Wideband communication is often expected to deal with a very wide spectrum, which in many environments of interest includes strong interferers. Thus receivers for the wideband communication systems often need to mitigate interferers to reduce the distortion caused by the amplifier nonlinearity and noise. Recently, a new architecture for communication receivers known as random modulation mixes a signal with different pseudorandom sequences using multiple branches of channels before sampling. While random modulation is used in these receivers to acquire the signal at low sampling rates, the modulation sequences used lack the ability to suppress interferers due to their flat spectra. In previous work, we introduced the design of a single spectrally shaped binary sequence that mitigates interferers to replace the pseudorandom sequence in a channel. However, the designed sequences cannot provide the stable recovery achieved by pseudorandom sequence approaches. In this paper, we extend our previous sequence design to guarantee stable recovery by designing a set of sequences to be orthogonal to each other. We show that it is difficult to find the necessary number of sequences featuring mutual orthogonality and introduce oversampling to the sequence set design to improve the recovery performance. We propose an algorithm for multi-branch sequence design as a binary optimization problem, which is solved using a semidefinite program relaxation and randomized projection. While it is common to model narrowband interferers as a subspace spanned by a subset of elements from the Fourier basis, we show that the Slepian basis provides an alternative and more suitable compact representation for signals with components contained in narrow spectrum bands. Numerical experiments using the proposed sequence sets show their advantages against pseudorandom sequences and our previous work.