On the least covering radius of the binary linear codes of

dimension 6

Tsonka Baicheva and Iliya BouyuklievInstitute of Mathematics and Informatics, Bulgaria

Basic definitions

Linear code C is a k-dimensional subspace of Fqn

[n,k,d]q linear code with length n, dimension k, minimum distance d, over Fq .

x+C={x+c | c ∈ C} a coset of the code C determined by the vector x ∈ Fq

n. Coset leader a vector with the smallest weight in

the coset.

Basic definitions

R(C) Covering radius of a code. The largest weight in the set of coset leaders.

[n,k,d]R or [n,k]R

tq[n,k] Least value of R(C) when C runs over the class of all linear [n,k] codes over Fq for given q.

Norm of the code

C0(i) the set of codewords in which i-th coordinate is 0

C1(i) the set of codewords in which i-th coordinate is 1

Norm of C with respect to the i-th coordinate

C has norm N if Nmin ≤ N

( ) ( ) ( )0 1( ) max{ ( , ) ( , )}

nq

i i i

x FN x d x C d x C

( )min min i

i

N N

Basic definitions

C is normal if it has norm 2R+1 If N(i) ≤ 2R+1 then the i-th coordinate is

acceptable with respect to 2R+1

Theorem If C is an [n,k,d] code with n≤15, k≤5 or n-k≤9, then C is normal.

ADS construction

IA

IB 0

0

[nA,kA]RA

IA

[nB,kB]RB

IB

.A⊕B= {(a,0,b)|(a,0)∈A,

(0,b)∈B} ∪ {(a,1,b)|(a,1)∈A, (1,b)∈B}

[nA+nB-1,kA+kB-1]R R≤RA+RB

t2[n,k], k≤5

Cohen, Karpovsky, Mattson, Jr., Schatz ’85

Graham and Sloane ’85

2

2

2

[ ,1] , 1.2

1[ , 2] , 2.

2

2[ ,3] , 3.

2

nt n for n

nt n for n

nt n for n

2 2

2 2

4[ ,4] , 4 5. [5,4] 1.

2

5[ ,5] , 5 6. [6,5] 1.

2

nt n for n and n t

nt n for n and n t

t2[n,6]

n 7 8 9 10 11 12 13 14 15 16

t2[n,6] 1 1 1 2 2 3 3 3 4 4

n 17 18 19 20 21 22 23 24 25 26

t2[n,6] 5 5 5 6 6 6-7 7 7-8 7-8 8-9

n 27 28 29 30 31 32 33 34 35 36

t2[n,6] 8-9 9-10 9-10 9-11 10-11

10-12

11-12

11-13 11-13

12-14

n 37 38 39 40 41 42 43 44 45 46

t2[n,6] 12-14

13-15

13-15

14-16

14-16

14-17

15-17

15-18 16-18

16-19

n 47 48 49 50 51 52 53 54 55 56

t2[n,6] 17-19

17-20

17-20

18-21

18-21

19-22

19-22

20-23 20-23

20-24

n 57 58 59 60 61 62 63 64

t2[n,6] 21-24

21-25

22-25

22-26

23-26

23-27

23-27

24-28

t2[n,6]

Graham and Sloane ‘85

2

2

8[ ,6] , 18

2

9[ ,7] , 19

2

nt n for n

nt n for n

t2[22,6]=6-7

If [22,6] code C contains a repeated coordinate R(C)≥t2[20,6]+1=7⇒ If a [22,6] code has covering radius 6 it must be

a projective one.

Bouyukliev ‘2006 Classification of all binary projective codes of dimension up to 6.

There are 2 852 541 [22,6] nonequavalent projective codes.

A heuristic algorithm for lower bound on the covering radius of a linear code

Idea of the algorithm. As fast as possible to find a coset leader of weight greater than R.

• Randomly chosen vector c from Kc={c+C}• N(c) set of neighbors of c which differ from c in one

coordinate• Evaluation function f=wt(Kc)2k-A(Kc)

• wt(Kc) weight of the coset Kc

• A(Kc) number of vectors of minimum weight in Kc

• Add some noise to c

t2[n,6]

We show the nonexistence of 236 779 414 projective codes of dimension 6 and even lengths 22 ≤ n ≤ 54

t2[22,6]=6-7 t2[22,6]=7

t2[24,6]=7-8 t2[24,6]=8

t2[25,6]=7-8 t2[25,6]≥t2[24,6] t2[25,6]=8

t2[56,6]=23-24

If [56,6] code C contains a repeated coordinate R(C) ≥ t2[54,6]+1=23+1=24 Otherwise, C is a shortened version of the [63,6]

Simplex code with covering radius 31 and R(C) ≥ 31-7=24

⇒ t2[56,6]=24 and t2[57,6]=24

For n≥64 every [n,6] code must contain repeated coordinate and t2[n,6] ≥ t2[n-2,6]+1

⇒ t2[n,6] ≥ |O (n-8)/2O| for all n≥18

t2[n,6]=|O (n-8)/2O| for all n≥18

n 7 8 9 10 11 12 13 14 15 16

t2[n,6] 1 1 1 2 2 3 3 3 4 4

n 17 18 19 20 21 22 23 24 25 26

t2[n,6] 5 5 5 6 6 6-7 7 7-8 7-8 8-9

n 27 28 29 30 31 32 33 34 35 36

t2[n,6] 8-9 9-10 9-10 9-11 10-11

10-12

11-12

11-13 11-13

12-14

n 37 38 39 40 41 42 43 44 45 46

t2[n,6] 12-14

13-15

13-15

14-16

14-16

14-17

15-17

15-18 16-18

16-19

n 47 48 49 50 51 52 53 54 55 56

t2[n,6] 17-19

17-20

17-20

18-21

18-21

19-22

19-22

20-23 20-23

20-24

n 57 58 59 60 61 62 63 64 65 66

t2[n,6] 21-24

21-25

22-25

22-26

23-26

23-27

23-27

24-28 28 29

Construction of codes of R=t2[n,6]

Theorem 20, Graham and Sloane ’85 If C is an [n,k]R normal code, there are [n+2i,k]R+i

normal codes for all i≥0.

.A⊕B

1 0 0 … … 1 0 0 1 0 0 1 1 1

A B

Construction of codes of R=t2[n,6]

There are 6 [7,6]1; 16 [8,6]1; 4 [9,6]1; 255 [10,6]2; 100

[11,6]2; 4126 [12,6]3; 2101 [13,6]3; 1 [14,6]3; 15376 [15,6]4 normal codes.

The [19,6,7]5 normal code constructed by Graham and Sloane is unique.

Thre are 22 [16,6]4; 51289 [17,6]5; 139 [18,6]5; 1195

[20,6]6; 3 [21,6]6; 6627 [22,6]7 projective codes.

Upper bounds for t2[n,8] and t2[n,9]

DS of two [9,4]2 normal codes gives an [18,8]4 normal code and [18+2i,8]4+i normal codes exists.

DS of [8,4]2 and [9,4]2 normal codes gives an [17,8]4 normal code and [17+2i,8]4+i normal codes exists.

t2[16,8]=3.

ADS of [7,4]1 and [14,6]3 normal codes gives an [20,9]4 normal code and [20+2i,9]4+i normal codes exists.

ADS of [7,4]1 and [19,6]5 normal codes gives an [25,9]6 normal code and [25+2i,9]6+i normal codes exists.

Upper bounds for t2[n,8] and t2[n,9]

Theorem 4

2

2

10[ ,8] , 16

2

12[ ,9] , 25

2

nt n for n

nt n for n