Best Known (81−35, 81, s)-Nets in Base 64
(81−35, 81, 593)-Net over F64 — Constructive and digital
Digital (46, 81, 593)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 18, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (28, 63, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- digital (1, 18, 80)-net over F64, using
(81−35, 81, 963)-Net in Base 64 — Constructive
(46, 81, 963)-net in base 64, using
- net defined by OOA [i] based on OOA(6481, 963, S64, 35, 35), using
- OOA 17-folding and stacking with additional row [i] based on OA(6481, 16372, S64, 35), using
- discarding factors based on OA(6481, 16386, S64, 35), using
- discarding parts of the base [i] based on linear OA(12869, 16386, F128, 35) (dual of [16386, 16317, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- linear OA(12869, 16384, F128, 35) (dual of [16384, 16315, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(12867, 16384, F128, 34) (dual of [16384, 16317, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- discarding parts of the base [i] based on linear OA(12869, 16386, F128, 35) (dual of [16386, 16317, 36]-code), using
- discarding factors based on OA(6481, 16386, S64, 35), using
- OOA 17-folding and stacking with additional row [i] based on OA(6481, 16372, S64, 35), using
(81−35, 81, 4582)-Net over F64 — Digital
Digital (46, 81, 4582)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6481, 4582, F64, 35) (dual of [4582, 4501, 36]-code), using
- 472 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 5 times 0, 1, 13 times 0, 1, 31 times 0, 1, 62 times 0, 1, 122 times 0, 1, 230 times 0) [i] based on linear OA(6469, 4098, F64, 35) (dual of [4098, 4029, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- linear OA(6469, 4096, F64, 35) (dual of [4096, 4027, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(6467, 4096, F64, 34) (dual of [4096, 4029, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- 472 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 5 times 0, 1, 13 times 0, 1, 31 times 0, 1, 62 times 0, 1, 122 times 0, 1, 230 times 0) [i] based on linear OA(6469, 4098, F64, 35) (dual of [4098, 4029, 36]-code), using
(81−35, 81, large)-Net in Base 64 — Upper bound on s
There is no (46, 81, large)-net in base 64, because
- 33 times m-reduction [i] would yield (46, 48, large)-net in base 64, but