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An algebraic complete decoding for double-error-correcting binary BCH codes of primitive lengthis derived.The decoding is done more quickly than the step-by-step decoding devised by Hartmann.And if an error pattern corresponding with syndromes s_1 and s_3 has weight 3,the decoding can find allerror patterns of weight 3 corresponding with these syndromes.At the same time,a discriminant for apolynomial of degree 3 over GF(2~m)has three distinct roots in GF(2~m)is also derived.The discrimi-nant is very important for complete decoding of triple-error-correcting binary BCH codes.