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

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Introduction

Let ❈ be a self-dual [♥, ❦, ❞]- code over Fq. Type I ❈ is ✷-divisible or even and q = ✷ Type II ❈ is ✹-divisible or doubly even and q = ✷ Type III ❈ is ✸-divisible and q = ✸ Type IV ❈ is ✷-divisible and q = ✹ In 1973 C.L. Mallows and N.J.A. Sloane proved that the mini- mum distance ❞ of a self-dual [♥, ❦, ❞]-code satisfies Type I ❞ ≤ ✷ ♥

✽

• + ✷

Type II ❞ ≤ ✹ ♥

✷✹

• + ✹

Type III ❞ ≤ ✸ ♥

✶✷

• + ✸

Type IV ❞ ≤ ✷ ♥

✻

• + ✷

Codes reaching the bound are called Extremal.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Introduction

Example: The extended ternary Golay code is a [✶✷, ✻, ✻]✸.         ■✻

• ✵

✶ ✶ ✶ ✶ ✶ ✶ ✵ ✶ ✷ ✷ ✶ ✶ ✶ ✵ ✶ ✷ ✷ ✶ ✷ ✶ ✵ ✶ ✷ ✶ ✷ ✷ ✶ ✵ ✶ ✶ ✶ ✷ ✷ ✶ ✵        

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Introduction

In 1969 Vera Pless discovered a family of self-dual ternary codes P(♣) of length ✷(♣ + ✶) for odd primes ♣ with ♣ ≡ −✶ (♠♦❞ ✻). Also the extended quadratic residue codes ❳◗❘(♣) of length ♣ + ✶, whenever ♣ prime ♣ ≡ ±✶ (♠♦❞ ✶✷), define a series of self-dual ternary codes of high minimum dis- tance. In fact for small values of ♣ both families define extremal codes.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

The known extremal ternary codes of length ✶✷♥.

Length ♥ P( ♥

✷ − ✶)

❳◗❘(♥ − ✶) Extremal Partial distance Classification∗ 12 6 6

• 24

9 9 9

• 36

12

• 12

♦(σ) ≥ ✺ 48 15 15 15 ♦(σ) ≥ ✺ 60 18 18 18 ♦(σ) ≥ ✶✶ 72

• 18

21 No extremal 84 21 21 24 Unknown

∗ σ ∈ ❆✉t(❈) of prime order.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Definitions

Given ❈ a [♥, ❦]−code over Fq and σ ∈ Aut(❈) of order ♣ a prime number, then we say that σ ∈ Sym(♥) has the type ♣ − (t, ❢ ) if σ has t ♣ cycles and ❢ fixed points. By the Maschke’s Theorem any code ❈ with an automorphism σ of prime order not dividing q is decomposable as ❈ = ❋σ(❈) ⊕ ❊σ(❈), where ❋σ(❈) denotes the Fixed code or submodule of words fixed by σ and ❊σ(❈) its σ−invariant complement.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Definitions

Let ❑ be a field, ♥ ∈ N. Then the monomial group ▼♦♥♥(❑ ∗) ∼ = (❑ ∗)♥ : ❙♥ ≤ ●▲♥(❑), the group of monomial ♥ × ♥-matrices over ❑, is the semidirect product of the subgroup (❑ ∗)♥ of diagonal matrices in ●▲♥(❑) with the group of permutation matrices. The monomial automorphism group of a code ❈ ≤ ❑ ♥ is ❆✉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 ● = ❙▲✷(♣) .

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Characterization of types

Theorem Let ❈ = ❈ ⊥ ≤ F♥

q, ♣ ∤ q(q −✶) and σ ∈ Aut(❈) of type ♣ −(t, ❢ )

with σ = Ω✶ · . . . · Ωt · Ωt+✶ · . . . · Ωt+❢ , where wlog we take

Ω✐ :=    (♣(✐ − ✶) + ✶, · · · , ✐♣) , ✐ ∈ {✶, · · · , t} (♣(✐ − ❢ ) + ❢ ) , ✐ ∈ {t + ✶, . . . , t + ❢ } .

