In coding theory, a standard array (or Slepian array) is a q^{n-k} by q^{k} array that lists all elements of a particular vector space. Standard arrays are used to decode linear codes; i.e. to find the corresponding codeword for any received vector.

Definition

A standard array for an [n,k]-code is a q^{n-k} by q^{k} array where: For example, the [5,2]-code C_{3} = { 0, 01101, 10110, 11011} has a standard array as follows: The above is only one possibility for the standard array; had 00011 been chosen as the first coset leader of weight two, another standard array representing the code would have been constructed. The first row contains the 0 vector and the codewords of C_{3} ( 0 itself being a codeword). Also, the leftmost column contains the vectors of minimum weight enumerating vectors of weight 1 first and then using vectors of weight 2. Also each possible vector in the vector space appears exactly once.

Constructing a standard array

Because each possible vector can appear only once in a standard array some care must be taken during construction. A standard array can be created as follows: Adding vectors is done mod q. For example, binary codes are added mod 2 (which equivalent to bit-wise XOR addition). For example, in Z_{2}, 11000 + 11011 = 00011. That selecting different coset leaders will create a slightly different but equivalent standard array, and will not affect results when decoding.

Construction example

Let C be the binary [4,2]-code. i.e. C = {0000, 1011, 0101, 1110}. To construct the standard array, we first list the codewords in a row. We then select a vector of minimum weight (in this case, weight 1) that has not been used. This vector becomes the coset leader for the second row. Following step 3, we complete the row by adding the coset leader to each codeword. We then repeat steps 2 and 3 until we have completed all rows. We stop when we have reached rows. In this example we could not have chosen the vector 0001 as the coset leader of the final row, even though it meets the criteria of having minimal weight (1), because the vector was already present in the array. We could, however, have chosen it as the first coset leader and constructed a different standard array.

Decoding via standard array

To decode a vector using a standard array, subtract the error vector - or coset leader - from the vector received. The result will be one of the codewords in C. For example, say we are using the code C = {0000, 1011, 0101, 1110}, and have constructed the corresponding standard array, as shown from the example above. If we receive the vector 0110 as a message, we find that vector in the standard array. We then subtract the vector's coset leader, namely 1000, to get the result 1110. We have received the codeword 1110. Decoding via a standard array is a form of nearest neighbour decoding. In practice, decoding via a standard array requires large amounts of storage - a code with 32 codewords requires a standard array with 2^{32} entries. Other forms of decoding, such as syndrome decoding, are more efficient. Decoding via standard array does not guarantee that all vectors are decoded correctly. If we receive the vector 1010, using the standard array above would decode the message as 1110, a codeword distance 1 away. However, 1010 is also distance 1 away from the codeword 1011. In such a case some implementations might ask for the message to be resent, or the ambiguous bit may be marked as an erasure and a following outer code may correct it. This ambiguity is another reason that different decoding methods are sometimes used.