NEW RESUL TS ON GRIESMER CODES AND ARCS Assia Rousseva Soa - - PowerPoint PPT Presentation

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NEW RESUL TS ON GRIESMER CODES AND ARCS Assia Rousseva Soa Universit y Ivan Landjev New Bulga rian Universit y ALCOMA 2015, Kloster Banz, 15.-20.03.2015 1. Linea r Co des over Finite Fields Linea r [ n,

slide-1
SLIDE 1 NEW RESUL TS ON GRIESMER CODES AND ARCS Assia Rousseva Soa Universit y Ivan Landjev New Bulga rian Universit y
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
slide-2
SLIDE 2 1. Linea r Co des
  • ver
Finite Fields

⋄

Linea r [n, k]q
  • de: C < Fn

q

, dim C = k

⋄ [n, k, d]q

  • o
de: d = min{d(u, v) | u, v ∈ C, u = v}.
  • n
  • the
length
  • f C
;
  • k
  • the
dimension
  • f C
;
  • d
  • the
minimum distan e
  • f C
.

⋄ Ai

  • numb
er
  • f
  • dew
  • rds
  • f
(Hamming) w eight i

⋄ (Ai)i≥0

  • the
sp e trum
  • f C
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 1
slide-3
SLIDE 3 The Main Problem in Co ding Theo ry . Optimize
  • ne
  • f
the pa rameters n , k , d , given the
  • ther
t w
  • .

nq(k, d)

  • minimal
length
  • f
a linea r
  • de
  • ver Fq
  • f
dimension k and minimum distan e d ;

Kq(n, d)

  • maximal
dimension
  • f
a linea r
  • de
  • ver Fq
  • f
length n and minimum distan e d ;

Dq(n, k)

  • maximal
minimum distan e
  • f
a linea r
  • de
  • ver Fq
  • f
length n and dimension k .
  • ptimalit
y with resp e t to n =

⇒

  • ptimalit
y with resp e t to k and d
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 2
slide-4
SLIDE 4

⋄

Griesmer b
  • und:
Let C b e an [n, k, d]q
  • o
de. Then

nq(k, d) ≥ gq(k, d) =

k−1

  • i=0

⌈ d qi⌉

Theo rem. Given the integer k and the p rime p
  • w
er q , Griesmer [gq(k, d), k, d]q
  • des
exist fo r all su iently la rge d . The p roblem
  • f
nding the exa t value
  • f nq(k, d)
is solved fo r
  • q = 2
: k ≤ 8 fo r all d ;
  • q = 3
: k ≤ 5 fo r all d ;
  • q = 4
: k ≤ 4 fo r all d ;
  • q = 5, 7, 8, 9: k ≤ 3
fo r all d ;
  • q = 5
: k = 4
  • four
values
  • f d
fo r whi h n5(4, d) is not kno wn.
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 3
slide-5
SLIDE 5 http://www.mi.s.
  • s
ak af u-u .a . jp / maruta/griesme r.h tm The Op en Cases fo r q = 5 , k = 4

d g5(4, d) n5(4, d) K K|H 81 103 103104 (103, 22) (22, 5)-a

r

82 104 104105 (104, 22)

in PG(2, 5)

161 203 203204 (203, 42) (42, 9)-a

r

162 204 204205 (204, 42)

in PG(2, 5)
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 4
slide-6
SLIDE 6 2. Divisible and Quasidivisibl e Ar s

⋄

A multiset in PG(k − 1, q) is a mapping

K :

  • P

→ N0, P → K(P). ⋄ K

(P )
  • multipli it
y
  • f
the p
  • int P
.

⋄ Q ⊂ P

: K(Q) =

P ∈Q K(P)

  • multipli it
y
  • f
the set Q .

⋄ K

(P )
  • the
a rdinalit y
  • f K
.

⋄

P
  • ints,
lines, ... ,hyp erplanes
  • f
multipli it y i a re alled i
  • p
  • ints, i
  • lines,
... ,

i

  • hyp
erplanes.

