ON THE ASYMPTOTIC TIGHTNESS OF THE GRIESMER BOUND Assia Rousseva - PowerPoint PPT Presentation

ON THE ASYMPTOTIC TIGHTNESS OF THE GRIESMER BOUND Assia Rousseva Sofia University (joint work with Ivan Landjev) – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 –

The Main Problem in Coding Theory Given the positive integers k and d , and a prime power q , find the smallest value of n for which there exists a linear [ n, k, d ] q -code. This value is denoted by n q ( k, d ) . The Griesmer bound: k − 1 ⌈ d � n q ( k, d ) ≥ g q ( k, d ) := q i ⌉ i =0 Griesmer code: an [ n, k, d ] q code with n = g q ( k, d ) . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 1

k , q - fixed, d → ∞ Theorem. For a given dimension k , there exists an integer d 0 such that for all d ≥ d 0 n q ( k, d ) − g q ( k, d ) = 0 . 2 ⌉ 2 k − 1 . • Baumert, McEliece: n 2 ( k, d ) − g 2 ( k, d ) = 0 for all d ≥ ⌈ k − 1 • V. I. Belov, V. N. Logachev, V. P. Sandimirov, R. Hill: n q ( k, d ) − g q ( k, d ) = 0 for all d ≥ ( k − 2) q k − 1 + 1 . • Maruta: n q ( k, d ) − g q ( k, d ) > 0 for d = ( k − 2) q k − 1 − ( k − 1) q k − 2 for q ≥ k , k = 3 , 4 , 5 , and for q ≥ 2 k + 3 , k ≥ 6 . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 2

d , q - fixed, k → ∞ Theorem. (S. Dodunekov) For every two integers t and d ≥ 3 , there exists an integer k 0 such that for all k ≥ k 0 n q ( k, d ) − g q ( k, d ) ≥ t. Idea of proof. d , q , R = ⌊ ( d − 1) / 2 ⌋ - fixed, k → ∞ R � g q ( k, d ) � | B g q ( k,d ) � ( q − 1) R − ( R ) | = → k →∞ ∞ . q i i =0 – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 3

Consider an optimal [ n q ( k, d ) , k, d ] q -code. From the sphere-packing bound: R � n q ( k, d ) � q k · � q n q ( k,d ) ( q − 1) i ≥ i i =0 R � g q ( k, d ) � q k · � ( q − 1) i ≥ i i =0 whence n q ( k, d ) − k ≥ log q | B g q ( k,d ) ( R ) | → ∞ . R – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 4

On the other hand d + ⌈ d q ⌉ + ⌈ d q 2 ⌉ + . . . + ⌈ d g q ( k, d ) = q k − 1 ⌉ d + d q + d d < q 2 + . . . + q k − 1 + k − 1 whence g q ( k, d ) − k < d q k − 1 q k − q k − 1 − 1 , and ( R ) | − d q k − 1 n q ( k, d ) − g q ( k, d ) > log q | B g q ( k,d ) q k − q k − 1 + 1 → ∞ . q – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 5

Problem A. Given the prime power q and the positive integer k , what is the smallest value of t , denoted t q ( k ) , such that there exists a [ g q ( k, d ) + t, k, d ] q -code for all d . Or, in other words, what is t q ( k ) := max d ( n q ( k, d ) − g q ( k, d )) . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 6

Known Results for Small k • t q (2) = 0 for all q • t q (3) = 1 for all q ≤ 19 ; • t q (3) ≤ 2 for q = 23 , 25 , 27 , 29 ; • t 3 (4) = 1 ; • t 4 (4) = 1 ; • t 5 (4) = 2 ( t = 2 for d = 25 only); • t 5 (5) ≤ 5 . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 7

The Geometric Approach to Linear Codes [ g q ( k, d ) + t, k, d ] q -code ∼ ( g q ( k, d ) + t, g q ( k, d ) + t − d ) -arc in PG( k − 1 , q ) . Write d = sq k − 1 − λ k − 2 q k − 2 − . . . − λ 1 q − λ 0 , ( ⋆ ) where 0 ≤ λ i < q . Then g q ( k, d ) = sv k − λ k − 2 v k − 1 − . . . − λ 1 v 2 − λ 0 v 1 , w q ( k, d ) = g q ( k, d ) − d = sv k − 1 − λ k − 2 v k − 2 − . . . − λ 1 v 1 , where v i = ( q i − 1) / ( q − 1) . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 8

Problem B. Find the smallest t for which there exists a ( g q ( k, d )+ t, w q ( k, d )+ t ) -arc in PG( k − 1 , q ) for all d . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 9

Let K be a ( g q ( k, d ) + t, w q ( k, d ) + t ) -arc in PG( k − 1 , q ) Denote the maximal point multiplicity in K by s 0 ≤ t + s . Construct the multiset F := s 0 PG( k − 1 , q ) − K . This multiset F is a minihyper with parameters ( σv k + λ k − 2 v k − 1 + . . . + λ 1 v 2 + λ 0 v 1 − t, σv k − 1 + λ k − 2 v k − 2 + . . . + λ 1 v 1 − t ) , where � s 0 − s if s < s 0 ≤ t + s, σ = 0 if s 0 ≤ s, with maximal point multiplicity σ + s . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 10

Problem C. For all d given by d = sq k − 1 − λ k − 2 q k − 2 − . . . − λ 1 q − λ 0 , find the minimum value of t for which there exists a minihyper in PG( k − 1 , q ) with parameters ( σv k + λ k − 2 v k − 1 + . . . + λ 1 v 2 + λ 0 v 1 − t, σv k − 1 + λ k − 2 v k − 2 + . . . + λ 1 v 1 − t ) . with maximal point multiplicity σ + s . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 11

