4.3 New Codes from Old
47
4.3 New Codes from Old
In this subsection, we describe some well-known techniques to obtain new codes from
old ones, namely, puncturing, extending, code expansion, direct sum, the (u|u + v)
construction and the direct product code construction. The reader who is interested
to investigate more details of such techniques can consult the textbooks [67, 114].
4.3.1 Puncturing Codes
We begin by the technique of puncturing codes.
Definition 4.3.1 Assume that C is an [n, k, d] q linear code over F q . Then if the
same coordinate i is deleted in each codewords of C we say that C was punctured.
The punctured code derived from C by deleting the ith coordinate will be denoted
by C
P
i .
Let C be as in Definition 4.3.1. If G is a generator matrix for C, then by deleting the
ith column of G and omitting a possible zero or duplicate row we obtain a generator
matrix G
P for C
P
i .
The following theorem gives us information about the parameters of the punctured
code.
Theorem 4.3.1 Let C be an [n, k, d] q linear code over F q . Assume that C
P
i is the
punctured code on the ith coordinate. Then the following hold:
(1) If d > 1, then C
P
i is an [n − 1, k, d i ] q code, where d i = d − 1 if C has a minimum weight codeword with a nonzero ith coordinate, and d i = d otherwise.
(2) If d = 1, then C
P
i is an [n − 1, k, 1] q code if C has no codeword of weight 1
whose nonzero entry is in the coordinate i; otherwise, if k > 1, then C
P
i is an
[n − 1, k − 1, d i ] q code with d i ≥ 1.
4.3.2 Code Extension
A linear code C over F q can be extended in several ways. However, the most common
method is by adding an extra symbol such that the sum (over F q ) of all coordinates
of the new code is equal to zero.
Definition 4.3.2 Assume that C is an [n, k, d] q linear code over F q . Then the
extended code C
e derived from C is defined as
C
e
= {(x 1 , . . . , x n , x n+1 ) ∈ F
n+1
q |(x 1 , . . . , x n ) ∈ C, x 1 + · · · + x n + x n+1 = 0}.
47
4.3 New Codes from Old
In this subsection, we describe some well-known techniques to obtain new codes from
old ones, namely, puncturing, extending, code expansion, direct sum, the (u|u + v)
construction and the direct product code construction. The reader who is interested
to investigate more details of such techniques can consult the textbooks [67, 114].
4.3.1 Puncturing Codes
We begin by the technique of puncturing codes.
Definition 4.3.1 Assume that C is an [n, k, d] q linear code over F q . Then if the
same coordinate i is deleted in each codewords of C we say that C was punctured.
The punctured code derived from C by deleting the ith coordinate will be denoted
by C
P
i .
Let C be as in Definition 4.3.1. If G is a generator matrix for C, then by deleting the
ith column of G and omitting a possible zero or duplicate row we obtain a generator
matrix G
P for C
P
i .
The following theorem gives us information about the parameters of the punctured
code.
Theorem 4.3.1 Let C be an [n, k, d] q linear code over F q . Assume that C
P
i is the
punctured code on the ith coordinate. Then the following hold:
(1) If d > 1, then C
P
i is an [n − 1, k, d i ] q code, where d i = d − 1 if C has a minimum weight codeword with a nonzero ith coordinate, and d i = d otherwise.
(2) If d = 1, then C
P
i is an [n − 1, k, 1] q code if C has no codeword of weight 1
whose nonzero entry is in the coordinate i; otherwise, if k > 1, then C
P
i is an
[n − 1, k − 1, d i ] q code with d i ≥ 1.
4.3.2 Code Extension
A linear code C over F q can be extended in several ways. However, the most common
method is by adding an extra symbol such that the sum (over F q ) of all coordinates
of the new code is equal to zero.
Definition 4.3.2 Assume that C is an [n, k, d] q linear code over F q . Then the
extended code C
e derived from C is defined as
C
e
= {(x 1 , . . . , x n , x n+1 ) ∈ F
n+1
q |(x 1 , . . . , x n ) ∈ C, x 1 + · · · + x n + x n+1 = 0}.