Improvement of the sunflower bound for 1-intersecting constant - - PowerPoint PPT Presentation

▶
improvement of the sunflower bound for 1 intersecting
SMART_READER_LITE
LIVE PREVIEW

Improvement of the sunflower bound for 1-intersecting constant - - PowerPoint PPT Presentation

Sep 22, 2022 164 likes •390 views

Improvement of the sunflower bound for 1-intersecting constant dimension subspace codes Leo Storme Ghent University Dept. of Mathematics Krijgslaan 281 9000 Ghent Belgium (joint work with D. Bartoli, A. Riet and P . Vandendriessche)

slide-1
SLIDE 1

Improvement of the sunflower bound for 1-intersecting constant dimension subspace codes

Leo Storme

Ghent University

  • Dept. of Mathematics

Krijgslaan 281 9000 Ghent Belgium (joint work with D. Bartoli, A. Riet and P . Vandendriessche)

Istanbul, November 6, 2015

Leo Storme Constant dimension subspace codes

slide-2
SLIDE 2

t-INTERSECTING CONSTANT DIMENSION SUBSPACE

CODES

t-Intersecting constant dimension subspace code: Codewords are k-dimensional vector spaces. Distinct codewords intersect in t-dimensional vector spaces. Classical example: Sunflower: all codewords pass through same t-dimensional vector space.

Leo Storme Constant dimension subspace codes

slide-3
SLIDE 3

SUNFLOWER

Leo Storme Constant dimension subspace codes

slide-4
SLIDE 4

LARGE t-INTERSECTING CONSTANT DIMENSION

SUBSPACE CODES

THEOREM Large t-intersecting constant dimension subspace codes are sunflowers. If |C| > qk − qt q − 1 2 + qk − qt q − 1

  • + 1,

then C is sunflower.

Leo Storme Constant dimension subspace codes

slide-5
SLIDE 5

CONJECTURE

Conjecture: Let C be t-intersecting constant dimension subspace code. If |C| > qk + qk−1 + · · · + q + 1, then C is sunflower.

Leo Storme Constant dimension subspace codes

slide-6
SLIDE 6

COUNTEREXAMPLES TO CONJECTURE

Code C of 1-intersecting 3-dimensional spaces in V(6, 2). Conjecture: If |C| > 15, then C is sunflower. Counterexample 1: (Etzion and Raviv) Code C of size 16 which is not sunflower. Counterexample 2: (Bartoli and Pavese) Code C of 1-intersecting 3-dimensional spaces in V(6, 2) has size at most 20, and unique example of size 20.

Leo Storme Constant dimension subspace codes

slide-7
SLIDE 7

IMPROVEMENT TO UPPER BOUND FOR t = 1

If |C| > qk − qt q − 1 2 + qk − qt q − 1

  • + 1,

then C is sunflower. For t = 1, if |C| > qk − q q − 1 2 + qk − q q − 1

  • + 1,

then C is sunflower. Question: Can this bound be improved?

Leo Storme Constant dimension subspace codes

slide-8
SLIDE 8

IMPROVEMENT TO UPPER BOUND FOR t = 1

Assumptions: C = 1-intersecting constant dimension code of k-spaces. C not sunflower. |C| = qk − q q − 1 2 + qk − q q − 1

  • + 1 − δ,

with δ = qk−2.

Leo Storme Constant dimension subspace codes

slide-9
SLIDE 9

IMPROVEMENT TO UPPER BOUND FOR t = 1

See codeword c ∈ C as PG(k − 1, q). Define S = ∪c∈Cc. LEMMA Point P ∈ S belongs to at most qk−1

q−1 codewords.

Leo Storme Constant dimension subspace codes

slide-10
SLIDE 10

IMPROVEMENT TO UPPER BOUND FOR t = 1

Leo Storme Constant dimension subspace codes

slide-11
SLIDE 11

IMPROVEMENT TO UPPER BOUND FOR t = 1

LEMMA If |C| > ( qk−q

q−1 )2, then every codeword in C has at least one

point in qk−1

q−1 codewords.

LEMMA If point P lies in qk−1

q−1 codewords, then line through P and other

point of S is completely contained in S.

Leo Storme Constant dimension subspace codes

slide-12
SLIDE 12

IMPROVEMENT TO UPPER BOUND FOR t = 1

Leo Storme Constant dimension subspace codes

slide-13
SLIDE 13

IMPROVEMENT TO UPPER BOUND FOR t = 1

LEMMA If point P lies in qk−1

q−1 codewords, then

|S| = | ∪c∈C c| = qk − q q − 1 2 + qk − q q − 1

  • + 1.

REMARK: qk − q q − 1 2 + qk − q q − 1

  • + 1 = |PG(T, q)|.

|PG(2k − 2, q)| < |S| < |PG(2k − 1, q)|.

Leo Storme Constant dimension subspace codes

slide-14
SLIDE 14

IMPROVEMENT TO UPPER BOUND FOR t = 1

LEMMA If more than q2k−3 + q2k−4 + · · · + q + 1 points of S lie in qk−1

q−1

codewords, then (2k − 2)-dimensional subspace contained in S.

Leo Storme Constant dimension subspace codes

slide-15
SLIDE 15

IMPROVEMENT TO UPPER BOUND FOR t = 1

Leo Storme Constant dimension subspace codes

slide-16
SLIDE 16

IMPROVEMENT TO UPPER BOUND FOR t = 1

Leo Storme Constant dimension subspace codes

slide-17
SLIDE 17

IMPROVEMENT TO UPPER BOUND FOR t = 1

Eventually PG(2k − 2, q) = c, P1, . . . , Pk−1 ⊂ S. But |S| > |PG(2k − 2, q)|. |S \ PG(2k − 2, q)| ≈ q2k−3.

Leo Storme Constant dimension subspace codes

slide-18
SLIDE 18

IMPROVEMENT TO UPPER BOUND FOR t = 1

If point of S \ PG(2k − 2, q) in qk−1

q−1 codewords, then

PG(2k − 1, q) ⊂ S. (FALSE) So all points of S \ PG(2k − 2, q) in less than qk−1

q−1

codewords.

Leo Storme Constant dimension subspace codes

slide-19
SLIDE 19

IMPROVEMENT TO UPPER BOUND FOR t = 1

Leo Storme Constant dimension subspace codes

slide-20
SLIDE 20

IMPROVEMENT TO UPPER BOUND FOR t = 1

All points of S \ PG(2k − 2, q) in less than qk−1

q−1 codewords.

|S \ PG(2k − 2, q)| ≈ q2k−3. So number of points of S in qk−1

q−1 codewords, is

approximately q2k−4. (TOO SMALL)

Leo Storme Constant dimension subspace codes

slide-21
SLIDE 21

IMPROVEMENT TO UPPER BOUND FOR t = 1

THEOREM (BARTOLI, RIET, STORME, VANDENDRIESSCHE) Every 1-intersecting constant dimension code C of codewords

  • f dimension k of size

|C| = qk − q q − 1 2 + qk − q q − 1

  • + 1 − δ,

with δ = qk−2, is sunflower.

Leo Storme Constant dimension subspace codes

slide-22
SLIDE 22

Thank you very much for your attention!

Leo Storme Constant dimension subspace codes