Then

❋σ(❈) := {❝ ∈ ❈ | σ(❝) = ❝ ⇔ ❝✶ = · · · = ❝♣, · · · , ❝♣(t−✶)+✶ = · · · = ❝t♣} ,

the Fixed Code has dimension

❢ +t ✷ and

❊σ(❈) :=

• ❝ ∈ ❈ |
• ✐∈Ω✶

❝✐ = · · · =

• ✐∈Ωt

❝✐ = ❝t♣+✶ = · · · = ❝t♣+❢ = ✵

• ,

the σ − invariant complement of ❋σ(❈) in ❈ has dimension

t(♣−✶) ✷

.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Characterization of types

Remark. If ❢ < d(❈), then t ≥ ❢ . There is a bound that is well known in coding theory, and it is the bound found by J. H. Griesmer in 1960. This bound states that: ♥ ≥

❦−✶

• ✐=✵

❞ q✐

• .

Using this bound we get a new inequality for the case in jump dimension where q | ♥. So we get this lemma. Lemma. Let ❈ be a [♥, ♥

✷, ❞]q−code. If ❈ is a type III code then

❞ ≤ ✷ ✸ ♥

• ✶ − ✸

✷−♥ ✷

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Case [60,30,18]

Theorem Let ❈ be an extremal type III code of length 60 with an au- tomorphism σ of order 29, then σ must be of type ✷✾ − (✷, ✷). Hence ❞✐♠(❋σ(❈)) = ✷ and ❞✐♠(❊σ(❈)) = ✷✽. In this scenario ❋σ(❈) ∼ = ✶✷✾ ✵✷✾ ✶ ✵ ✵✷✾ ✶✷✾ ✵ ✶

• And

❊σ(❈) = ❊σ(❈)⊥ ≤ (F✸✷✽)✷.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Case [60,30,18]

Theorem (Nebe, Villar) Let ❈ = ❈ ⊥ ≤ F✻✵

✸ , σ ∈ ❆✉t(❈) of order 29. Then

❈ ∼ = P(✷✾), ❈ ∼ = ❳◗❘(✺✾) or ❈ ∼ = V(✷✾), where | Aut(V(✷✾)) |= ✷✸ · ✸ · ✺ · ✼ · ✷✾ and contains ❙▲✷(✷✾). The later even lead us in 2013 to a generalization of the Pless symmetry code over Fq and to find a new family of codes invariant under a monomial representation of ❙▲✷(♣) of degree ✷(♣ + ✶), ♣ a prime so that ♣ ≡ ✺♠♦❞✽.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

The new series of Codes

Minimum distance of ternary V(♣) computed with Magma: ♣ ✺ ✶✸ ✷✾ ✸✼ ✺✸ ✷(♣ + ✶) ✶✷ ✷✽ ✻✵ ✼✻ ✶✵✽ ❞(V(♣)) ✻ ✾ ✶✽ ✶✽ ✷✹ ❆✉t(V(♣)) ✷.▼✶✷ ❙▲✷(✶✸) ❙▲✷(✷✾) ≥ ❙▲✷(✸✼) ≥ ❙▲✷(✺✸) For q = ✺, ✼, and ✶✶ and small lengths we computed ❞(Vq(♣)) with Magma: (♣, q) (✶✸, ✺) (✷✾, ✺) (✺, ✼) (✶✸, ✼) (✺, ✶✶) (✶✸, ✶✶) ✷(♣ + ✶) ✷✽ ✻✵ ✶✷ ✷✽ ✶✷ ✷✽ ❞(V(♣)) ✶✵ ✶✻ ✻ ✾ ✼ ✶✶

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Case [✺✷, ✷✻, ✶✺]

Theorem Let ❈ ≤ F✺✷

✸

be an extremal type III code and ♣ a prime such that ♣ divides the order of ❆✉t(❈). Then ♣ ≤ ✶✸. Moreover if σ ∈ ❆✉t(❈) is of order 13, then it is of type ✶✸−(✹, ✵). Therefore ❞✐♠(❋σ(❈)) = ✷ and ❞✐♠(❊σ(❈)) = ✷✹. We know that ❈ is a 3-divisible code, then we may assume, up to equivalence, that ❋σ(❈) is generated by

