On a conjecture regarding the symmetric difference of certain sets

Let n be a positive integer and n = {1, 2, . . . , n}. A conjecture arising from certain polynomial near-ring codes states that if k ≤ 1 and a1, a2, . . . , ak are distinct positive integers, then the symmetric difference a1n δ a2n δ δ akn contains at least n elements. Here, ain = {ai, 2ai, . . . , nai} for each i. We prove this conjecture for arbitrary n and for k = 1, 2, 3.