On the lifted Zetterberg code
The even-weight subcode of a binary Zetterberg code is a cyclic code with generator polynomial g ( x ) = ( x + 1 ) p ( x ) , where p ( x ) is the minimum polynomial over GF (2) of an element of order 2 m + 1 in G F ( 2 2 m ) and m is even. This even binary code has parameters [ 2 m + 1 , 2 m - 2 m ,...
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| Vydané v: | Designs, codes, and cryptography Ročník 80; číslo 3; s. 561 - 576 |
|---|---|
| Hlavní autori: | , , , , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
New York
Springer US
01.09.2016
|
| Predmet: | |
| ISSN: | 0925-1022, 1573-7586 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | The even-weight subcode of a binary Zetterberg code is a cyclic code with generator polynomial
g
(
x
)
=
(
x
+
1
)
p
(
x
)
, where
p
(
x
) is the minimum polynomial over
GF
(2) of an element of order
2
m
+
1
in
G
F
(
2
2
m
)
and
m
is even. This even binary code has parameters
[
2
m
+
1
,
2
m
-
2
m
,
6
]
. The quaternary code obtained by lifting the code to the alphabet
Z
4
=
{
0
,
1
,
2
,
3
}
is shown to have parameters
[
2
m
+
1
,
2
m
-
2
m
,
d
L
]
, where
d
L
≥
8
denotes the minimum Lee distance. The image of the Gray map of the lifted code is a binary code with parameters
(
2
m
+
1
+
2
,
2
k
,
d
H
)
, where
d
H
≥
8
denotes the minimum Hamming weight and
k
=
2
m
+
1
-
4
m
. For
m
=
6
these parameters equal the parameters of the best known binary linear code for this length and dimension. Furthermore, a simple algebraic decoding algorithm is presented for these
Z
4
-codes for all even
m
. This appears to be the first infinite family of
Z
4
-codes of length
n
=
2
m
+
1
with
d
L
≥
8
having an algebraic decoding algorithm. |
|---|---|
| ISSN: | 0925-1022 1573-7586 |
| DOI: | 10.1007/s10623-015-0118-y |