $n$-th root of unity. Definition 1. Let $v$ = ($v_0$, ...
0
votes
1answer
54 views

Berlekamp Massey and DFT

I was looking into the Berlekamp Massey algortihm, for LFSR, over GF(2) wondering if there was any DFT(alternately FFT), for the above scheme. Also, is there any generalization to Fn, ie, start ...
1
vote
1answer
127 views

Coding Theory - Fourier/Walsh/Hadamard Transform

Hi guys these questions are from my homework. I am not asking you to solve my homework. Instead, i just need some help in getting started. Please list down some steps as I am very lost. Also, if you ...
8
votes
3answers
344 views

Using Hensel's Lemma to Factor a Polynomial over $\mathbb{Z}_4[x]$

We recently learned about codes over $\mathbb{Z}_4$, and Hensel's Lemma. The lemma is as follows: Let $f(x) \in \mathbb{Z}_4[x]$. Suppose $\mu(f(x)) = h_1(x)h_2(x) \cdots h_k(x)$, where $h_1(x), ...
2
votes
2answers
112 views

Encode the message $[1,1,0,1,1,0,1]$ in BCH code based on the field $\mathbb F = \frac{\mathbb Z_{2}[x]}{x^4+x+1}$

So here's what I understand so far: $\mathbb F = \frac{\mathbb Z_{2}[x]}{x^4+x+1} = GF(16)$ The code is written as $[x^{14},x^{13},x^{12},x^{11},x^{10},x^{9},x^{8}$ $|$ ...
0
votes
1answer
657 views

Reed-Solomon Code calculation

I have a Reed-Solomon Code which can correct t=2 errors. The generator polynomial is $p(X) = X^3 + X + 1$ and $p(a) = a^3 + a + 1 = 0$ this means $a^3 = a + 1$ What is the degree of generator ...
1
vote
1answer
73 views

Unimodular matrices without stable sub-spaces of even weight?

For each N, is there an N×N invertible matrix T over ℤ/2ℤ which does not have a stable subspace of "even weight" -- i.e.  such that there does not exist a set of vectors over ...
6
votes
2answers
5k views

Reed Solomon Polynomial Generator

I am developing a sample program to generate a 2D Barcode. And i am using reed solomon error correction code. By Going through this article i am developing the program. But i couldn't understand how ...
1
vote
1answer
747 views

CRC computation

I would like to understand the CRC computation using CCITT CRC-16 $x^{16} + x^{12} +x^{5} +1$. I was able to successfully implement it in a program but I would like to understand the computation ...