Unions of intervals in codes based on powers of sets

We prove that for every integer $d\ge 2$ there exists a dense collection of subsets of $[n{]}^d$ such that no two of them have a symmetric difference that may be written as the $d$th power of a union of at most $\lfloor d\mathrm{/}2\rfloor$ intervals. This provides a limitation on reasonable tightenings of a question of Alon from 2023 and of a conjecture of Gowers from 2009, and investigates a direction analogous to that of recent works of Conlon, Kamčev, Leader, Räty and Spiegel on intervals in the Hales-Jewett theorem.