Best Known (40−18, 40, s)-Nets in Base 64
(40−18, 40, 457)-Net over F64 — Constructive and digital
Digital (22, 40, 457)-net over F64, using
- net defined by OOA [i] based on linear OOA(6440, 457, F64, 18, 18) (dual of [(457, 18), 8186, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6440, 4113, F64, 18) (dual of [4113, 4073, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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 Ce(17) ⊂ Ce(11) [i] based on
- OA 9-folding and stacking [i] based on linear OA(6440, 4113, F64, 18) (dual of [4113, 4073, 19]-code), using
(40−18, 40, 517)-Net in Base 64 — Constructive
(22, 40, 517)-net in base 64, using
- base change [i] based on digital (12, 30, 517)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (2, 20, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- digital (1, 10, 258)-net over F256, using
- (u, u+v)-construction [i] based on
(40−18, 40, 2720)-Net over F64 — Digital
Digital (22, 40, 2720)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6440, 2720, F64, 18) (dual of [2720, 2680, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(6440, 4113, F64, 18) (dual of [4113, 4073, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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 Ce(17) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(6440, 4113, F64, 18) (dual of [4113, 4073, 19]-code), using
(40−18, 40, 7012569)-Net in Base 64 — Upper bound on s
There is no (22, 40, 7012570)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 1 766847 662816 624204 091598 473225 754788 233202 750633 286166 692592 647128 383833 > 6440 [i]