Let's define a codeword to have the following structure (each X _i is a bit in the data set):

• X ₁ X ₂ X ₃ X ₄ X ₅ X ₆ X ₇ X ₈ X ₉ X ₁₀ X ₁₁ X ₁₂ ...

Also, let every bit with a subscript that is a power of 2 be a check bit, denoted by r _j, which will contain redundant information for error correction. Thus, X ₁, X ₂, X ₄, and X ₈ are check bits and we replace them with r ₁, r ₂, r ₄, and r ₈, respectively.

• r ₁ r ₂ X ₃ r ₄ X ₅ X ₆ X ₇ r ₈ X ₉ X ₁₀ X ₁₁ X ₁₂ ...

Since the X _i bits are predetermined (they represent user data), we need a rule for forming check bits. This rule is based on parity and is defined as follows:

Each check bit, r _j, is formed by collecting all corresponding X _i bits. These are determined by expressing the subscript of each X _i as the sum of a power of 2. For example, X ₃ corresponds to r ₁ and r ₂ because its subscript, 3, is equal to the sum of check bits r ₁'s and r ₂'s subscripts (i.e., 3 = 1 + 2). Once the...