Reed-Solomon codes

Reed-Solomon codes are a class of non-binary cyclic error-correcting codes invented by Irving Reed and Gustave Solomon in 1960. They operate by treating a message as the coefficients of a polynomial over a finite field and transmitting the polynomial's values at selected points; decoding recovers the polynomial even when some values are corrupted, using the algebraic fact that a degree-k polynomial is uniquely determined by any k+1 correct evaluations.

Reed-Solomon codes are the workhorse of deep-space and optical communication: the Voyager probes, compact discs, QR codes, and RAID storage systems all depend on them. Their error-correcting power comes from the algebraic structure of finite fields, not from probabilistic iteration — making them a fundamentally different architectural family from turbo codes or LDPC codes. The BCH codes, which generalize Reed-Solomon to binary alphabets, share this algebraic lineage and represent a rival tradition in coding theory: one that trusts mathematical structure over iterative approximation.