⋄ ai

  • the
numb er
  • f
hyp erplanes H with K(H) = i

⋄ (ai)i≥0

  • the
sp e trum
  • f K
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 5
slide-7
SLIDE 7
  • Denition. (n, w)-a
r in PG(k − 1, q): a multiset K with 1) K(P) = n ; 2) fo r every hyp erplane H : K(H) ≤ w ; 3) there exists a hyp erplane H0 : K(H0) = w .
  • Denition. (n, w)-blo
king set in PG(k − 1, q) (o r (n, w)-minihyp er): a multiset K with 1) K(P) = n ; 2) fo r every hyp erplane H : K(H) ≥ w ; 3) there exists a hyp erplane H0 : K(H0) = w .
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 6
slide-8
SLIDE 8 Denition. An (n, w)-a r K in PG(k − 1, q) is alled t
  • extendable,
if there exists an (n + t, w)-a r K′ in PG(k − 1, q) with K′(P) ≥ K(P) fo r every p
  • int

P ∈ P

. An 1-extendable a r is alled extendable. Denition. An a r K in PG(k − 1, q) with K(P) = n and sp e trum (ai) is said to b e divisible with diviso r ∆ , ∆ > 1, if ai = 0 fo r all i ≡ n (mod ∆). Denition. An a r K with K(P) = n and sp e trum (ai) is said to b e t
  • quasidivisible
with diviso r ∆ , ∆ > 1, (o r t
  • quasidivisible
mo dulo ∆ ) if ai = 0 fo r all i ≡ n, n + 1, . . . , n + t (mod ∆).
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 7
slide-9
SLIDE 9 3. Linea r
  • des
as multisets
  • f
p
  • ints

[n, k, d]q

  • o
de C

⇔ (n, w = n − d)-a

r K
  • f
full length in PG(k − 1, q)

0 = u ∈ C

, wt(u) = u

⇔

a hyp erplane H with K(H) = n − u , extendable [n, k, d]q
  • o
de C

⇔

extendable (n, n − d)-a r K divisible [n, k, d]q
  • o
de

⇔

divisible (n, n − d)-a r in PG(k − 1, q)

Ai = 0

fo r all i ≡ 0 (mod ∆)

ai = 0

fo r all i ≡ n (mod ∆)

t

  • quasidivisible [n, k, d]q
  • o
de

⇔ t

  • quasidivisible (n, n − d)-a
r

Ai = 0

fo r all i ≡ −j (mod q) in PG(k − 1, q) ai = 0 fo r all

j ∈ {0, 1, . . . , t} i ≡ n + j (mod q)

  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 8
slide-10
SLIDE 10

⋄

Griesmer a r s: a r s asso iated with
  • des
meeting the Griesmer b
  • und
Griesmer [n, k, d]q
  • des

⇔

Griesmer (n, w)-a r s in PG(k − 1, q)

n = k−1

i=0 ⌈d/qi⌉

n = k−1

i=0 ⌈(n − w)/qi⌉

⋄

If d = n − w = sqk−1 − εk−2qk−2 − . . . − ε1q − ε0 , and

wi :=

maximal multipli it y
  • f
a subspa e
  • f
  • dimension i
, i = 0, . . . , k − 1. Then

wi = svk−i − εk−2vk−i−1 − . . . − εi+1v2 − εiv1,

where vk = (qk − 1)/(q − 1).
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 9
slide-11
SLIDE 11 4. Some Extension Results Theo rem. (R. Hill, P . Lizak, 1995, geometri version) Let K b e a (n, w)-a r in

PG(k − 1, q)

with gcd(n − w, q) = 1 . Let further K(H) ≡ n
  • r w (mod q)
fo r all hyp erplanes H . Then K is extendable to a divisible (n + 1, w)-a r in

PG(k − 1, q).

In pa rti ula r, every 1-quasidivisible a r with diviso r q is extendable. Theo rem. (T. Ma ruta, 2004, geometri version) Let K b e a 2-quasidivisible

(n, w)-a

r in PG(k − 1, q), q ≥ 5,
  • dd,
with diviso r q . Then K is extendable to an (n + 1, w)-a r in PG(k − 1, q).
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 10
slide-12
SLIDE 12

⋄ K

  • t
  • quasidivisible (n, w)-a
r in Σ = PG(k − 1, q), i.e. fo r every hyp erplane

H

, w e have K(H) ≡ n, n + 1, . . . , n + t (mod q), where 0 < t < q is an integer
  • nstant.

