What is dual codes?

Dual codes are pairs of linear codes that are related in a specific way. A linear code is a group of codewords, each of which is a linear combination of a set of basis vectors. The dual code is created by taking the orthogonal complement of the original code, which is the set of all vectors that are orthogonal to all the codewords in the original code.

The dual code of a linear code has a number of important properties. First, the dimension of the dual code is equal to the difference between the length of the original code and its dimension. Second, a codeword in the dual code is orthogonal to all codewords in the original code, and vice versa. Finally, the minimum distance of the dual code is equal to the minimum weight of any non-zero codeword in the original code.

Dual codes have a variety of applications in coding theory and cryptography. For example, they are used in the construction of error-correcting codes and in the design of secure cipher systems. Dual codes are also used in studying the complexity of linear programming problems and in the analysis of combinatorial designs.