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Let  be a linear code of block length  over the finite field
 . Then the set
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is the dual code of  . Here,  denotes either the standard dot product or the Hermitian dot product.
This definition is reminiscent of orthogonal complements of finite dimensional vector spaces over the real or complex numbers. Indeed,  is also a linear code and it is true that if  is the dimension of
 , then the dimension of  is  . It is, however, not necessarily true that
 . For example, if  is the binary code of block length  spanned by the codeword  then
 , that is,
 . In fact,  equals  in this case. In general, if  ,  is called self-dual. Furthermore  is called self-orthogonal if
 .
Famous examples of self-dual codes are the extended binary Hamming code of block length  and the extended binary Golay code of block length  .
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"dual code" is owned by GrafZahl.
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Cross-references: extended binary golay code, binary, codes, binary code, complex numbers, real, vector spaces, orthogonal complements, Hermitian dot product, dot product, finite field, block length, linear code
There are 5 references to this entry.
This is version 3 of dual code, born on 2005-05-01, modified 2005-05-13.
Object id is 6989, canonical name is DualCode.
Accessed 3992 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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