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Given a subgroup $H$ of a group $(G,+)$, a $(G,H,k,1)$ difference family (DF) is a set $\mathcal F$ of $k$-subsets of $G$ such that $\{f-f':f,f'\in F, f\neq f',F\in \mathcal F\}=G\setminus H$. Let $g\mathbb Z_{gh}$ is the subgroup of order $h$ in $\mathbb Z_{gh}$ generated by $g$. A $(\mathbb Z_{gh},g\mathbb Z_{gh},k,1)$-DF is called cyclic and written as a $(gh,h,k,1)$-CDF. This paper shows that for $h\in\{2,3,6\}$, there exists a $(gh,h,4,1)$-CDF if and only if $gh\equiv h\pmod{12}$, $g\geq 4$ and $(g,h)\not\in\{(9,3),(5,6)\}$. As a corollary, it is shown that a 1-rotational S$(2,4,v)$ exists if and only if $v\equiv4\pmod{12}$ and $v\neq 28$. This solves the long-standing open problem on the existence of a 1-rotational S$(2,4,v)$. As another corollary, we establish the existence of an optimal $(v,4,1)$-optical orthogonal code with $\lfloor(v-1)/12\rfloor$ codewords for any positive integer $v\equiv 1,2,3,4,6\pmod{12}$ and $v\neq 25$. We also give applications of our results to cyclic group divisible designs with block size four and optimal cyclic $3$-ary constant-weight codes with weight four and minimum distance six.

Self-doping can not only suppress the photogenerated charge recombination of semiconducting quantum dots by self-introducing trapping states within the bandgap, but also provide high-density catalytic active sites as the consequence of abundant non-saturated bonds associated with the defects. Here, we successfully prepared semiconducting copper selenide (CuSe) confined quantum dots with abundant vacancies and systematically investigated their photoelectrochemical characteristics. Photoluminescence characterizations reveal that the presence of vacancies reduces the emission intensity dramatically, indicating a low recombination rate of photogenerated charge carriers due to the self-introduced trapping states within the bandgap. In addition, the ultra-low charge transfer resistance measured by electrochemical impedance spectroscopy implies the efficient charge transfer of CuSe semiconducting quantum dots-based photoelectrocatalysts, which is guaranteed by the high conductivity of their confined structure as revealed by room-temperature electrical transport measurements. Such high conductivity and low photogenerated charge carriers recombination rate, combined with high-density active sites and confined structure, guaranteeing the remarkable photoelectrocatalytic performance and stability as manifested by photoelectrocatalysis characterizations. This work promotes the development of semiconducting quantum dots-based photoelectrocatalysis and demonstrates CuSe semiconducting quantum confined catalysts as an advanced photoelectrocatalysts for oxygen evolution reaction.