Timeline for How large can the dimension of a 'Span of powers of a finite field basis' be?
Current License: CC BY-SA 4.0
8 events
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| Jul 31, 2021 at 15:52 | comment | added | actcon | I proved the statement. I cannot understand how I couldn't see this before. Please check the edited question. | |
| Jul 31, 2021 at 8:25 | comment | added | actcon | Although your proof may be wrong, I find your statement "$\gcd (d,p^k-1)=1 \implies M_{p^k}^{(d)}=k$" very compelling. Thanks for the insight. I will add it to the question as a conjecture. | |
| Jul 31, 2021 at 8:23 | comment | added | Donggeon Yhee | Sorry (i) has big error. dim=q is not justified yet, my mistake. | |
| Jul 31, 2021 at 7:58 | comment | added | Donggeon Yhee | (i) Under the assumption gcd$(d,...)$, $\beta_1^d\neq \beta_2^d$ so that the subspace has dimension at least 2. It should be then $q$. (ii) In the first case, (if I'm correct) $d$-power preserves basis property, at each $B_j$'s. Consequently, a basis of $\mathbb{F}_{p^k}$ is replaced by another basis by the $d$-power operation : the product of new $\mathcal{B}$'s gives a new basis of $\mathbb{F}_{p^k}$. Thus $M_{p^k}^{(d)}=k$. | |
| Jul 31, 2021 at 7:47 | history | edited | Donggeon Yhee | CC BY-SA 4.0 |
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| Jul 31, 2021 at 7:27 | comment | added | actcon | Thanks for your answer:) Your inspection on $\gcd(d,p^k-1)$ seems to be correct, comparing with the experimental results. However, I could not follow your answer. Can you kindly elaborate more on the following? (i) Why is the dimension of $\langle \beta_1^d, \cdots, \beta_q^d \rangle$ equals to $q$? (ii) How can we deduce that $M_{p^k}^{(d)}$ from construction of a basis? | |
| Jul 31, 2021 at 6:09 | history | edited | Donggeon Yhee | CC BY-SA 4.0 |
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| Jul 31, 2021 at 5:12 | history | answered | Donggeon Yhee | CC BY-SA 4.0 |