##

Generalization

The lexicographic construction may be extended to produce codes
with desired characteristics using the generalized lexicographic
construction, which we abbreviate as the
-construction.
The
-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.

**Definition 1**
The generalized lexicographic construction is initialized 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}^{*}. |
(1) |

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 Table 1
is
, as can be verified by hand.

Despite its generality, there are many linear codes that cannot
be constructed using a non-trivial application of the
-construction. The
code given by the following basis vectors is one such example:

1110000

0001111

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