Autocorrelation - cyclic vs acyclic
If the cyclic autocorrelation output has the properties you need in your application (e.g., it's peak-to-average ratio is high enough for picking out the peaks), then it's probably okay to substitute the cyclical correlation for the linear one. For example, consider a length 7 maximal-length sequence:
$$
\{+1, -1, -1, +1, +1, +1, -1\}
$$
The cyclical and acyclical autocorrelation functions are plotted below.


Notice that both functions have a peak value of 7. The acyclical autocorrelation function has a largest off-peak magnitude of 3, whereas the cyclical autocorrelation function has a largest off-peak magnitude of 1. The peak-to-off-peak amplitude ratios are $7$ and $7/3$ for cyclical vs. acyclical, respectively. This means that the cyclical autocorrelation function is basically three times more peaky, making it much easier to detect. If the extra peak-to-off-peak amplitude ratio is not necessary for your application, it would be okay to correlate using linear correlation instead.

If instead, you have something whose acyclical autocorrelation properties are optimal (such as a Barker code), and you would like to calculate that acyclical correlation but don't want to use the computationally inefficient direct form of convolution, there are algorithms that calculate the acyclical autocorrelation using the computational savings of FFTs. Two of these I know of are called the overlap-add and overlap-save methods. They work on blocks of the input data at a time, and using FFTs of the blocks of data and multiplication in the transformed domain, calculate the acyclical correlations/convolution of the data against your given sequence.