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A new method of finding binary cyclic codes from the finite-field transform domain is presented. These cyclic codes have sparse parity-check matrix and thus are suitable for iterative decoding. Some interesting properties of the proposed construction method include the knowledge of the minimum distance and the ability to trade the increase in code dimension with a reduction in the parity-check matrix sparsity. By means of simulations, we show that the error correcting performance of the codes under iterative decoding is very close to the sphere-packing-bound constrained for binary transmission.