Patent ID: 7774679

Claim:
An integrated circuit having a memory and a microprocessor that performs Galois field arithmetic on 12-bit numbers read from a digital data stored on a disk, the integrated circuit comprising: an arithmetic unit operable by the microprocessor to extend for the 12-bit numbers a base field GF(2) to GF(2 3 ) using a polynomial a3+a+1 over the base field GF(2), wherein the arithmetic unit is operable by the microprocessor to extend for the 12-bit numbers GF(2 3 ) to a first quadratic extension GF(2 6 ) using a polynomial b2+b+ai, and values for ai are selected from a first list comprising {trace(ai), i=0, . . . , 6}=[1, 0, 0, 1, 0, 1, 1] wherein trace(ai) =1, and wherein the arithmetic unit is operable by the microprocessor to extend for the 12-bit numbers GF(2 6 ) to a second quadratic extension GF(2 12 ) using a polynomial c2+c+bi, and values for bi are selected from a second list comprising {trace(trace(bi), i=0, . . . , 62}=[0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0], wherein trace(trace(bi))=1; wherein the values for ai and the first list produce a first set of irreducible polynomials, and the values for bi and the second list produce a second set of irreducible polynomials; and wherein the first set of irreducible polynomials and the second set of irreducible polynomials are used to detect errors in digital data stored on a disk.