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

NEW RESUL TS ON GRIESMER CODES AND ARCS Assia Rousseva So�a 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, k ] q o de : C < F n , dim C = k - o de : d = min { d ( u , v ) | u , v ∈ C, u � = v } . - n - the length of C ; ⋄ q - k - the dimension of C ; ⋄ [ n, k, d ] q - d - the minimum distan e of C . � numb er of o dew o rds of (Hamming) w eight i � the sp e trum of C ⋄ A i � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 1 ⋄ ( A i ) i ≥ 0

The Main Problem in Co ding Theo ry . Optimize one of the pa rameters n , k , d , given the other t w o. - minimal length of a linea r o de over F q of dimension k and minimum distan e d ; - maximal dimension of a linea r o de over F q of length n and minimum distan e d ; n q ( k, d ) - maximal minimum distan e of a linea r o de over F q of length n and dimension k . K q ( n, d ) optimalit y with resp e t to n = optimalit y with resp e t to k and d D q ( n, k ) ⇒ � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 2

Griesmer b ound: Let C b e an [ n, k, d ] q - o de. Then ⋄ Theo rem. Given the integer k and the p rime p o w er q , Griesmer [ g q ( k, d ) , k, d ] q - k − 1 � o des exist fo r all su� iently la rge d . ⌈ d n q ( k, d ) ≥ g q ( k, d ) = q i ⌉ The p roblem of �nding the exa t value of n q ( k, d ) is solved fo r i =0 : k ≤ 8 fo r all d ; : k ≤ 5 fo r all d ; : k ≤ 4 fo r all d ; • q = 2 • q = 5 , 7 , 8 , 9 : k ≤ 3 fo r all d ; • q = 3 : k = 4 � four values of d fo r whi h n 5 (4 , d ) is not kno wn. • q = 4 � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 3 • q = 5

http://www.mi.s. os ak af u-u .a . jp / maruta/griesme r.h tm The Op en Cases fo r q = 5 , k = 4 103 � 104 (22 , 5) -a r 104 � 105 in PG(2 , 5) 203 � 204 (42 , 9) -a r d g 5 (4 , d ) n 5 (4 , d ) K K| H 204 � 205 in PG(2 , 5) 81 103 (103 , 22) 82 104 (104 , 22) 161 203 (203 , 42) 162 204 (204 , 42) � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 4

2. Divisible and Quasidivisibl e Ar s A multiset in PG( k − 1 , q ) is a mapping ⋄ ( P ) � multipli it y of the p oint P . � P → N 0 , : K ( Q ) = � � multipli it y of the set Q . K : P → K ( P ) . ( P ) � the a rdinalit y of K . ⋄ K P oints, lines, ... ,hyp erplanes of multipli it y i a re alled i -p oints, i -lines, ... , -hyp erplanes. ⋄ Q ⊂ P P ∈Q K ( P ) � the numb er of hyp erplanes H with K ( H ) = i ⋄ K � the sp e trum of K ⋄ i � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 5 ⋄ a i ⋄ ( a i ) i ≥ 0

De�nition. ( 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 H 0 : K ( H 0 ) = w . De�nition. ( 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 H 0 : K ( H 0 ) = w . � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 6

De�nition. 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 oint . An 1-extendable a r is alled extendable. De�nition. An a r K in PG( k − 1 , q ) with K ( P ) = n and sp e trum ( a i ) is P ∈ P said to b e divisible with diviso r ∆ , ∆ > 1 , if a i = 0 fo r all i �≡ n (mod ∆) . De�nition. An a r K with K ( P ) = n and sp e trum ( a i ) is said to b e t - quasidivisible with diviso r ∆ , ∆ > 1 , (o r t -quasidivisible mo dulo ∆ ) if a i = 0 fo r all i �≡ n, n + 1 , . . . , n + t (mod ∆) . � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 7

3. Linea r o des as multisets of p oints - o de C ( n, w = n − d ) -a r K of full length in PG( k − 1 , q ) , wt( u ) = u a hyp erplane H with K ( H ) = n − u , [ n, k, d ] q ⇔ 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 ) 0 � = u ∈ C ⇔ fo r all i �≡ 0 (mod ∆) fo r all i �≡ n (mod ∆) ⇔ -quasidivisible [ n, k, d ] q - o de -quasidivisible ( n, n − d ) -a r fo r all i �≡ − j (mod q ) in PG( k − 1 , q ) a i = 0 fo r all ⇔ A i = 0 a i = 0 t ⇔ t � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 8 A i = 0 j ∈ { 0 , 1 , . . . , t } i �≡ n + j (mod q )

