1. On the least covering radius of the binary linear codes of dimension 6 Tsonka Baicheva and Iliya Bouyukliev Institute of Mathematics and Informatics, Bulgaria

2. Basic definitions  Linear code C is a k-dimensional subspace of F q n  [n,k,d] q linear code with length n, dimension k, minimum distance d, over F q.  x+C={x+c | c ∈ C} a coset of the code C determined by the vector x ∈ F q n.  Coset leader a vector with the smallest weight in the coset.

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

4. Norm of the code  C 0 (i) the set of codewords in which i-th coordinate is 0  C 1 (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 N min ≤ N

5. 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.

6. ADS construction I A I B 0 0 [n A,k A ]R A I A [n B,k B ]R B I B. A⊕BA⊕B = {(a,0,b)|(a,0) ∈ A, (0,b) ∈ B} ∪ {(a,1,b)|(a,1) ∈ A, (1,b) ∈ B} [n A +n B -1,k A +k B -1]R R ≤ R A +R B

7. t 2 [n,k], k≤5 Cohen, Karpovsky, Mattson, Jr., Schatz ’85 Graham and Sloane ’85

8. t 2 [n,6] n78910111213141516 t 2 [n,6]1112233344 n17181920212223242526 t 2 [n,6]555666-777-8 8-9 n27282930313233343536 t 2 [n,6]8-99-10 9-1110-1110-1211-1211-13 12-14 n37383940414243444546 t 2 [n,6]12-1413-15 14-16 14-1715-1715-1816-1816-19 n47484950515253545556 t 2 [n,6]17-1917-20 18-21 19-22 20-23 20-24 n5758596061626364 t 2 [n,6]21-2421-2522-2522-2623-2623-27 24-28

9. t 2 [n,6] Graham and Sloane ‘85

10. t 2 [22,6]=6-7  If [22,6] code C contains a repeated coordinate R(C)≥t 2 [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.

11. 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 K c ={c+C} N(c) set of neighbors of c which differ from c in one coordinate Evaluation function f=wt(K c )2 k -A(K c ) wt(K c ) weight of the coset K c A(K c ) number of vectors of minimum weight in K c Add some noise to c

12. t 2 [n,6]  We show the nonexistence of 236 779 414 projective codes of dimension 6 and even lengths 22 ≤ n ≤ 54  t 2 [22,6]=6-7 t 2 [22,6]=7  t 2 [24,6]=7-8 t 2 [24,6]=8  t 2 [25,6]=7-8  t 2 [25,6]≥t 2 [24,6] t 2 [25,6]=8

13. t 2 [56,6]=23-24  If [56,6] code C contains a repeated coordinate R(C) ≥ t 2 [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 ⇒ t 2 [56,6]=24 and t 2 [57,6]=24  For n≥64 every [n,6] code must contain repeated coordinate and t 2 [n,6] ≥ t 2 [n-2,6]+1 ⇒ t 2 [n,6] ≥ |̱ (n-8)/2̱| for all n≥18

14. t 2 [n,6]=|̱ (n-8)/2̱| for all n≥18 n78910111213141516 t 2 [n,6]1112233344 n17181920212223242526 t 2 [n,6]555666-777-8 8-9 n27282930313233343536 t 2 [n,6]8-99-10 9-1110-1110-1211-1211-13 12-14 n37383940414243444546 t 2 [n,6]12-1413-15 14-16 14-1715-1715-1816-1816-19 n47484950515253545556 t 2 [n,6]17-1917-20 18-21 19-22 20-23 20-24 n57585960616263646566 t 2 [n,6]21-2421-2522-2522-2623-2623-27 24-282829

15. Construction of codes of R=t 2 [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⊕BA⊕B 1 0 0 … … 1 0 0 1 1 1 A B

16. Construction of codes of R=t 2 [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.

17. Upper bounds for t 2 [n,8] and t 2 [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. t 2 [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.

18. Upper bounds for t 2 [n,8] and t 2 [n,9] Theorem 4