Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv Joint - - PowerPoint PPT Presentation

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Subspace Polynomials and Cyclic Subspace Codes

Netanel Raviv

June 2014 1

Joint work with:

• Prof. Tuvi Etzion
• Prof. Eli Ben-Sasson
• Dr. Ariel Gabizon

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Motivation – Subspace Codes for Network Coding

June 2014 2

“The Butterfly Example”

• A and B are two information

sources.

• A sends
• B sends

A,B

The values of A,B are the solution of:

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Errors in Network Coding.

Motivation – Subspace Codes for Network Coding

June 2014 3

The values of A,B are the solution of:

Solution:

Both Wrong…

A,B

Even a single error may corrupt the entire message.

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Motivation – Subspace Codes for Network Coding

4

Received message Sent message Transfer matrix Transfer matrix Error vectors Metric Metric Set Term Setting Coherent Network Coding known to the receiver. chosen by adversary. Kschischang, Silva 09’ Noncoherent Network Coding chosen by adversary. Koetter, Kshischang 08’

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

For and -

Define .

The Forbenius Automorphisms -

For define Generator of the Galois group - Define

Let .

is called cyclic if for all and all ,

Cyclic Subspace Codes – Definitions

5

A cyclic shift

• f V

A Forbenius shift of V

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Motivation – Cyclic Subspace Codes

6

• T. Etzion, A. Vardy: Error-correcting

codes in projective space. IEEE Transactions on Information Theory,

• vol. 57(2), 1165-1173, (2011)
• A. Kohnert and S. Kurz: Construction
• f large constant dimension codes

with a prescribed minimum distance, Lecture Notes Computer Science, 5393, 31-42, (2008)

• M. Braun, T. Etzion, P. R. J. Ostergard,
• A. Vardy, and A. Wassermann:

Existence of q-Analogs of Steiner Systems, arxiv.org/abs/1304.1462, (2013)

Several small examples for good codes turned out to be cyclic:

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

All examples were found using (clever) computer search.

Motivation – Cyclic Subspace Codes

7

Explicit Construction of Cyclic Subspace Codes?

Encoding? Decoding? List Dec’?

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Let be a primitive element and . A Cyclic code is a union of orbits of the form: Note:

. .

Corollary 1: is a subfield. Corollary 2:

Cyclic Codes – Structure

8

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Let be a subfield of . Define:

Size: (one orbit) Minimum distance:

Conjecture [Rosenthal et al. 13]:

For every there exists a cyclic code of size and minimum distance . More than one orbit?

Cyclic Codes – a Trivial Construction

9

Yes!

(For large enough n…)

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

A Linearized Polynomial:

, where . The roots form a subspace.

A monic linearized polynomial is a Subspace Polynomial (w.r.t ) if:

splits completely in with all roots of multiplicity 1. for some .

Background – Linearized and Subspace Polynomials

10

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

is nonzero. For each there exists a unique . Lemma: Proof:

Subspace Polynomials – Observations

11

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Lemma: If , then Proof: For ,

Subspace Polynomials – Observations

12

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Let with , .

Define:

Lemma: If then Proof:

Subspace Polynomials – Gap

13

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Corollary: If then . Observation:

, , and have the same support.

Corollary: for all ,

Subspace Polynomials – Gap

14

A subspace with a big gap yields a cyclic code with a large distance.

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Subspace Polynomials – Size of Orbit

15

Lemma: Let .

If with , Then

Proof: Assume .

0th and sth coefficients are equal:

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Fact: Corollary:

Subspace Polynomials – Size of Orbit

16

Equivalence relation: Size of Classes:

• No. of Classes:

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Subspace Polynomials – Summary

17

Big Gap – Large distance

Nonzero with small - Big orbit.

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Consider the polynomial - . Let be its splitting field. for some .

Cyclic Codes With A Single Orbit

18

Proof of Conjecture for large enough n.

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

One special case:

Fact: is the product of all monic irreducible polynomials over with degree dividing . Lemma: If is irreducible over then is a subspace polynomial w.r.t , Proof: Problem: When is irreducible? Empirically: For , .

Cyclic Codes With A Single Orbit

19

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Use Forbenius automorphism to add orbits -

Cyclic Codes With Multiple Orbits

20

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Problem: Lemma:

Let be a primitive element of . Consider the polynomial . Let be its splitting field and the corresponding subspace. Then -

Cyclic Codes With Multiple Orbits

21

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Let . Define: .

Claim: If then . Proof: By enumeration,

• If then there exists ,
• Such that

Cyclic Codes With Degenerate Orbits

22

are dependent over

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

is cyclic:

If are independent over , So are , for all .

Subspace polynomial structure:

Let

Cyclic Codes With Degenerate Orbits

23

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Alternative definition: is the image of the following embedding –

Cyclic Codes With Degenerate Orbits

24

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Corollary: Let . Proof:

Cyclic Codes With Degenerate Orbits - Union

25

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Construction: Let Minimum Distance: Size: Inclusion-Exclusion formula.

Cyclic Codes With Degenerate Orbits - Union

26

Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv

Questions?

27

Thank you!