Griesmer a r s: a r s asso iated with o des meeting the Griesmer b ound Griesmer [ n, k, d ] q o des Griesmer ( n, w ) -a r s in PG( k − 1 , q ) ⋄ ⇔ n = � k − 1 n = � k − 1 If d = n − w = sq k − 1 − ε k − 2 q k − 2 − . . . − ε 1 q − ε 0 , and i =0 ⌈ d/q i ⌉ i =0 ⌈ ( n − w ) /q i ⌉ maximal multipli it y of a subspa e of o dimension i , i = 0 , . . . , k − 1 . Then ⋄ w i := where v k = ( q k − 1) / ( q − 1) . w i = sv k − i − ε k − 2 v k − i − 1 − . . . − ε i +1 v 2 − ε i v 1 , � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 9

4. Some Extension Results Theo rem. (R. Hill, P . Lizak, 1995, geometri version) Let K b e a ( n, w ) -a r in with gcd( n − w, q ) = 1 . Let further K ( H ) ≡ n o 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 PG( k − 1 , q ) ( n, w ) -a r in PG( k − 1 , q ) , q ≥ 5 , o 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

- t -quasidivisible ( n, w ) -a r in Σ = PG( k − 1 , q ) , i.e. fo r every hyp erplane , w e have K ( H ) ≡ n, n + 1 , . . . , n + t (mod q ) , where 0 < t < q is an integer onstant. De�ne an a r � in the dual spa e � ⋄ K H ⋄ K Σ where H is the set of all hyp erplanes of Σ . � H → N 0 , � K : � H → K ( H ) := n + t − K ( H ) (mod q ) . � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 11

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 fo r some a r � and c not ne essa rily di�erent hyp erplanes � then K � c � H i + � is c -extendable. In pa rti ula r, if � ontains a hyp K ′ erplane in its supp o rt then K is K = χ e extendable. i =1 H 1 , . . . , � K ′ H c Theo rem. Let � b e a subspa e of � of p ositive dimension. Then � K (mod q ) . K ( � S Σ S ) ≡ t � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 12

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 Let further ε 0 = t, . . . ε k − 2 < √ q . Then K is t -extendable. d = n − w = sq k − 1 − ε k − 2 q k − 2 − . . . − ε 1 q − ε 0 . is a ( tv k − 1 , tv k − 2 ) -a r , where v k = q k − 1 is a sum of t hyp erplanes ⋄ � K q − 1 ⋄ � K � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 13

5. ( t mod q ) -Ar s De�nition. An a r F is alled a ( t mod q )-a r if � all p oints have multipli it y ≤ t ; � all subspa es S of p ositive dimension have multipli it y F ( S ) ≡ t (mod q ) . Theo rem A. The sum of a ( t 1 mod q ) -a r s and a ( t 2 mod q ) -a r is a ( t mod q ) -a r with t = t 1 + t 2 . In pa rti ula r, the sum of t hyp erplanes in is a ( t mod q )-a r . � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 14 PG( k − 1 , q )

Theo rem B. Let F 0 b e a ( t mod q )-a r in a hyp erplane H ∼ = PG( k − 2 , q ) . of Σ = PG ( k − 1 , q ) . F o r a �xed p oint P ∈ Σ \ H , de�ne an a r F in Σ as follo ws: � F ( P ) = t ; � fo r ea h p oint Q � = P : F ( Q ) = F 0 ( R ) where R = � P, Q � ∩ H . Then the a r F is a ( t mod q ) -a r in PG( k − 1 , q ) of size q |F 0 | + t . De�nition. ( t mod q ) -a r s obtained b y Theo rem B a re alled lifted a r s. � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 15

F ( P ) = t P F ( Q ) = F 0 ( R ) Q R F 0 � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 16 H ∼ = PG( k − 2 , q )

: an a r in Σ = PG( k − 1 , q ) � the set of all hyp erplanes in Σ - a fun tion su h that σ ( F ( H )) is a non-negative integer fo r all H ∈ H . F The a r F σ in � H σ is alled the σ -dual of F . Σ � Theo rem C. Let F b e a ( t mod q ) -a r in PG(2 , q ) of size mq + t . Then the H → N 0 F σ : a r F σ with σ ( x ) = ( x − t ) /q is an (( m − t ) q + m, m − t ) -blo king set in the H → σ ( F ( H )) dual plane with line multipli ities m − t, m − t + 1 , . . . , m . � ALCOMA 2015, Kloster Banz, 15.-20.03.2015 � 17