# Explanation of vector in Visual Cryptography

I'm starting to study about visual cryptography, but I am stuck with the following problem.

On paper "Visual Cryptography" by Moni Naor and Adi Shamir, on page 6. How to read/what is the meaning of

$$J^0_i = 0^{i-1}10^{k-i}$$

and

$$J^0_k = 1^{k-1}0$$

where $1 \le i \le k$?

Can someone explain how to read it? Because I think I misunderstood.. And also I'm happy if anyone gives me a reference to learn it.

This is string notation: $J_i^0=0^{i-1}10^{k-i}$ means i-1 consecutive 0's followed by a 1 (we don't write $1^1$) which is then followed by k-i consecutive 0's. So $0^{3}10^{4}$ is $00010000$.
As for $S^t[i,x]=\langle J_i^t,x\rangle$, with $t\in \{0,1\}$ this is an inner product of $J_i^t$ with $x=(x_1,\ldots,x_k) \in \{0,1\}^k$ so in general $S^t[i,x]=x_i$ (only the 1 component in $J_i^0$ picks up a nontrivial value. Thus $S^0[2,x]=0\times x_1+1\times x_2+0\times x_3=x_2$ and $S^0[2,101]=0,$ while $S^0[2,010]=1.$ I agree it is quite cumbersome.