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We propose a new family of spatially coupled product codes, called sub-block rearranged staircase (SR-staircase) codes. Each code block of SR-staircase codes is obtained by encoding rearranged preceding code blocks and new information block, where the rearrangement involves sub-blocks decomposition and transposition. The proposed codes can be constructed to have each code block size of $1/q$ to that of the conventional staircase codes while having the same rate and component codes, for any positive integer $q$. In this regard, we can use strong algebraic component codes to construct SR-staircase codes with a similar or the same code block size and rate as staircase codes with weak component codes. We characterize the decoding threshold of the proposed codes under iterative bounded distance decoding (iBDD) by using density evolution. We also derive the conditions under which they achieve a better decoding threshold than that of staircase codes. Further, we investigate the error floor performance by analyzing the contributing error patterns and their multiplicities. Both theoretical and simulation results show that the designed SR-staircase codes outperform staircase codes in terms of waterfall and error floor while the performance can be further improved by using a large coupling width.