Subspace Polynomials and Cyclic Subspace Codes Netanel Raviv Joint work with: Prof. Tuvi Etzion Prof. Eli Ben-Sasson Dr. Ariel Gabizon Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 1 June 2014
Motivation – Subspace Codes for Network Coding “ The Butterfly Example ” A,B A and B are two information • sources. A sends • B sends • The values of A,B are the solution of: Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 2 June 2014
Motivation – Subspace Codes for Network Coding Errors in Network Coding. A,B The values of A,B are the solution of: Solution: Even a single error Both Wrong … may corrupt the entire message. Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 3 June 2014
Motivation – Subspace Codes for Network Coding Received Error message vectors Transfer Sent Transfer matrix message matrix Setting Term Metric Set Metric Kschischang, known to the receiver. Coherent Network Coding Silva 09 ’ chosen by adversary. Koetter, chosen by Noncoherent Network Kshischang adversary. Coding 08 ’ Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 4
Cyclic Subspace Codes – Definitions For and - A cyclic shift Define . of V The Forbenius Automorphisms - For define A Forbenius Generator of the Galois group - shift of V Define Let . is called cyclic if for all and all , Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 5
Motivation – Cyclic Subspace Codes Several small examples for good codes turned out to be cyclic : A. Kohnert and S. Kurz: Construction M. Braun, T. Etzion, P. R. J. Ostergard, of large constant dimension codes A. Vardy, and A. Wassermann: with a prescribed minimum distance, Existence of q-Analogs of Steiner Lecture Notes Computer Science, Systems, arxiv.org/abs/1304.1462, 5393, 31-42, (2008) (2013) T. Etzion, A. Vardy: Error-correcting codes in projective space. IEEE Transactions on Information Theory, vol. 57(2), 1165-1173, (2011) Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 6
Motivation – Cyclic Subspace Codes All examples were found using (clever) computer search. Explicit Construction of Cyclic Subspace Codes? Encoding? Decoding? List Dec ’ ? Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 7
Cyclic Codes – Structure 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: Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 8
Cyclic Codes – a Trivial Construction 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 . Yes! More than one orbit? (For large enough n … ) Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 9
Background – Linearized and Subspace Polynomials 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 . Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 10
Subspace Polynomials – Observations is nonzero. For each there exists a unique . Lemma: Proof: Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 11
Subspace Polynomials – Observations Lemma: If , then Proof: For , Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 12
Subspace Polynomials – Gap Let with , . Define: Lemma: If then Proof: Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 13
Subspace Polynomials – Gap Corollary: If then . Observation: , , and have the same support. Corollary: for all , A subspace with a big gap yields a cyclic code with a large distance . Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 14
Subspace Polynomials – Size of Orbit Lemma: Let . If with , Then Proof: Assume . 0 th and s th coefficients are equal: Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 15
Subspace Polynomials – Size of Orbit Fact: Equivalence relation: Size of Classes: No. of Classes: Corollary: Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 16
Subspace Polynomials – Summary Big Gap – Nonzero with small - Large Big orbit. distance Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 17
Cyclic Codes With A Single Orbit Consider the polynomial - . Let be its splitting field. for some . Proof of Conjecture for large enough n. Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 18
Cyclic Codes With A Single Orbit 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 , . Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 19
Cyclic Codes With Multiple Orbits Use Forbenius automorphism to add orbits - Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 20
Cyclic Codes With Multiple Orbits Problem: Lemma: Let be a primitive element of . Consider the polynomial . Let be its splitting field and the corresponding subspace. Then - Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 21
Cyclic Codes With Degenerate Orbits Let . Define: . Claim: If then . Proof: By enumeration, • If then there exists , • Such that are dependent over Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 22
Cyclic Codes With Degenerate Orbits is cyclic: If are independent over , So are , for all . Subspace polynomial structure: Let Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 23
Cyclic Codes With Degenerate Orbits Alternative definition: is the image of the following embedding – Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 24
Cyclic Codes With Degenerate Orbits - Union Corollary: Let . Proof: Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 25
Cyclic Codes With Degenerate Orbits - Union Construction: Let Minimum Distance: Size: Inclusion-Exclusion formula. Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 26
Questions? Thank you! Netanel Raviv Subspace Polynomials and Cyclic Subspace Codes 27
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