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research
An Algebraic-Coding Equivalence to the Maximum Distance Separable Conjecture
Authors
Safal Bora
Steven B. Damelin
Daniel Kaiser
Jeffrey Sun
Publication date
6 May 2018
Publisher
View
on
arXiv
Abstract
We formulate an Algebraic-Coding Equivalence to the Maximum Distance Separable Conjecture. Specifically, we present novel proofs of the following equivalent statements. Let
(
q
,
k
)
(q,k)
(
q
,
k
)
be a fixed pair of integers satisfying
q
q
q
is a prime power and
2
β€
k
β€
q
2\leq k \leq q
2
β€
k
β€
q
. We denote by
P
q
\mathcal{P}_q
P
q
β
the vector space of functions from a finite field
F
q
\mathbb{F}_q
F
q
β
to itself, which can be represented as the space
P
q
:
=
F
q
[
x
]
/
(
x
q
β
x
)
\mathcal{P}_q := \mathbb{F}_q[x]/(x^q-x)
P
q
β
:=
F
q
β
[
x
]
/
(
x
q
β
x
)
of polynomial functions. We denote by
O
n
β
P
q
\mathcal{O}_n \subset \mathcal{P}_q
O
n
β
β
P
q
β
the set of polynomials that are either the zero polynomial, or have at most
n
n
n
distinct roots in
F
q
\mathbb{F}_q
F
q
β
. Given two subspaces
Y
,
Z
Y,Z
Y
,
Z
of
P
q
\mathcal{P}_q
P
q
β
, we denote by
β¨
Y
,
Z
β©
\langle Y,Z \rangle
β¨
Y
,
Z
β©
their span. We prove that the following are equivalent. [A] Suppose that either: 1.
q
q
q
is odd 2.
q
q
q
is even and
k
βΜΈ
{
3
,
q
β
1
}
k \not\in \{3, q-1\}
k
ξ
β
{
3
,
q
β
1
}
. Then there do not exist distinct subspaces
Y
Y
Y
and
Z
Z
Z
of
P
q
\mathcal{P}_q
P
q
β
such that: 3.
d
i
m
(
β¨
Y
,
Z
β©
)
=
k
dim(\langle Y, Z \rangle) = k
d
im
(β¨
Y
,
Z
β©)
=
k
4.
d
i
m
(
Y
)
=
d
i
m
(
Z
)
=
k
β
1
dim(Y) = dim(Z) = k-1
d
im
(
Y
)
=
d
im
(
Z
)
=
k
β
1
. 5.
β¨
Y
,
Z
β©
β
O
k
β
1
\langle Y, Z \rangle \subset \mathcal{O}_{k-1}
β¨
Y
,
Z
β©
β
O
k
β
1
β
6.
Y
,
Z
β
O
k
β
2
Y, Z \subset \mathcal{O}_{k-2}
Y
,
Z
β
O
k
β
2
β
7.
Y
β©
Z
β
O
k
β
3
Y\cap Z \subset \mathcal{O}_{k-3}
Y
β©
Z
β
O
k
β
3
β
. [B] Suppose
q
q
q
is odd, or, if
q
q
q
is even,
k
βΜΈ
{
3
,
q
β
1
}
k \not\in \{3, q-1\}
k
ξ
β
{
3
,
q
β
1
}
. There is no integer
s
s
s
with
q
β₯
s
>
k
q \geq s > k
q
β₯
s
>
k
such that the Reed-Solomon code
R
\mathcal{R}
R
over
F
q
\mathbb{F}_q
F
q
β
of dimension
s
s
s
can have
s
β
k
+
2
s-k+2
s
β
k
+
2
columns
B
=
{
b
1
,
β¦
,
b
s
β
k
+
2
}
\mathcal{B} = \{b_1,\ldots,b_{s-k+2}\}
B
=
{
b
1
β
,
β¦
,
b
s
β
k
+
2
β
}
added to it, such that: 8. Any
s
Γ
s
s \times s
s
Γ
s
submatrix of
R
βͺ
B
\mathcal{R} \cup \mathcal{B}
R
βͺ
B
containing the first
s
β
k
s-k
s
β
k
columns of
B
\mathcal{B}
B
is independent. 9.
B
βͺ
{
[
0
,
0
,
β¦
,
0
,
1
]
}
\mathcal{B} \cup \{[0,0,\ldots,0,1]\}
B
βͺ
{[
0
,
0
,
β¦
,
0
,
1
]}
is independent. [C] The MDS conjecture is true for the given
(
q
,
k
)
(q,k)
(
q
,
k
)
.Comment: This is version: 5.6.18. arXiv admin note: substantial text overlap with arXiv:1611.0235
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Last time updated on 19/06/2017