###### Abstract  
We present a new set of near orthogonal codes which we call Hadamard Aperiodic Interval (HAPI) codes and demonstrate their utility for parallel multi-transmitter synthetic aperture imaging. The codes are tri-state and sparse. Locations of non-zero bits are based on marks in a sequence of aperiodic intervals, also known as a Golomb ruler. The values of non-zero bits are selected from Hadamard sequences that are mutually orthogonal. This ensures that cross-correlation sidelobe magnitudes between differing codes are bounded by unity while the autocorrelation approaches a delta function with mainlobe-to-sidelobe levels scaling with the number of non-zero bits. We use simulations to demonstrate the potential of the codes for synthetic aperture imaging. A multiplicity of transmitter elements is used to transmit codes simultaneously, with a different code for each element. Echo signals are received from a multiplicity of transducer elements in parallel. Channel data from each receiver element are cross-correlated with respective HAPI codes to estimate the transmit-receive signature associated with each transmitter-receiver pair while minimizing crosstalk. This estimate of the full transmit-receive synthetic aperture dataset is then used to form high-quality images demonstrating image quality and signal-to-noise ratio improvements over multiple flash angle imaging and synthetic aperture imaging methods for linear arrays. We also demonstrate simulated full volume synthetic aperture imaging with random sparse arrays, possible with one extended HAPI code-set transmit event. c 2022122022122022120221220221220221220221222222222222222222222222222222222  
Goday codes are another type of binary code that are used for increasing SNR and for improving penetration depth in ultrasound imaging [8; 9; 10]. They are pairs of binary codes belonging to a set of codes called complementary codes [11]. These complementary codes consist of two codes of length N, whose autocorrelation function have sidelobe levels equal in magnitude but opposite in sign. When the autocorrelation functions of the complementary codes are added together, they result in a composite autocorrelation function that has a mainlobe peak of 2N and sidelobe levels of zero. The sidelobes completely cancel out because they are equal but opposite in sign. This means Golay complementary codes produce an ideal autocorrelation function which is essential for imaging with good quality. Golay codes have been successfully used to image and have shown an increase in SNR of 32 dB over sine burst excitation [12]. Binary codes such as Barker codes, Golay codes, and others are good for increasing penetration depth and signal-to-noise ratio, but they do not constitute a near orthogonal code set, and hence cannot be used for parallel transmission in ultrasound imaging. Orthogonal codes have been investigated for use in multi-user communications [13; 14; 15] and are successfully used in radio communications technologies such as CDMA (code division multiple access) [16; 17]; however, most of these are meant for synchronous coded information transfer rather than high dynamic range imaging. The imaging problem can be considered asynchronous because locations of scatterers and hence echo-time delays are unknown and different for an ensemble of scatterers. Two main types of orthogonal codes that are commonly used for asynchronous communications but are not commonly used in imaging are pseudorandom codes [18; 19; 20; 21] and Kasami codes [22; 23; 24]. The sidelobe levels in the autocorrelation function of Kasami codes are small compared to the peak but are not guaranteed to be bounded by a minimum level. Kasami codes are specially constructed binary sequences with good cross-correlation properties. Unlike other orthogonal codes, the small Kasami code set meets the Welch lower bound [25]. The Welch bound is a lower bound on how small the cross-correlation and autocorrelation can simultaneously be at non-zero lags [26]. In the field of medical imaging, [PERSON] et al.