⋄

Dene an a r

K

in the dual spa e

Σ

  • K :

H → N0, H →

  • K(H) := n + t − K(H)

(mod q).

where H is the set
  • f
all hyp erplanes
  • f Σ
.
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 11
slide-13
SLIDE 13 Theo rem. Let K b e an (n, w)-a r in Σ = PG(k−1, q) whi h is t
  • quasidivisible
mo dulo q , t < q . Let
  • K =

c

  • i=1

χ e

Hi +

K′

fo r some a r

K′

and c not ne essa rily dierent hyp erplanes

H1, . . . , Hc

then K is c -extendable. In pa rti ula r, if

K

  • ntains
a hyp erplane in its supp
  • rt
then K is extendable. Theo rem. Let

S

b e a subspa e
  • f

Σ

  • f
p
  • sitive
dimension. Then

K( S) ≡ t (mod q).

  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 12
slide-14
SLIDE 14 Theo rem. (Landjev, Rousseva, Sto rme, 2014) Let K b e a t
  • quasidivisible
Griesmer a r in PG(k − 1, q) with pa rameters (n, w), where

d = n − w = sqk−1 − εk−2qk−2 − . . . − ε1q − ε0.

Let further ε0 = t, . . . εk−2 < √q . Then K is t
  • extendable.

⋄ K

is a (tvk−1, tvk−2)-a r , where vk = qk − 1

q − 1 ⋄ K

is a sum
  • f t
hyp erplanes
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 13
slide-15
SLIDE 15
  • 5. (t mod q)
  • Ar s
Denition. An a r F is alled a (t mod q )-a r if
  • all
p
  • ints
have multipli it y ≤ t ;
  • all
subspa es S
  • f
p
  • sitive
dimension have multipli it y F(S) ≡ t (mod q). Theo rem A. The sum
  • f
a (t1 mod q)-a r s and a (t2 mod q)-a r is a

(t mod q)-a

r with t = t1 + t2 . In pa rti ula r, the sum
  • f t
hyp erplanes in

PG(k − 1, q)

is a (t mod q )-a r .
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 14
slide-16
SLIDE 16 Theo rem B. Let F0 b e a (t mod q )-a r in a hyp erplane H ∼

= PG(k − 2, q).

  • f Σ = PG(k − 1, q).
F
  • r
a xed p
  • int P ∈ Σ \ H
, dene an a r F in Σ as follo ws: F(P) = t ;
  • fo
r ea h p
  • int Q = P
: F(Q) = F0(R) where R = P, Q ∩ H . Then the a r F is a (t mod q)-a r in PG(k − 1, q)
  • f
size q|F0| + t .
  • Denition. (t mod q)-a
r s
  • btained
b y Theo rem B a re alled lifted a r s.
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 15
slide-17
SLIDE 17

P F(P ) = t R Q F(Q) = F0(R) H ∼ = PG(k − 2, q) F0

  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 16
slide-18
SLIDE 18

F

: an a r in Σ = PG(k − 1, q)

H

  • the
set
  • f
all hyp erplanes in Σ

σ

  • a
fun tion su h that σ(F(H)) is a non-negative integer fo r all H ∈ H . The a r Fσ in

Σ Fσ :

  • H

→ N0 H → σ(F(H))

is alled the σ
  • dual
  • f F
. Theo rem C. Let F b e a (t mod q)-a r in PG(2, q)
  • f
size mq + t . Then the a r Fσ with σ(x) = (x − t)/q is an ((m − t)q + m, m − t)-blo king set in the dual plane with line multipli ities m − t, m − t + 1, . . . , m .
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 17
slide-19
SLIDE 19

(3 mod 5)-a

r s in PG(2, 5)

(18, {3, 8, 13, 18})-a

r s
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 18
slide-20
SLIDE 20

F

: (23, {3, 8})-a r

Fσ

:

