- ... non-trivial
^{2.1}
- For our purposes, a
trivial linear code is one whose generator matrix contains an
all-zero column. Trivial codes can be simply reduced to non-trivial
codes by deleting the zero columns from their generator matrix.
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- ... code.
^{2.2}
- We do have to take
some care that when we delete the generator the resulting code
is non-trivial. If the generator of the resulting code has an
all-zero column, making the code trivial, we may simply delete
the offending column.
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- ... Tanner
^{3.1}
**Note on terminology.**
The term *Tanner graph* was first used
by Wiberg, Loeliger, and Kötter [66]
to refer to the more general graphs introduced
in [66]. These were later termed TWL graphs by Forney [33],
although *TWLK graphs* would have been more appropriate.
By now, the term *factor graphs* is almost universally
used in this context, which leaves *Tanner graphs*
available to refer to the kind of factor
graphs actually studied by Tanner [55].
The emphasis in this chapter (as in all of the
literature [17,40,41,50,55] on the subject)
is on a special type of Tanner graphs that come with
simple parity-check constraints. These Tanner graphs include
the graphs underlying Gallager's low-density
parity-check codes [25,26].
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