1. No category

Technion Coding Seminar
Technion, May 2015
Some Gabidulin Codes cannot be List
Decoded Efficiently at any Radius
Netanel Raviv
Joint work with:
Dr. Antonia Wachter-Zeh
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
1
Background – Gabidulin Codes




Codewords from
Rank metric
Gabidulin Codes – rank metric equivalent of Reed-Solomon codes.
Recall –


Linearized polynomials Gabidulin codes [Delsarte], [Gabidulin], [Roth].
May 2015
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
2
Background – MRD Codes and List Decoding
Gabidulin codes [Delsarte], [Gabidulin], [Roth].




Any rank-metric code over
of length , distance ,
and size satisfies
If
the code is called a Maximum Rank Distance (MRD) code.
Gabidulin codes are Linear MRD codes.
Combinatorial Bound on L.D
List decoding –

Given a word

For
May 2015
output all words in
the problem reduces to unique decoding.
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
3
Background – Subspace polynomials

A monic linearized polynomial
is called a
Subspace Polynomial w.r.t
,
if any of the following equivalent conditions hold:

All roots of are in
and have multiplicity 1.
 There exists a
dimensional subspace
such that


There exists a 1:1 correspondence
May 2015
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
4
Background – Cyclic Shifts of Subspaces




For a subspace
and
the set
is a cyclic shift of
A cyclic shift is a subspace of the same dimension.
Useful for constructing subspace codes.
A connection between the subspace polynomials of and
Same support
May 2015
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
5
History
Bounds for Reed-Solomon
codes using subspace
polynomials.
Subcodes of Gabidulin
codes can be list decoded
efficiently.
& C. Wang
Y. Ding
Random rank-metric
codes can be list decoded
efficiently.
Construction of Subspace
Codes using subspace
polynomials.
Bounds for Gabidulin
Codes using subspace
polynomials.
Can Gabidulin codes be
list decoded efficiently?
There are non-linear rank
metric codes that cannot
be list decoded efficiently.
May 2015
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
6
Bound on L.D. from Sets of Subspace Polynomials
Theorem [Ben-Sasson, Kopparty, Radhakrishnan]
For Reed-Solomon codes –
A “large” set of subspace polynomials with mutual top
coefficients provides a bound on the maximal radius which
allows efficient list-decoding.
Our Contribution:
Theorem [Wachter-Zeh]
For Gabidulin codes –
A “large” set of subspace polynomials with mutual top
coefficients provides a bound on the maximal radius which
allows efficient list-decoding.
May 2015
Both theorems use a
counting argument to
show that such sets
indeed exist.
1.
2.
Better existential bound using
a counting argument on a
restricted set.
Better explicit bound using
q-associates.
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
7
L.D. bounds from Mutual Top Coefficients
Theorem [Wachter-Zeh]
Consider
a word
over
such that

then for
there exists




Proof Sketch:
There are
subspace of dimension
of
By a counting argument, there exists a set
subspace polynomials of size at least
Let be the linearized p’ which
has these coefficients, and let
be the resulting word.
For any

Hence, the evaluation of
is a codeword in

The radius
of
results in exp’ list if
whose polynomials “agree” on the top
coefficients.
May 2015
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
8
Existential bound - Special Sets of Subspace Polynomials


Goal – Improve the bound on the decoding radius.
Tool – Sets of subspace polynomials with mutual coefficients.
Theorem [Ben-Sasson, Etzion, Gabizon, Raviv]
• Let
and let
• Subspaces of dimension of
which are also subspace over
the following subspace polynomial structure –
• Nonzero coefficients occur in “jumps” of
• May alternatively be described as the image of
• Corollary –
• There are exactly
of those.
May 2015
, have
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
9
Existential Bound


Apply the Theorem of Wachter-Zeh on
Restrict the counting argument to the set
under certain division constraints.
Theorem [Raviv, Wachter-Zeh]
Consider
over
radius such that
Then there exists a word
such that
, and let
be a
Drawbacks:
such that
(due to technical limitations of the proof)


Corollary
For any
the code
cannot be list-decoded
efficiently at all.
May 2015
Rate at least
The field
must
satisfy
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
10
Explicit Bound

Goals –
Present a “problematic set” explicitly.
 Find another applicable set of parameters.


Tool – q-associates.
Definition
The following polynomial are called
q-associates of each other:
May 2015
Theorem [Lidl, Niederreiter]
If
are polynomials with q-associates
then
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
11
Explicit Bound


Lemma – If
,
, and
, then
is a subspace polynomial of a subspace of dim
Proof –

From lemma: The -associates divide each other.
 Complies with one of the definitions of a subspace polynomial.

May 2015
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
12
Explicit Bound
is a subspace polynomial.

Corollary –




Let be the corresponding subspace.
The set
has subspace polynomials
For some easily chosen set (representatives of cyclic shifts of
An explicit set of subspace polynomials.
).
Theorem [Raviv, Wachter-Zeh]
The code
efficiently at all.
May 2015
cannot be list-decoded
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
13
Implications to Subspace Codes
Theorem – Lifting [Silva, Koetter, Kschischang]
A rank-metric code of




A subspace code of
All results apply for lifted Gabidulin codes as well.
By lifting the non-linear codes of Wachter-Zeh, a similar result is achieved.
No known notion of linearity for subspace codes.
However,


May 2015
L.D. of Lifted Gabidulin codes (a.k.a Koetter-Kschischang codes) was
extensively studied.
[Guruswami, Wang], [Vardy, Mahdavifar].
There are non-linear rank
metric codes that cannot
be List decoded efficiently.
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
14
Future Research

Achieve similar bounds for:

Rates less than



Other sets of parameters.
List-decode Gabidulin codes that avoid these parameter sets.
May 2015
Netanel Raviv Some Gabidulin Codes cannot be List Decoded Efficiently in any Radius
15