On extremal type III codes Darwin Villar RWTH-Aachen ALCOMA 15 - PowerPoint PPT Presentation

On extremal type III codes Darwin Villar RWTH-Aachen ALCOMA 15

Introduction ALgebraic Let ❈ be a self-dual [ ♥ , ❦ , ❞ ]- code over F q . COMbinatorics and Applications -ALCOMA Type I ❈ is ✷ -divisible or even and q = ✷ 2015- Type II ❈ is ✹ -divisible or doubly even and q = ✷ D. Villar Type III ❈ is ✸ -divisible and q = ✸ Introduction Type IV ❈ is ✷ -divisible and q = ✹ New extremal type III codes In 1973 C.L. Mallows and N.J.A. Sloane proved that the mini- Definitions The [60,30,18] ✸ code mum distance ❞ of a self-dual [ ♥ , ❦ , ❞ ]-code satisfies The [52,26,15] ✸ code Conclusions � ♥ � Type I ❞ ≤ ✷ + ✷ � ♥ � ✽ Type II ❞ ≤ ✹ + ✹ � ♥ � ✷✹ Type III ❞ ≤ ✸ + ✸ � ♥ � ✶✷ Type IV ❞ ≤ ✷ + ✷ ✻ Codes reaching the bound are called Extremal .

Introduction ALgebraic COMbinatorics and Applications -ALCOMA 2015- Example: D. Villar The extended ternary Golay code is a [ ✶✷ , ✻ , ✻ ] ✸ . Introduction �   � ✵ ✶ ✶ ✶ ✶ ✶ New extremal � type III codes  �  ✶ ✵ ✶ ✷ ✷ ✶  �  Definitions  �  The [60,30,18] ✸ code ✶ ✶ ✵ ✶ ✷ ✷  �  The [52,26,15] ✸ code ■ ✻  �  ✶ ✷ ✶ ✵ ✶ ✷ Conclusions  �  �   ✶ ✷ ✷ ✶ ✵ ✶ � � ✶ ✶ ✷ ✷ ✶ ✵

Introduction ALgebraic COMbinatorics and In 1969 Vera Pless discovered a family of self-dual ternary Applications -ALCOMA codes P ( ♣ ) of length ✷ ( ♣ + ✶ ) for odd primes ♣ with 2015- D. Villar ♣ ≡ − ✶ ( ♠♦❞ ✻ ) . Introduction New extremal type III codes Also the extended quadratic residue codes ❳◗❘ ( ♣ ) of length Definitions The [60,30,18] ✸ code ♣ + ✶ , whenever ♣ prime The [52,26,15] ✸ code Conclusions ♣ ≡ ± ✶ ( ♠♦❞ ✶✷ ) , define a series of self-dual ternary codes of high minimum dis- tance. In fact for small values of ♣ both families define extremal codes.

The known extremal ternary codes of length ✶✷ ♥ . ALgebraic COMbinatorics and Applications Extremal Partial -ALCOMA P ( ♥ Length ♥ ✷ − ✶ ) ❳◗❘ ( ♥ − ✶ ) Classification ∗ 2015- distance D. Villar 12 6 6 � 24 9 9 9 � Introduction 36 12 - 12 ♦ ( σ ) ≥ ✺ New extremal type III codes 48 15 15 15 ♦ ( σ ) ≥ ✺ Definitions 60 18 18 18 ♦ ( σ ) ≥ ✶✶ The [60,30,18] ✸ code The [52,26,15] ✸ code 72 - 18 21 No extremal Conclusions 84 21 21 24 Unknown ∗ σ ∈ ❆✉t ( ❈ ) of prime order.

Definitions ALgebraic COMbinatorics and Applications -ALCOMA Given ❈ a [ ♥ , ❦ ] − code over F q and σ ∈ Aut( ❈ ) of order ♣ a 2015- prime number, then we say that σ ∈ Sym( ♥ ) has the type D. Villar ♣ − ( t , ❢ ) if σ has t ♣ cycles and ❢ fixed points. Introduction New extremal By the Maschke’s Theorem any code ❈ with an automorphism type III codes Definitions σ of prime order not dividing q is decomposable as The [60,30,18] ✸ code The [52,26,15] ✸ code Conclusions ❈ = ❋ σ ( ❈ ) ⊕ ❊ σ ( ❈ ) , where ❋ σ ( ❈ ) denotes the Fixed code or submodule of words fixed by σ and ❊ σ ( ❈ ) its σ− invariant complement.

