3.7.1 Standard Array Decoding

Let V ₁ = 0, V ₂, . . ., V _{2 ^k} be the code vectors (codewords) of an ( n, k) binary linear block code. In the presence of channel errors of 0's and 1's, the received vector can take on any of the 2 ⁿ n-component binary vectors. We shall partition the 2 ⁿ binary vectors into 2 ^k disjoint subsets such that the code vector V _i is in the i-th subset for 1 ? i ? 2 ^k. If the received vector is found in the i-th subset, the received vector is decoded into V _i. Based on the linear structure of the code, the partitioning of the 2 ⁿ binary vectors is done as follows:

1. Place all code vectors in the first row of a 2 ^{n ? k}-by-2 ^k array with V ₁ as the left-most element.

2. For l = 2, 3, . . ., 2 ^{n ? k}, choose a distinct n-component error vector E _l of smallest possible weight from the remaining n-component binary vectors and place it as the left-most element in the l-th row of the array. Fill the remaining entries in that row by adding E _l to V _i.

The resultant array is called standard array of a linear code.