Best Known (33−16, 33, s)-Nets in Base 64
(33−16, 33, 513)-Net over F64 — Constructive and digital
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
(33−16, 33, 514)-Net in Base 64 — Constructive
(17, 33, 514)-net in base 64, using
- 1 times m-reduction [i] based on (17, 34, 514)-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
- (6, 23, 257)-net in base 64, using
- 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- base change [i] based on digital (0, 18, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- base change [i] based on digital (0, 18, 257)-net over F256, using
- 1 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- (3, 11, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(33−16, 33, 1819)-Net over F64 — Digital
Digital (17, 33, 1819)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6433, 1819, F64, 2, 16) (dual of [(1819, 2), 3605, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6433, 2052, F64, 2, 16) (dual of [(2052, 2), 4071, 17]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(6433, 4104, F64, 16) (dual of [4104, 4071, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(6433, 2052, F64, 2, 16) (dual of [(2052, 2), 4071, 17]-NRT-code), using
(33−16, 33, 1685935)-Net in Base 64 — Upper bound on s
There is no (17, 33, 1685936)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 401735 324689 581869 497779 169109 772219 853771 463969 678979 001735 > 6433 [i]