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Often in coding theory, a code's alphabet is taken to be a finite field. In particular, if  is the finite field with two (resp. three, four, etc.) elements, we call  a binary (resp. ternary, quaternary, etc.) code. In particular, when our alphabet is a finite field then the set  is a vector space over  , and we define a linear code over  of block length  to be a subspace (as opposed to merely a subset) of  . We define the dimension of  to be its dimension as a vector space over  .
Though not sufficient for unique classification, a linear code's block length, dimension, and minimum distance are three crucial parameters in determining the strength of the code. For referencing, a linear code with block length  , dimension  , and minimum distance  is referred to as an  -code.
Some examples of linear codes are Hamming Codes, BCH codes, Goppa codes, Reed-Solomon codes, and the Golay code.
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"linear code" is owned by mathcam.
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Cross-references: parameters, minimum distance, sufficient, dimension, subset, subspace, block length, vector space, binary, finite field, alphabet, code's
There are 7 references to this entry.
This is version 4 of linear code, born on 2004-05-05, modified 2004-06-08.
Object id is 5836, canonical name is LinearCode.
Accessed 9720 times total.
Classification:
| AMS MSC: | 94B05 (Information and communication, circuits :: Theory of error-correcting codes and error-detecting codes :: Linear codes, general) |
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Pending Errata and Addenda
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