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A class of quasi-cyclic multiple-burst-error-correcting codes are constructed in which threshold decoding is used. These codes resemble the interlaced self-orthogonal quasi-cyclic random-error-correcting codes constructed by Townsend and Weldon, but the interlacing depends on the parity-check equations used. These parity-check equations are based on difference triangles that were introduced by Robinson and Bernstein in connection with convolutional codes. A restriction on these codes is that the maximum error burst length allowable is a multiple of a subperiod of the codes. It is shown that in many cases these codes have shorter length than the equivalent interlaced random-error-correcting quasi-cyclic codes.