Standard array: Difference between revisions

Revision as of 02:42, 12 May 2012

Revision as of 20:02, 26 January 2013

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In [[coding theory]], a '''standard array''' (or Slepian array) is a <math>q^{n-k}</math> by <math>q^{k}</math> array that lists all elements of a particular <math>\mathbb{F}_q^n</math> [[vector space]]. Standard arrays are used to [[Decoding_methodsDecoding methods|decode]] [[linear code]]s; i.e. to find the corresponding [[codeword]] for any received vector.

== Definition ==

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{| class="wikitable"

|-

| <u>[[Zero_vectorZero vector|0]]</u>

| 01101

| 10110

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Note that 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.

Note that the first row contains the <u>0</u> vector and the codewords of <math>C_{3}</math> (<u>0</u> itself being a codeword). Also, the leftmost column contains the vectors of [[Hamming_weightHamming weight|minimum weight]] enumerating vectors of weight 1 first and then using vectors of weight 2. Note also that each possible vector in the vector space appears exactly once.

== Constructing a standard array ==

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Note that 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. This ambiguity is another reason that different decoding methods are sometimes used.

== See Alsoalso ==

* [[Linear code]]