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A bound and construction are presented for high-rate burst-error-correcting recurrent codes. The bound is an upper bound on the block length in terms of the total redundancy used in decoding, the redundancy per block, and the burst length. The construction uses a block-code parity-check matrix as the first block of the recurrent code parity-check matrix. For a block code it is typical to find that only a portion of the redundancy need be used to detect a burst. Any block code for which this is true can be used in the construction. The recurrent code is then related as follows to the block code from which it is constructed. 1) The recurrent code block length is the same as the block-code block length. 2) The total redundancy used in decoding the recurrent code is the same as the block-code redundancy per block. 3) The recurrent code redundancy per block is the same as the block-code redundancy required for burst detection only. 4) The recurrent code is of higher rate than the block code. 5) The recurrent code requires a guard space between bursts but otherwise corrects the same bursts as the block code. It is shown that, when certain well-known cyclic codes are used in the construction, the resulting recurrent codes are close to the upper bound.