Definitions ALgebraic COMbinatorics Let ❑ be a field, ♥ ∈ N . Then the monomial group and Applications -ALCOMA ▼♦♥ ♥ ( ❑ ∗ ) ∼ = ( ❑ ∗ ) ♥ : ❙ ♥ ≤ ●▲ ♥ ( ❑ ) , 2015- D. Villar the group of monomial ♥ × ♥ -matrices over ❑ , is the semidirect Introduction product of the subgroup ( ❑ ∗ ) ♥ of diagonal matrices in ●▲ ♥ ( ❑ ) New extremal type III codes with the group of permutation matrices. Definitions The monomial automorphism group of a code ❈ ≤ ❑ ♥ is The [60,30,18] ✸ code The [52,26,15] ✸ code Conclusions ❆✉t ( ❈ ) := { ❣ ∈ ▼♦♥ ♥ ( ❑ ∗ ) | ❈❣ = ❈ }. The idea to construct good self-dual codes is to investigate codes that are invariant under a given subgroup ● of ▼♦♥ ♥ ( ❑ ∗ ) . A very fruitful source are monomial representations, for some prime ♣ , of ● = ❙▲ ✷ ( ♣ ) .

Characterization of types ALgebraic COMbinatorics Theorem and Applications Let ❈ = ❈ ⊥ ≤ F ♥ q , ♣ ∤ q ( q − ✶ ) and σ ∈ Aut( ❈ ) of type ♣ − ( t , ❢ ) -ALCOMA 2015- with σ = Ω ✶ · . . . · Ω t · Ω t + ✶ · . . . · Ω t + ❢ , where wlog we take  D. Villar  ( ♣ ( ✐ − ✶ ) + ✶ , · · · , ✐♣ ) , ✐ ∈ { ✶ , · · · , t } Introduction Ω ✐ := . ( ♣ ( ✐ − ❢ ) + ❢ ) , ✐ ∈ { t + ✶ , . . . , t + ❢ } New extremal  type III codes Definitions Then The [60,30,18] ✸ code The [52,26,15] ✸ code ❋ σ ( ❈ ) := { ❝ ∈ ❈ | σ ( ❝ ) = ❝ ⇔ ❝ ✶ = · · · = ❝ ♣ , · · · , ❝ ♣ ( t − ✶ )+ ✶ = · · · = ❝ t♣ } , Conclusions the Fixed Code has dimension ❢ + t ✷ and � � � � ❊ σ ( ❈ ) := ❝ ∈ ❈ | ❝ ✐ = · · · = ❝ ✐ = ❝ t♣ + ✶ = · · · = ❝ t♣ + ❢ = ✵ , ✐ ∈ Ω ✶ ✐ ∈ Ω t the σ − invariant complement of ❋ σ ( ❈ ) in ❈ has dimension t ( ♣ − ✶ ) . ✷

Characterization of types ALgebraic Remark. COMbinatorics and If ❢ < d( ❈ ) , then t ≥ ❢ . Applications -ALCOMA 2015- There is a bound that is well known in coding theory, and it is D. Villar the bound found by J. H. Griesmer in 1960. This bound states Introduction � ❞ � that: ❦ − ✶ � New extremal ♥ ≥ . type III codes q ✐ Definitions ✐ = ✵ The [60,30,18] ✸ code The [52,26,15] ✸ code Using this bound we get a new inequality for the case in jump Conclusions dimension where q | ♥ . So we get this lemma. Lemma. Let ❈ be a [ ♥ , ♥ ✷ , ❞ ] q −code. If ❈ is a type III code then ❞ ≤ ✷ ♥ � � ✸ ✷ − ♥ ✶ − ✸ ✷

Case [60,30,18] ALgebraic COMbinatorics and Theorem Applications -ALCOMA Let ❈ be an extremal type III code of length 60 with an au- 2015- tomorphism σ of order 29, then σ must be of type ✷✾ − ( ✷ , ✷ ) . D. Villar Hence Introduction New extremal type III codes ❞✐♠ ( ❋ σ ( ❈ )) = ✷ and ❞✐♠ ( ❊ σ ( ❈ )) = ✷✽ . Definitions The [60,30,18] ✸ code The [52,26,15] ✸ code In this scenario Conclusions � ✶ ✷✾ � ✵ ✷✾ ✶ ✵ ❋ σ ( ❈ ) ∼ = ✵ ✷✾ ✶ ✷✾ ✵ ✶ And ❊ σ ( ❈ ) = ❊ σ ( ❈ ) ⊥ ≤ ( F ✸ ✷✽ ) ✷ .

