Best Known (43−19, 43, s)-Nets in Base 64
(43−19, 43, 457)-Net over F64 — Constructive and digital
Digital (24, 43, 457)-net over F64, using
- 641 times duplication [i] based on digital (23, 42, 457)-net over F64, using
- net defined by OOA [i] based on linear OOA(6442, 457, F64, 19, 19) (dual of [(457, 19), 8641, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6442, 4114, F64, 19) (dual of [4114, 4072, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,9]) ⊂ C([0,6]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(6442, 4114, F64, 19) (dual of [4114, 4072, 20]-code), using
- net defined by OOA [i] based on linear OOA(6442, 457, F64, 19, 19) (dual of [(457, 19), 8641, 20]-NRT-code), using
(43−19, 43, 518)-Net in Base 64 — Constructive
(24, 43, 518)-net in base 64, using
- (u, u+v)-construction [i] based on
- (6, 15, 259)-net in base 64, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on (6, 16, 259)-net in base 64, using
- (9, 28, 259)-net in base 64, using
- base change [i] based on digital (2, 21, 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, 21, 259)-net over F256, using
- (6, 15, 259)-net in base 64, using
(43−19, 43, 3295)-Net over F64 — Digital
Digital (24, 43, 3295)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6443, 3295, F64, 19) (dual of [3295, 3252, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(6443, 4116, F64, 19) (dual of [4116, 4073, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(11) [i] based on
- linear OA(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(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(18) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(6443, 4116, F64, 19) (dual of [4116, 4073, 20]-code), using
(43−19, 43, large)-Net in Base 64 — Upper bound on s
There is no (24, 43, large)-net in base 64, because
- 17 times m-reduction [i] would yield (24, 26, large)-net in base 64, but