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## Is there a name for these sets in a finite field/ring?

Let $\mathbb{F}_{p^{mn}}$ be a Finite Field with Char $p$ with $\zeta$ a primitive element in the field.

$\forall {k} \in $$\{$$1,\cdots,m$$\}, \forall {i}_{k} \in$$\{$$0,\cdots,p-1$$\}$, consider the two sets of $p^{m}$ elements of the form

$(1)$$\displaystyle \sum_{j=1}^{n} \displaystyle \sum_{k=1}^{m}(ji_{k} \mod p)p^{k+m(j-1)-1} \in \mathbb{Z}_{p^{mn}}. (2)$$\displaystyle \sum_{j=1}^{n} \displaystyle \sum_{k=1}^{m}(ji_{k} \mod p)\zeta^{k+m(j-1)-1} \in \mathbb{F}_{p^{mn}}$.

$(3)$$\zeta^{\displaystyle \sum_{j=1}^{n} \displaystyle \sum_{k=1}^{m}(ji_{k} \mod p)p^{k+m(j-1)-1}} \in \mathbb{F}_{p^{mn}}$ where the exponent is in $\mathbb{Z}$.

Is there a name for these sets?

When $p$ is not prime we will not be in a finite field, however is there a term for $(1)$ in this case and is there something analogous to $(2)$ and $(3)$ in this case?

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Why should there be? Please provide more context and motivation for these sums. – Martin Brandenburg Jul 16 2011 at 8:28
seems to arise as patterns that needs to be avoided/used in certain constrained codes. – unknown (yahoo) Jul 16 2011 at 20:26