Best Known (18, 18+16, s)-Nets in Base 64
(18, 18+16, 513)-Net over F64 — Constructive and digital
Digital (18, 34, 513)-net over F64, using
- 641 times duplication [i] based on digital (17, 33, 513)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 513, F64, 16, 16) (dual of [(513, 16), 8175, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(6433, 4104, F64, 16) (dual of [4104, 4071, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- OA 8-folding and stacking [i] based on linear OA(6433, 4104, F64, 16) (dual of [4104, 4071, 17]-code), using
- net defined by OOA [i] based on linear OOA(6433, 513, F64, 16, 16) (dual of [(513, 16), 8175, 17]-NRT-code), using
(18, 18+16, 515)-Net in Base 64 — Constructive
(18, 34, 515)-net in base 64, using
- (u, u+v)-construction [i] based on
- (3, 11, 257)-net in base 64, using
- 1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
- base change [i] based on digital (0, 9, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 9, 257)-net over F256, using
- 1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
- (7, 23, 258)-net in base 64, using
- 1 times m-reduction [i] based on (7, 24, 258)-net in base 64, using
- base change [i] based on digital (1, 18, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 18, 258)-net over F256, using
- 1 times m-reduction [i] based on (7, 24, 258)-net in base 64, using
- (3, 11, 257)-net in base 64, using
(18, 18+16, 2053)-Net over F64 — Digital
Digital (18, 34, 2053)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6434, 2053, F64, 2, 16) (dual of [(2053, 2), 4072, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6434, 4106, F64, 16) (dual of [4106, 4072, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(6434, 4107, F64, 16) (dual of [4107, 4073, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(6434, 4107, F64, 16) (dual of [4107, 4073, 17]-code), using
- OOA 2-folding [i] based on linear OA(6434, 4106, F64, 16) (dual of [4106, 4072, 17]-code), using
(18, 18+16, 2835396)-Net in Base 64 — Upper bound on s
There is no (18, 34, 2835397)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 25 711020 856547 877009 162981 964585 329978 535043 774405 427321 903448 > 6434 [i]