Standard array

In coding theory, a standard array (or Slepian array) is a 𝔽qn vector space. Standard arrays are used to decode linear codes; i.e. to find the corresponding codeword for any received vector.

qk array where:

1. The first row lists all codewords (with the 0 codeword on the extreme left)
2. Each row is a coset with the coset leader in the first column
3. The entry in the i-th row and j-th column is the sum of the i-th coset leader and the j-th codeword.

For example, the [5,2]-code C, starting with 0, as the first row

Choose any vector of minimum weight not already in the array. Write this as the first entry of the next row. This vector is denoted the 'coset leader'.

Fill out the row by adding the coset leader to the codeword at the top of each column. The sum of the i-th coset leader and the j-th codeword becomes the entry in row i, column j.

Repeat steps 2 and 3 until all rows/cosets are listed and each vector appears exactly once.

Adding vectors is done mod q. For example, binary codes are added mod 2 (which equivalent to bit-wise XOR addition). For example, in