The present paper introduces a new approach to the construction of a sequence set with a zero-correlation zone for both periodic and aperiodic correlation functions. The proposed sequences can be constructed from a pair of Hadamard matrices of orders n₀ and n₁. The constructed sequence set consists of n₀n₀ ternary sequences, each of length n₀^(m+2)(n₁+Δ), for a non-negative integer m and τ≥2. The zero-correlation zone of the proposed sequences is $|\\ au|\\le \\ablen^{m+1}-1$, where τ is the phase shift. The proposed sequence set consists of n₀ subsets, each with a member size n₁. The correlation function of the sequences of a pair of different subsets, referred to as the inter-subset correlation function, has a zero-correlation zone with a width that is approximately τ times that of the correlation function of sequences of the same subset (intra-subset correlation function). The inter-subset zero-correlation zone of the proposed sequences is $|\\ au|\\le \\slen\\ablen^{m+1}$, where τ is the phase shift. The wide inter-subset zero-correlation enables performance improvement during application of the proposed sequence set.