Cryptography Reference
In-Depth Information
Generalized Srivastava codes.
Definition 4.
Let
(
α
1
,...,α
n
)
,
(
ω
1
,...,ω
s
)
are
n
+
s
distinct elements of
F
q
m
,
and
(
z
1
,...,z
n
)
are nonzero elements of
F
q
m
.The
Generalized Srivastava code
is an
[
n, k
≥
n
−
mst, d
≥
st
+1]
code over
F
q
, is also an alternant code, and is
defined by the parity-check matrix
⎛
⎝
⎞
⎠
H
1
H
2
.
H
s
H
=
where
⎛
⎝
⎞
⎠
z
1
α
1
−ω
l
z
2
α
2
−ω
l
z
n
α
n
−ω
l
...
z
1
(
α
1
−ω
l
)
2
z
2
(
α
2
−ω
l
)
2
z
n
(
α
n
−ω
l
)
2
...
H
l
=
...
...
...
...
z
1
(
α
1
−ω
l
)
t
z
2
(
α
2
−ω
l
)
t
z
n
(
α
n
−ω
l
)
t
...
for
l
=1
,...,s
. The original Srivastava codes are the case
t
=1
,
z
i
=
α
i
for
some
μ
.
For more details about Generalized Srivastava codes, see [MS77][Ch. 12,
§
6].
3
Quasi-monoidic Codes
Monoidic matrices.
Definition 5.
Let
R
be a commutative ring,
A
=
{
a
0
,
···
,a
N−
1
}
a finite abelian
group of size
R
asequence
indexed by
A
.The
A
-adic matrix
M
(
h
)
associated with this sequence is one for
which
M
i,j
=
h
(
a
i
−
|
A
|
=
N
with neutral element
a
0
=0
,and
h
:
A
−→
a
j
)
holds, i.e.,
⎛
⎞
h
(0)
h
(
−
a
1
)
···
h
(
−
a
N−
1
)
⎝
⎠
h
(
a
1
)
h
(0)
···
h
(
a
1
−
a
N−
1
)
M
=
.
.
.
.
.
.
.
h
(
a
N−
1
)
h
(
a
N−
1
−
a
1
)
···
h
(0)
All
A
-adic matrices form a ring that is isomorphic to the monoid ring
R
[
A
],
which is studied in abstract algebra [Lan02]. We use the additive notation for
the finite abelian group
A
here for practical purposes, but the definition can
be generalized to all groups, in which case one might prefer the multiplicative
notation.
Some
A
-adic matrices have special names, for example the
2
-adic matrices
Z
3
-adic matrices are triadic. If we do not want to specify
the group
A
explicitly, we will say the matrix is monoidic. So, to identify all
Goppa codes with a monoidic representation, we continue by giving necessary
and sucient conditions for Cauchy matrices to be monoidic and show that the
case for Cauchy power matrices follows from that.
aredyadicandthe
Z
