##

Generalization

The lexicographic construction may be extended to produce codes with
desired characteristics using the *generalized lexicographic
construction*, abbreviated G-construction. The G-construction
replaces the ``lexicographically earliest'' heuristic used in building
lexicodes with an arbitrary function. This allows us to generate
arbitrary greedy codes in which various properties are grafted upon
the good code parameters of the lexicodes.
More formally,

**Definition 1**
The generalized lexicographic construction is boot-strapped with a
linear

(*n*, *k*, *d* ) seed code

and iteratively constructs the
family of codes

using a mapping from codes to vectors:

*f* (^{ . }) : _{q}^{*} *v*_{q}^{n} |
(9) |

where we define

The construction follows the scheme:

We will call *f* (^{ . }) the generating mapping of the
construction. The familiar lexicode family of minimum distance *d* is
thus the simple special case
, where we
use the trivial seed code
_{d}{0^{d}, 1^{d}}.
The code in Figure 2.2
is
, as can be verified by hand.

The G-construction serves to homogenize its seed code, since each
iteration builds a subcode with the same minimum distance. Thus,
there are many linear codes which do not fit into the mold of this
construction. For example, the linear code given by the following
generator matrix cannot be constructed with the G-construction:

1110000

0001111

For sake of simplicity we shall concern ourselves only with binary
codes in the remainder of this chapter,
though the extension to *q*-ary codes is fairly straighforward.

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