Best Known (83−36, 83, s)-Nets in Base 64
(83−36, 83, 593)-Net over F64 — Constructive and digital
Digital (47, 83, 593)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 19, 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, 64, 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, 19, 80)-net over F64, using
(83−36, 83, 910)-Net in Base 64 — Constructive
(47, 83, 910)-net in base 64, using
- net defined by OOA [i] based on OOA(6483, 910, S64, 36, 36), using
- OA 18-folding and stacking [i] based on OA(6483, 16380, S64, 36), using
- discarding factors based on OA(6483, 16386, S64, 36), using
- discarding parts of the base [i] based on linear OA(12871, 16386, F128, 36) (dual of [16386, 16315, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(34) [i] based on
- linear OA(12871, 16384, F128, 36) (dual of [16384, 16313, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- 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(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(35) ⊂ Ce(34) [i] based on
- discarding parts of the base [i] based on linear OA(12871, 16386, F128, 36) (dual of [16386, 16315, 37]-code), using
- discarding factors based on OA(6483, 16386, S64, 36), using
- OA 18-folding and stacking [i] based on OA(6483, 16380, S64, 36), using
(83−36, 83, 4529)-Net over F64 — Digital
Digital (47, 83, 4529)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6483, 4529, F64, 36) (dual of [4529, 4446, 37]-code), using
- 419 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 4 times 0, 1, 13 times 0, 1, 26 times 0, 1, 55 times 0, 1, 108 times 0, 1, 204 times 0) [i] based on linear OA(6471, 4098, F64, 36) (dual of [4098, 4027, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(34) [i] based on
- linear OA(6471, 4096, F64, 36) (dual of [4096, 4025, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- 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(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(35) ⊂ Ce(34) [i] based on
- 419 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 4 times 0, 1, 13 times 0, 1, 26 times 0, 1, 55 times 0, 1, 108 times 0, 1, 204 times 0) [i] based on linear OA(6471, 4098, F64, 36) (dual of [4098, 4027, 37]-code), using
(83−36, 83, large)-Net in Base 64 — Upper bound on s
There is no (47, 83, large)-net in base 64, because
- 34 times m-reduction [i] would yield (47, 49, large)-net in base 64, but