Best Known (26, 26+21, s)-Nets in Base 64
(26, 26+21, 411)-Net over F64 — Constructive and digital
Digital (26, 47, 411)-net over F64, using
- 641 times duplication [i] based on digital (25, 46, 411)-net over F64, using
- net defined by OOA [i] based on linear OOA(6446, 411, F64, 21, 21) (dual of [(411, 21), 8585, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6446, 4111, F64, 21) (dual of [4111, 4065, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6446, 4114, F64, 21) (dual of [4114, 4068, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(6429, 4097, F64, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6446, 4114, F64, 21) (dual of [4114, 4068, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6446, 4111, F64, 21) (dual of [4111, 4065, 22]-code), using
- net defined by OOA [i] based on linear OOA(6446, 411, F64, 21, 21) (dual of [(411, 21), 8585, 22]-NRT-code), using
(26, 26+21, 518)-Net in Base 64 — Constructive
(26, 47, 518)-net in base 64, using
- (u, u+v)-construction [i] based on
- (6, 16, 259)-net in base 64, using
- base change [i] based on digital (2, 12, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- base change [i] based on digital (2, 12, 259)-net over F256, using
- (10, 31, 259)-net in base 64, using
- 1 times m-reduction [i] based on (10, 32, 259)-net in base 64, using
- base change [i] based on digital (2, 24, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256 (see above)
- base change [i] based on digital (2, 24, 259)-net over F256, using
- 1 times m-reduction [i] based on (10, 32, 259)-net in base 64, using
- (6, 16, 259)-net in base 64, using
(26, 26+21, 2961)-Net over F64 — Digital
Digital (26, 47, 2961)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6447, 2961, F64, 21) (dual of [2961, 2914, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6447, 4116, F64, 21) (dual of [4116, 4069, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(6441, 4096, F64, 21) (dual of [4096, 4055, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(646, 20, F64, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,64)), using
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- Reed–Solomon code RS(58,64) [i]
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(6447, 4116, F64, 21) (dual of [4116, 4069, 22]-code), using
(26, 26+21, large)-Net in Base 64 — Upper bound on s
There is no (26, 47, large)-net in base 64, because
- 19 times m-reduction [i] would yield (26, 28, large)-net in base 64, but