1. A method for efficiently calculating syndromes in Reed-Solomon coding, which is adapted to be implemented in a processor having a parallel processing instruction set, the method comprising the following steps: (a) configuring the processor to initialize a syndrome vector; (b) configuring the processor to obtain a symbol from a Reed-Solomon block code; (c) configuring the processor to find a lookup index based on the symbol; (d) configuring the processor to use the parallel processing instruction set, to obtain a finite field product vector corresponding to the lookup index from a finite field vector multiplication table that includes at least one finite field product vector in which the finite field product vector may be expressed as [γ j ×(α 1 ) j ,γ j ×(α 2 ) j ,γ j ×(α 3 ) j ,. .. , γ j ×(α 2t ) j ], where 0<j<n, γ j represents the j th symbol obtained from the Reed-Solomon block code, t represents the maximum number of correctable errors, α is a known constant, n represents the total number of symbols of the Reed-Solomon block code, and both γ j and α are finite field elements; (e) configuring the processor to use the parallel processing instruction set to perform vector finite field addition on the finite field product vector corresponding to the lookup index and the syndrome vector, thereby obtaining an updated syndrome vector; and (f) configuring the processor to output the updated syndrome vector.