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The generator polynomial of the BCH code is defined as the least common multiple g ( x ) lcm( m 1( x m d 1( x ). There is no need to calculate the error values in this example, as the only possible value. Since jdisplaystyle alpha j are the zeros of g(x displaystyle g(x of which C(x)displaystyle C(x) is a multiple, C(j)0.displaystyle C(alpha j)0.

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(This is used in QR codes.) Let the message to be transmitted be, or in polynomial notation, M(x)x4x3x1.displaystyle M(x)x4x3x1. Decoding with unreadable characters edit Suppose the same scenario, but the received word has two unreadable characters 1 0 0? Factor error locator polynomial edit Now that you have the (x)displaystyle Lambda (x) polynomial, its roots can be found in the form Lambda (x alpha i_1x-1 alpha i_2x-1)cdots (alpha i_vx-1) by brute force for example using the Chien search algorithm. Endaligned We have reached polynomial of degree at most 3, and as beginpmatrix-(alpha 4alpha -5x) alpha -3alpha 5xalpha 7x2alpha 3alpha -5xalpha 6x2 -(alpha 7xalpha 5x2alpha 7xalpha 5x2alpha 3x3 alpha -3alpha 5xalpha 7x2alpha 3alpha -5xalpha 6x2 alpha 4alpha we get beginpmatrix-(alpha 4alpha -5x) alpha -3alpha 5xalpha.

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Therefore, g(x)displaystyle g(x) is the least common multiple of at most d/2displaystyle d/2 minimal polynomials mi(x)displaystyle m_i(x) for odd indices i,displaystyle i, each of degree at most mdisplaystyle. For the goal of locating error positions we could change the set of syndromes in the similar way to reflect all unreadable characters. Therefore, the least common multiple of d1displaystyle d-1 of them has degree at most (d1)mdisplaystyle (d-1)m.

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Calculate the error location polynomial edit If there are nonzero syndromes, then there are errors. (2004 Modern Algebra with Applications (2nd. Vleqslant d-1,lambda _0neq 0qquad Lambda (x)sum _i0vlambda _ixilambda _0prod _k0v(alpha -i_kx-1). Recall that c,cd2displaystyle alpha c,ldots,alpha cd-2 are roots of g(x displaystyle g(x hence of p(x)displaystyle p(x). Edu/class/ee387/ Gorenstein, Daniel ; Peterson,.

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Suppose there is unreadable letter on position k1,displaystyle k_1, we could replace set of syndromes sc, scd2displaystyle s_c,cdots,s_cd-2 by set of syndromes tc, tcd3displaystyle t_c,cdots,t_cd-3 defined by equation tik1sisi1.displaystyle t_ialpha k_1s_i-s_i1. If the determinant of matrix Svvdisplaystyle S_vtimes v is nonzero, then we can actually find an inverse of this matrix and solve for the values of unknown displaystyle Lambda values.

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(Using log notation which is independent on GF(24) isomorphisms. We will consider different values. The roots are 2,displaystyle alpha 2, and 10displaystyle alpha 10 (after finding for example 2displaystyle alpha 2 we can divide displaystyle Lambda by corresponding monom (x2)displaystyle (x-alpha 2) and the root of resulting monom could be found easily). Let (x)31x.displaystyle Lambda (x)alpha 3alpha. If (x)displaystyle Lambda (x) denotes the polynomial eliminating the influence of these coordinates, we obtain S(x x x)kv, d20.displaystyle S(x)Gamma (x)Lambda (x)stackrel kv, cdots,d-20.

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In Forney formula, (x)displaystyle Lambda (x) could be multiplied by a scalar giving the same result. Next, apply the Peterson procedure by row-reducing the following augmented matrix.