Example. Let d = 85 , k = 4 , q = 5 . Then s = 1 , λ 2 = 1 , λ 1 = 3 , λ 0 = 0 , g 5 (4 , 85) = 107 , w = 22 . As a code: Find the smallest t so that there exists a [107 + t, 4 , 85] 5 -code. As an arc: Find the smallest t so that there exists a (107 + t, 22 + t ) -arc in PG(3 , 5) As a minihyper: σ + s 1 2 3 4 t 0 (49 , 9) 1 (48 , 8) (204 , 39) 2 (47 , 7) (203 , 38) (359 , 69) 3 (46 , 6) (202 , 37) (358 , 68) (514 , 99) – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 12

Theorem. Let d = sq k − 1 − λ k − 2 q k − 2 − . . . − λ 1 q − λ 0 , and let the multiset F be a minihyper in PG( k − 1 , q ) with parameters ( σv k + λ k − 2 v k − 1 + . . . + λ 0 v 1 − τ 1 , σv k − 1 + λ k − 2 v k − 2 + . . . + λ 1 v 1 − τ 1 ) . Define the multiset F ′ by � F ( x ) if F ( x ) ≤ σ + s, F ′ ( x ) = σ + s if F ( x ) > σ + s. Let N = |F| and N ′ = |F ′ | . If F − F ′ is an ( N − N ′ , τ 2 ) -arc then there exists a ( g q ( k, d ) + t, w q ( k, d ) + t ) -arc in PG( k − 1 , q ) , or, equivalently, a code with parameters [ g q ( k, d ) + t, k, d ] q , with t = τ 1 + τ 2 . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 13

Example. k = 4 , d = 2 q 3 − 4 q 2 ; s = 2 , λ 2 = 4 , λ 1 = 0 s 0 2 3 4 t 0 (4 v 3 , 4 v 2) 1 (4 v 3 − 1 , 4 v 2 − 1) ( v 4 + 4 v 3 − 1 , v 3 + 4 v 2 − 1) 2 (4 v 3 − 2 , 4 v 2 − 2) ( v 4 + 4 v 3 − 2 , v 3 + 4 v 2 − 2) (2 v 4 + 4 v 3 − 2 , 2 v 3 + 4 v 2 − 2) 3 (4 v 3 − 3 , 4 v 2 − 3) ( v 4 + 4 v 3 − 3 , v 3 + 4 v 2 − 3) (2 v 4 + 4 v 3 − 3 , 2 v 3 + 4 v 2 − 3) (4 v 3 , 4 v 2 ) -minihyper, σ = 0 , τ 1 = 0 – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 14

(4 v 3 − 3 , 4 v 2 − 3) -minihyper with maximal point multiplicity 2 [ g q (4 , d ) + 3 , 4 , d ] q -code F − F ′ F ′ F – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 15

(4 v 3 − 2 , 4 v 2 − 2) -minihyper with maximal point multiplicity 2: Take the four planes to have a common point, but no three with a common line F − F ′ is a (2 , 2) -arc, i.e. t = 2 [ g q (4 , d ) + 2 , 4 , d ] q -code for all q F − F ′ F ′ F For q = 5 there exists a code with t = 1 , i.e. a [189 , 4 , 150] 5 -code, which is optimal. – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 16

D ( t ) q ( k ) := { d ∈ Z | 1 ≤ d ≤ q k − 1 , n q ( k, d ) = g q ( k, d ) + t } . Lemma. Let d 1 < d 2 be integers from D ( t ) q ( k ) . Then for every integer d with d 1 < d < d 2 k − 2 ( ⌈ d 2 q i ⌉ − ⌈ d 1 � n q ( k, d ) ≤ g q ( k, d ) + t + q i ⌉ ) . i =1 Theorem. t q ( k ) ≤ q k − 2 . Theorem. t q (4) ≤ q − 1 . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 17

The Case k = 3 Problem D. (S. Ball): For a fixed n − d , is there always a 3-dimensional linear [ n, 3 , d ] -code meeting the Griesmer bound (or at least close to the Griesmer bound, maybe a constant or log q away)? The answer to the first part of the question in Problem D is NO. Take w = n − d = q + 2 . Then an optimal arcs has n = q 2 + q + 2 , but a [ q 2 + q + 2 , 3 , q 2 ] q -code is NOT a Griesmer code. Lemma. Let K be an ( n, w ) -arc in PG(2 , q ) with n = ( w − 1) q + w − α and let C K be the [ n, 3 , d ] q -code associated with K . Then n = t + g q (3 , d ) with t = ⌊ α/q ⌋ . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 18

Theorem. For all d ≥ q 2 there exist Griesmer [ n, 3 , d ] q codes (arcs). In fact, Griesmer codes do exist for all d ≥ q 2 − 2 q + 1 For q 2 − 3 q + 1 ≤ d ≤ q 2 − 2 q we have t = 0 or t = 1 . Theorem. If q = 2 h then t q (3) ≤ log 2 q − 1 = h − 1 . The proof is based on the following two lemmas. – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 19

Lemma. Let q = 2 h . The sum of r maximal arcs is equivalent to a linear [ n, 3 , d ] q -code whose length satisfies n = g q (3 , d ) + ( r − 1) , i.e. its length exceeds by r − 1 the corresponding Griesmer bound. Lemma. Let q = 2 h . Every integer m ≤ q − 1 can be represented as m = 2 a 1 + . . . + 2 a r − r, for some a i ∈ { 1 , . . . , h − 1 } and some r ≤ h = log 2 q . – Fifth Irsee Conference “Finte Geometries”, Kloster Irsee, 10.–16.09.2017 – 20