... 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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