Fσ

: (9,1)-blo king set with line multipli ities 1, 2, 3, 4

F

: (28, {3, 8})-a r

Fσ

: (15, 2)-blo king set with line multipli ities 2, 3, 4, 5

Fσ

: the
  • mplement
  • f
the unique (16, 4)-a r without external lines
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 19
slide-21
SLIDE 21 The (23, {3, 8})-a r
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 20
slide-22
SLIDE 22 The (28, {3, 8})-a r
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 21
slide-23
SLIDE 23

F

: (33, {3, 8, 13})-a r

Fσ

: (21, 3)-blo king set with line multipli ities 3, 4, 5, 6

Fσ

is
  • ne
  • f
the follo wing: (1) the
  • mplement
  • f
the seven non-isomo rphi (10, 3)-a r s; Λ2 = 0 (2) the
  • mplement
  • f
the (11, 3)-a r with four external lines; a p
  • int
not
  • n
an external line is doubled; Λ2 = 1 (3)
  • ne
double p
  • int
whi h fo rms an
  • val
with ve
  • f
the 0-p
  • ints;
the tangent in the 2-p
  • int
is a 3-line; Λ2 = 1 (4) PG(2, 5) minus a triangle with verti es
  • f
multipli it y 2, 2, 1; Λ2 = 2
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 22
slide-24
SLIDE 24 (2) The rst (33, {3, 8, 13})-a r with
  • ne 13-line
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 23
slide-25
SLIDE 25 (3) the se ond (33, {3, 8, 13})-a r with
  • ne
13-line
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 24
slide-26
SLIDE 26 (4) (33, {3, 8, 13})-a r with t w
  • 13-lines
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 25
slide-27
SLIDE 27 Theo rem D. Every (3 mod 5)-a r F in PG(3, 5) with |F| ≤ 153 is a lifted a r (obtained b y Theo rem B). In pa rti ula r, |F| = 93, 118,
  • r 143.
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 26
slide-28
SLIDE 28 6. The Nonexisten e
  • f
Some Griesmer Co des A. The Nonexisten e
  • f [104, 4, 82]5
  • Co
des

d = 82 = 53 − 52 − 3 · 5 − 3 s = 1, ε2 = 1, ε1 = 3, ε0 = t = 3

;

ε0, ε1 ≥ √q w3 = 1, w2 = 5, w1 = 22, w0 = n = 104

  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 27
slide-29
SLIDE 29 Theo rem E. Let K b e a (104, 22)-a r in PG(3, 5). Then (i) K is a Griesmer 3-quasidivisible p roje tive a r ; (ii)

K

is a (3 mod 5)-a r in PG(3, 5); (iii) there is no 18-plane π su h that

K|π

is the sum
  • f
three
  • pies
  • f
the same line; (iv) |

K| ≤ 143.

  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 28
slide-30
SLIDE 30
  • Hen e |

K| = 93

and

K

is a sum
  • f
three planes.
  • Hen e K
is 3-extendable to a (non-existent) (107, 22)-a r .
  • There
is no [104, 4, 82]5
  • o
de and

n5(4, 82) = 105

.
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 29
slide-31
SLIDE 31 B. The Nonexisten e
  • f [q3 − 3q − 6, 4, q3 − q2 − 3q − 3]q
  • Co
des

⋄ q = 5

: (104, 22)-a r s in PG(3, 5)

⋄ q = 7

: (316, 46)-a r s in PG(3, 7) Plane multipli ities: 1, 8, 15, 22, 29, 36, 4349

⋄ q = 8

: (482, 61)-a r s in PG(3, 8) Plane multipli ities: 10, 42, 5861

⋄ q = 9

: (696, 78)-a r s in PG(3, 9) Plane multipli ities: 7578

⋄ q ≥ 11:

a re 3-extendable sin e εi < √q .
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 30
slide-32
SLIDE 32 C. The Nonexisten e
  • f [204, 4, 162]5
  • Co
des
  • ALCOMA
2015, Kloster Banz, 15.-20.03.2015
  • 31