## 1 Introduction  
A frame is a set of vectors in Hilbert space that spans the whole space [1]. The concept of a frame is useful to represent an overcomplete spanning system, which means that a frame has more vectors than the dimension of the space [2]. Frames have been widely applied to various fields of quantum physics/computing [3], communications [4], graph theory [5], algebraic geometry [6], spherical codes [7], and so on. A frame with incoherent characteristics, where the frame vectors have low similarity to each other, is desired in many applications. The mutual coherence [8] is used as an indicator of how similar each frame vector is to each other. For example, the low mutual coherence of a frame enables sparse signal recovery from incomplete measurements [9] in compressed sensing (CS), accurate target scene recovery in radar sensing [10], low multiple access interference in synchronous code-division multiple access (CDMA) [11], and reliable angular momentum analysis in quantum mechanics [12]. However, the mutual coherence cannot be reduced indefinitely close to 0 when the number of frame vectors \(N\) is larger than the frame length \(m\). There exists a theoretical lower bound on mutual coherence, which is called the Welch bound [13]. If all the vectors in a frame have unit norm, and the magnitude of the inner product of each pair meets the Welch bound equality, the frame is called an equiangular tight frame (ETF) [14]. ETFs have been studied for a long time, but they are known to be notoriously difficult to design [15; 16; 17]. Therefore, a frame with nearly optimal coherence, which is a good alternative to ETF, has been actively studied. A variety of algebraic methods have been exploited in [18; 19; 20; 21; 22; 23; 24; 25] to design various incoherent frames with nearly optimal coherence in a constructive way. In addition, there have been numerous algorithmic approaches for obtaining incoherent frames. Alternating projection methods were used in [26; 27; 28; 29], which solved the singular value decomposition (SVD) to minimize the maximum magnitude of the Gram matrix of a frame. To reduce the computational complexity of implementing SVD, the algorithms that optimize the coherence of a frame without solving SVD problems [30, 31, 32, 33] or using only matrix-vector multiplications [34] were proposed. In [35], the authors utilized the discrete stochastic approximation method, using the Kronecker product of two Fourier matrices for near-optimal codebook design under spatially correlated 3D channels. In [7], incoherent tight frames were designed by the distance optimization algorithm for spherical code construction. In [36], incoherent unit-norm frames were designed by the alternating minimization penalty method. In [37], high-dimensional incoherent frames were designed by linear programming exploiting the minorization maximization technique [38]. In [39], the authors optimized the coherence of a dictionary matrix by a greedy algorithm based on a quasi-Newton method [40]. Finally, real/complex-valued, sparse, and harmonic incoherent frames were designed in [41] by using sequential iterative decorrelation convex optimization (SIDCO) or iterative reweighted \(l_{1}\) optimization (IRL1) [42]. Especially, the incoherent harmonic frames were designed by subsampling the indices of the Fourier matrix by solving an \(l_{1}\)-optimization problem using IRL1 and exploiting a local search method. In this paper, we design incoherent tight frames by subsampling rows from the matrices generated by the Kronecker product with unitary and Fourier matrices. In this _Kronecker-product-based frame_, each frame vector has the elements that have a small number of phases, which allows low implementation complexity. We show that the Gram matrix of the frame is block-wise circulant due to the structure of the Fourier matrix. Using this feature, we derive an objective function that calculates the mutual coherence of the Kronecker-product-based frame equivalently.