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Given $\,f(t)=t^3+t+1,\,$ let $\,\mathbb{F}_{2^3}=\frac{\mathbb{F}_2\left[t\right]}{\langle f\,\rangle}\,$ be a finite field and $\,\xi=t\,\left(\textrm{mod}\,f\right).$ Let also $\,\mathcal{C}\,$ be a cyclic code where $\,g(x)=\left(x-\xi\right)\left(x-\xi^2\right)\left(x-\xi^3\right)\left(x-\xi^4\right)\,$ is his generator polynomial.

Question: Find a generator matrix of $\,\mathcal{C}\,$ in the basis $\,B=\left\{1,\xi,\xi^2\right\}\,$ of $\,\mathbb{F}_{2^3}\,$ as an $\,\mathbb{F}_{2}$- vector space.

The fisrt thing they ask is to prove that $\,\xi\,$ is a $\,7^{\,th}\,$ primitive root of the unity (quite easy). Then, prove that the length of $\,\mathcal{C}\,$ is $\,n=7$.

I found out that $\,\mathcal{C}\,$ is a Reed-Solomon code, a type of BCH code whose length is always $\,n=q-1\,$ over a finite field $\,\mathbb{F}_q,\,$ so this question is also easy to answer.

However, I do not know how to proceed to find the generator matrix of $\,\mathcal{C}.$ I tried to expand the generator polynomial $$g(x)=g_0+g_1x+\dots+g_4x^4$$

and define the generator matrix as $$G=\begin{pmatrix}g_0&g_1&g_2&g_3&g_4 & 0 & 0\\ 0 & g_0&g_1&g_2&g_3&g_4 & 0\\ 0 & 0 & g_0&g_1&g_2&g_3&g_4\end{pmatrix}$$

in the basis $\,B=\left\{1,\xi,\xi^2\right\}\,$ but I guess I failed.

Somebody please, help me! Thanks in advance.

  • No, its fine. You need to calculate the coefficients $g_i$. – Wuestenfux Jan 11 '19 at 11:31
  • Basic Galois theory tells us that the zeros of $f(t)$ are $\xi,\xi^2$ and $\xi^4$. So $$f(x)=x^3+x+1=(x-\xi)(x-\xi^2)(x-\xi^4).$$ All you need to do is to calculate $$g(x)=f(x)(x-\xi^3).$$ Observe that $\xi^3=\xi+1$ because $\xi$ is a zero of $f$. – Jyrki Lahtonen Jan 11 '19 at 11:41
  • @Jyrki Lahtonen: I guess you mean $f(t)$ is the minimal polynomial of $\xi$ over $\mathbb{F}_{2}$. I didn't realize, thank you very much. However, when I calculate the coefficients of $g(x)$ as you said I get $$g(x)=x^4 - \left(\xi +1\right)^3+x^2-\xi x-\left(\xi+1\right)$$ so, I can't express the coefficients of $x^3$ and $1$ in the basis $B$, or maybe I just don't know how. –  Jan 11 '19 at 17:45
  • Given that $f(x)$ has no quadratic term the coefficient of $x^3$ in $g(x)$ should be just $\xi^3=\xi+1$, no? – Jyrki Lahtonen Jan 11 '19 at 17:51
  • @Jyrki Lahtonen: Sorry, I didn't write it properly. I get this $$g(x)=x^4-\left(\xi+1\right)x^3+x^2-\xi x-\left(\xi+1\right)$$ –  Jan 11 '19 at 17:57
  • If you want to bursh up on doing arithmetic in $\Bbb{F}_8$ you can take a peek at this post I prepared with referrals like this in mind. The second section is relevant to this field. The difference is that I denoted your $\xi$ with $\alpha$ there. – Jyrki Lahtonen Jan 11 '19 at 17:58
  • That seems correct to me. But I don't understand. $\xi+1$ most emphatically IS written using the basis $1,\xi,\xi^2$. It is an $\Bbb{F}_2$-linear combination of the basis elements. The first basis element $1$ appears with coefficient $1$, the second basis element $\xi$ appears with coeffcient $1$ and the last basis element $\xi^2$ appears with coefficient $0$. – Jyrki Lahtonen Jan 11 '19 at 18:01
  • Or, are you worried about the minus signs? Remember that in characteristic two $-a=a$ for all $a$. – Jyrki Lahtonen Jan 11 '19 at 18:02
  • @Jyrki Lahtonen: What I can't see is how to write the matrix elements. I mean, for $x^3$ I have the coefficient $\xi +1$ which would be $1 1 0$ in the basis $B$. But, I am given just one element of the matrix, $g_3$ for that coefficient. –  Jan 11 '19 at 18:05
  • Ahh, that was the problem. Most definitely the entries of $G$ shoud be elements of $\Bbb{F}_8$. This is because the components of the codewords will also be free to range over $\Bbb{F}_8$. You can turn this into a generator matrix with entries in $\Bbb{F}_2$, but then you get nine rows (and are only allowed to use the $2^9$ $\Bbb{F}_2$-linear combinations of them. Then the rows of the generator matrix should (probably?) be the coefficients of $g(x)$, $\xi g(x)$ and $\xi^2 g(x)$ (and their cyclic shifts). If you want to treat the codewords as bit vectors, then you will need $21$ bits per word. – Jyrki Lahtonen Jan 11 '19 at 18:30
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    Effectively you would then replace each entry of the $3\times7$ matrix with a $3\times3$ block. – Jyrki Lahtonen Jan 11 '19 at 18:30

1 Answers1

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Lots of comments. We have $$g = (x-\xi)(x-\xi^2)(x-\xi^3)(x-\xi^4).$$ If $\xi$ is a zero of the primitive polynomial $f=x^3+x+1$, then $f$ has the zeros $\xi,\xi^2,(\xi^2)^2=\xi^4$. Thus $$g = (x^3+x+1)(x+\xi^3) = x^4+x^2+x+\xi^3x^3+\xi^3 x +\xi^3\\ = x^4 + \xi^3 x^3 +x^2+(\xi^3+1)x+\xi^3.$$ Since $\xi$ is a zero of $f$, $\xi^3+\xi+1=0$ and so $\xi=\xi^3+1$ Hence,
$$g= = x^4 + \xi^3 x^3 +x^2+\xi x+\xi^3.$$

Wuestenfux
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