Error correcting system

Claim 1

An error correcting system which receives double correction BCH codes each consisting of M data symbols and four parity symbols, each consisting of m bits and represented by the following generated polynomial: g.sub.(x)=( x+1)(x+α)(x+α.sup.2)(x+α.sup.3), where α is the origin element of Galois field GF(2 m ), and which find the roots of the following error location polynomial: f.sub.(x) =x.sup.2 +σ.sub.1 x+σ.sub.2, thereby correcting errors in the double correction BCH codes, said system comprising: means for storing the double correction BCH codes; syndrome generating means for generating four syndromes S 0 , S 1 , S 2 and S 3 from the double correction BCH codes; error location calculating means comprising: a first multiplying/adding section for multiplying and adding the four syndromes S 0 , S 1 , S 2 and S 3 to obtain three syndromes S a , S b and S c which are given as follows: S.sub.a =S.sub.1.sup.2 +S.sub.0 S.sub.2, S.sub.b =S.sub.0 S.sub.3 +S.sub.1 S.sub.2, S.sub.c =S.sub.1 S.sub.3 +S.sub.2.sup.2 a second multiplying/adding section for performing multiplication and addition by substituting α 0 -α M+3 for x in the following polynomial obtained by transforming said error location polynomial f.sub.(x) : f'.sub.(x) =S.sub.a f.sub.(x) =S.sub.a x.sup.2 +S.sub.b x+S.sub.c, thereby obtaining two products S a ·α 2 and S b ·α and a sum S a α 2 +S b α+S c , utilizing the fact that the coefficients σ 1 , σ 2 of said error location polynomial f.sub.(x) and the three syndromes S a , S b and S c have the following relations according to the theory on the correction of said double correction BCH codes: σ.sub.1 =S.sub.b /S.sub.a =α.sup.i +α.sup.j, σ.sub.2 =S.sub.c /S.sub.a =α.sup.i α.sup.j ; an α i , α j detecting section for detecting α i and α j which satisfy f'.sub.(x) =0; and a third multiplying/adding section for adding α i and α j and multiplying α i by α j , thereby finding the coefficients σ 1 , σ 2 which define error locations; error pattern calculating means comprising a fourth multiplying/adding means for obtaining an error pattern e i from α i and α j by performing the following multiplication: e.sub.1 =(S.sub.0 α.sup.j +S.sub.1)(α.sup.i +α.sup.j).sup.-1, said equation having been derived from the following division: ##EQU22## and for obtaining another error pattern e j by performing the following addition: e.sub.j =S.sub.0 +e.sub.i ; and correction means for correcting the errors in said double correction codes stored by said code storing means in accordance with the coefficients σ 1 , σ 2 and the error patterns e i , e j .