Case [60,30,18] ALgebraic Theorem (Nebe, Villar) COMbinatorics and Let ❈ = ❈ ⊥ ≤ F ✻✵ Applications ✸ , σ ∈ ❆✉t ( ❈ ) of order 29. Then -ALCOMA 2015- ❈ ∼ = P ( ✷✾ ) , ❈ ∼ = ❳◗❘ ( ✺✾ ) or ❈ ∼ D. Villar = V ( ✷✾ ) , Introduction where New extremal | Aut( V ( ✷✾ )) | = ✷ ✸ · ✸ · ✺ · ✼ · ✷✾ type III codes Definitions The [60,30,18] ✸ code and contains ❙▲ ✷ ( ✷✾ ) . The [52,26,15] ✸ code Conclusions The later even lead us in 2013 to a generalization of the Pless symmetry code over F q and to find a new family of codes invariant under a monomial representation of ❙▲ ✷ ( ♣ ) of degree ✷ ( ♣ + ✶ ), ♣ a prime so that ♣ ≡ ✺ ♠♦❞ ✽ .

The new series of Codes ALgebraic COMbinatorics and Minimum distance of ternary V ( ♣ ) computed with Magma : Applications -ALCOMA 2015- ♣ ✺ ✶✸ ✷✾ ✸✼ ✺✸ D. Villar ✷ ( ♣ + ✶ ) ✶✷ ✷✽ ✻✵ ✼✻ ✶✵✽ Introduction ❞ ( V ( ♣ )) ✻ ✾ ✶✽ ✶✽ ✷✹ New extremal ❆✉t ( V ( ♣ )) ✷ . ▼ ✶✷ ❙▲ ✷ ( ✶✸ ) ❙▲ ✷ ( ✷✾ ) ≥ ❙▲ ✷ ( ✸✼ ) ≥ ❙▲ ✷ ( ✺✸ ) type III codes Definitions The [60,30,18] ✸ code For q = ✺ , ✼ , and ✶✶ and small lengths we computed ❞ ( V q ( ♣ )) The [52,26,15] ✸ code Conclusions with Magma : ( ♣ , q ) ( ✶✸ , ✺ ) ( ✷✾ , ✺ ) ( ✺ , ✼ ) ( ✶✸ , ✼ ) ( ✺ , ✶✶ ) ( ✶✸ , ✶✶ ) ✷ ( ♣ + ✶ ) ✷✽ ✻✵ ✶✷ ✷✽ ✶✷ ✷✽ ❞ ( V ( ♣ )) ✶✵ ✶✻ ✻ ✾ ✼ ✶✶

Case [ ✺✷ , ✷✻ , ✶✺ ] ALgebraic Theorem COMbinatorics and Applications Let ❈ ≤ F ✺✷ be an extremal type III code and ♣ a prime such -ALCOMA ✸ 2015- that ♣ divides the order of ❆✉t ( ❈ ) . Then ♣ ≤ ✶✸ . Moreover if D. Villar σ ∈ ❆✉t ( ❈ ) is of order 13, then it is of type ✶✸ − ( ✹ , ✵ ) . Therefore Introduction ❞✐♠ ( ❋ σ ( ❈ )) = ✷ and ❞✐♠ ( ❊ σ ( ❈ )) = ✷✹ . New extremal type III codes Definitions We know that ❈ is a 3-divisible code, then we may assume, The [60,30,18] ✸ code The [52,26,15] ✸ code up to equivalence, that ❋ σ ( ❈ ) is generated by Conclusions � ✶ ✶✸ � ✵ ✶✸ − ✶ ✶✸ ✶ ✶✸ ● ✵ := , ✵ ✶✸ ✶ ✶✸ ✶ ✶✸ ✶ ✶✸ as ❈ ′ � �� � ❋ σ ( ❈ ) ∼ = � ( ✶ , ✵ , − ✶ , ✶ ) , ( ✵ , ✶ , ✶ , ✶ ) � ⊗� ( ✶ , ✶ , ✶ , ✶ , ✶ , ✶ , ✶ , ✶ , ✶ , ✶ , ✶ , ✶ , ✶ ) � , and there is a unique ( ✹ , ✷ , ✸ ) ✸ -code ❈ ′ , up to equivalence.