Before revision:  Our lower-bound proofs proceed in three steps: (i) a classic connection due to Crapo and Rota [1970] between the number of tuples of codewords of full support and the Tutte polynomial of the matroid associated with the code;
After revision:  Our lower-bound proofs proceed in three steps: (i) a connection due to Crapo and Rota [1970] between the number of tuples of codewords of full support and the Tutte polynomial of the matroid associated with the code;