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Filter banks over finite fields have found applications in digital signal processing and error-control coding. One method to design a filter bank is to factor its polyphase matrix into the product of elementary building blocks that are fully parameterized. It has been shown that this factorization is always possible for one-dimensional (1-D) paraunitary filter banks. In this paper, we focus on two-channel two-dimensional (2-D) paraunitary filter banks that are defined over fields of characteristic two. We generalize the 1-D factorization method to this case. Our approach is based on representing a bivariate finite-impulse-response paraunitary matrix as a polynomial in one variable whose coefficients are matrices over the ring of polynomials in the other variable. To perform the factorization, we extend the definition of paraunitariness to the ring of polynomials. We also define two new building blocks in the ring setting. Using these elementary building blocks, we can construct FIR two-channel 2-D paraunitary filter banks over fields of characteristic two. We also present the connection between these 2-D filter banks and 2-D error-correcting codes. We use the synthesis bank of a 2-D filter bank over the finite field to design 2-D lattice-cyclic codes that are able to correct rectangular erasure bursts. The analysis bank of the corresponding 2-D filter bank is used to construct the parity check matrix. The lattice-cyclic property of these codes provides very efficient decoding of erasure bursts for these codes.