Using Hash Table and Cyclotomic Coset Method for Decoding the Quadratic Residue Code

A method for decoding of the quadratic residue (QR) code with hash tables is presented. The method can be applied in decoding the (31, 16, 7) QR code that the generator polynomial can be factored. In other words, the mapping between elements of syndrome S1 and all correctable error patterns is not one-to-one. Therefore, the decoding of the (31, 16, 7) QR code needs to stuck S1,S5, S7 known syndrome mapping an error pattern that has one-to-one nature, where a subscript 1, 5, 7 are cyclotomic cosets. Furthermore, the algorithm determines the error locations by hash tables without operating the additions and multiplications over finite field. To decode QR code result, the hash table method for the (31, 16, 7) QR code is dramatically reduced the memory size above 97%. It is very suitable for high speed in modern communication system.