• ✵ :=

✶✶✸ ✵✶✸ −✶✶✸ ✶✶✸ ✵✶✸ ✶✶✸ ✶✶✸ ✶✶✸

• ,

as

❋σ(❈) ∼ =

❈ ′

• (✶, ✵, −✶, ✶), (✵, ✶, ✶, ✶) ⊗(✶, ✶, ✶, ✶, ✶, ✶, ✶, ✶, ✶, ✶, ✶, ✶, ✶),

and there is a unique (✹, ✷, ✸)✸-code ❈ ′, up to equivalence.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Case [✺✷, ✷✻, ✶✺]

In F✸[①] we have that

(①✶✸ − ✶) = (① − ✶)(①✸ − ① − ✶)(①✸ + ①✷ − ✶)(①✸ + ①✷ + ① − ✶)(①✸ − ①✷ − ① − ✶) = (① − ✶) · ♣✶ · ♣✷ · ♣✸ · ♣✹.

Then, F✸σ ∼ = F✸ ⊕ F✸✸ ⊕ F✸✸ ⊕ F✸✸ ⊕ F✸✸. Let ❡✵, ❡✶, ❡✷, ❡✸, ❡✹ ∈ F✸σ denote the primitive idempotent elements, thus we get ❈ = ❈❡✵ ⊕ ❈❡✶ ⊕ ❈❡✷ ⊕ ❈❡✸ ⊕ ❈❡✹. Here ❋σ(❈) = ❈❡✵ = ❈❡⊥

✵ of dimension 2 over F✸,

❈❡✶ = ❈❡⊥

✷ ≤ (F✸✸ ⊕ F✸✸ ⊕ F✸✸ ⊕ F✸✸)✹

and ❈❡✸ = ❈❡⊥

✹ ≤ (F✸✸ ⊕ F✸✸ ⊕ F✸✸ ⊕ F✸✸)✹.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

One gets that ❈❡✐, ✐ = ✶, ✷, ✸, ✹, are 2 dimensional codes over F✸✸. Thus we choose generator matrices

• ✶ =

✶ ✵ ❛ ❜ ✵ ✶ ❝ ❞

• , ●✷ =

−❛ −❝ ✶ ✵ −❜ −❞ ✵ ✶

• ,
• ✸ =

✶ ✵ ❡ ❢ ✵ ✶ ❣ ❤

• , ●✹ =

−❡ −❣ ✶ ✵ −❢ −❤ ✵ ✶

• ,

Put then s✐ := (①✶✸ − ✶)/♣✐, ✐ = ✶, ✷, ✸, ✹ and let ❈✐ be the ternary cyclic code generated by s✐. We compute the action of σ and represent this as a left multiplication with ③✶✶ ∈ F✸×✸

✸

• n the basis of ❈✶, ❈✸

respectively.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

By taking orbit representatives of the action of −③✶✶ on F∗

✸✸

and considering at the same time the action of ❆✉t(❋σ(❈)) on (F✸✸ ⊕ F✸✸ ⊕ F✸✸ ⊕ F✸✸)✹ we obtained two non-equivalent

• codes. One equivalent to the found by Gaborit in 2002 with

|Aut(❈)| = ✷✺ · ✶✸ and a new one with |Aut(❈)| = ✷✷ · ✸ · ✶✸, both can be written as the product of cyclic groups. Something interesting about this codes is that they are related to lattices of norm 5 also thanks to a work done by Gaborit.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Conclusions

Extremal type III codes of length 60 with an automorphism

• f order 29: P(✷✾), ❳◗❘(✺✾) and V(✷✾)

Series V(♣), ♣ ≡ ✺♠♦❞✽ of good type III codes. Two extremal type III codes of length 52 with an automor- phism of order 13 and associated to a unimodular extremal lattice with norm 5 and dimension 52.

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Some related research topics are:

1

Properties of the weight distribution of codes invariant un- der a big automorphism group.

2

Is the new [52,26,15] extremal type III code part of an in- finite series of good codes?

ALgebraic COMbinatorics and Applications

• ALCOMA

2015-

• D. Villar

Introduction New extremal type III codes

Definitions The [60,30,18]✸ code The [52,26,15]✸ code

Conclusions

Thanks for your attention