We consider codes consisting of arrays over an alphabet F, in which certain intersecting subsets of n/spl times/m coordinates are required to form codewords of length n in prescribed codes over the alphabet F/sup m/. Two specific cases are studied. In the first case, referred to as a singly-intersecting coding scheme, the user data is mapped into n/spl times/(2m-1) arrays over an alphabet F, such that the n/spl times/m subarray that consists of the left (respectively, right) m columns forms a codeword of a prescribed code of length n over F/sup m/; in particular, the center column is shared by the left and right subarrays. Bounds are obtained on the achievable redundancy region of singly-intersecting coding schemes, and constructions are presented that approach-and sometimes meet-these bounds. It is shown that singly-intersecting coding schemes can be applied in a certain model of broadcast channels to guarantee reliable communication. The second setting, referred to as a fully-intersecting coding scheme, maps the user data into n/spl times/m/spl times/m three-dimensional arrays in which parallel n/spl times/m subarrays are all codewords of the same prescribed code over F/sup m/. Bounds and constructions are presented for these codes, with the analysis based on representing the n/spl times/m/spl times/m arrays as vectors over certain algebras on m/spl times/m matrices.