Computation of cyclic redundancy checks may use whichever form is more convenient. Parallel update for a byte or a word at a time can also be done explicitly, without a table. This is normally used in high-speed hardware implementations. For each bit an equation is solved after 8 bits have been shifted in. The following tables list the equations for some commonly used polynomials, using following symbols: As the CRC-32 polynomial has a large number of terms, when computing the remainder a byte at a time each bit depends on several bits of the previous iteration. In byte-parallel hardware implementations this calls for either multiple-input
Stochastic computing bit operation that can compute this function; the usual solution involves producing correlated output bits, which, as we have seen above, can cause a host of problems. Other functions (such as the averaging operator formula_25) require either stream decimation or inflation. Tradeoffs between precision and memory can be challenging. Although stochastic computing has a number of defects when considered as a method of general computation, there are certain applications that highlight its strengths. One notable case occurs in the decoding of certain error correcting codes. In developments unrelated to stochastic computing, highly effective methods of decoding LDPC codes using the