Best Known (74−31, 74, s)-Nets in Base 64
(74−31, 74, 578)-Net over F64 — Constructive and digital
Digital (43, 74, 578)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 15, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (28, 59, 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 (0, 15, 65)-net over F64, using
(74−31, 74, 1092)-Net in Base 64 — Constructive
(43, 74, 1092)-net in base 64, using
- 642 times duplication [i] based on (41, 72, 1092)-net in base 64, using
- net defined by OOA [i] based on OOA(6472, 1092, S64, 31, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(6472, 16381, S64, 31), using
- discarding factors based on OA(6472, 16386, S64, 31), using
- discarding parts of the base [i] based on linear OA(12861, 16386, F128, 31) (dual of [16386, 16325, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- linear OA(12861, 16384, F128, 31) (dual of [16384, 16323, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(12859, 16384, F128, 30) (dual of [16384, 16325, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(30) ⊂ Ce(29) [i] based on
- discarding parts of the base [i] based on linear OA(12861, 16386, F128, 31) (dual of [16386, 16325, 32]-code), using
- discarding factors based on OA(6472, 16386, S64, 31), using
- OOA 15-folding and stacking with additional row [i] based on OA(6472, 16381, S64, 31), using
- net defined by OOA [i] based on OOA(6472, 1092, S64, 31, 31), using
(74−31, 74, 5486)-Net over F64 — Digital
Digital (43, 74, 5486)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6474, 5486, F64, 31) (dual of [5486, 5412, 32]-code), using
- 1375 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 6 times 0, 1, 17 times 0, 1, 41 times 0, 1, 93 times 0, 1, 201 times 0, 1, 389 times 0, 1, 618 times 0) [i] based on linear OA(6461, 4098, F64, 31) (dual of [4098, 4037, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- linear OA(6461, 4096, F64, 31) (dual of [4096, 4035, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(6459, 4096, F64, 30) (dual of [4096, 4037, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(30) ⊂ Ce(29) [i] based on
- 1375 step Varšamov–Edel lengthening with (ri) = (5, 1, 0, 1, 6 times 0, 1, 17 times 0, 1, 41 times 0, 1, 93 times 0, 1, 201 times 0, 1, 389 times 0, 1, 618 times 0) [i] based on linear OA(6461, 4098, F64, 31) (dual of [4098, 4037, 32]-code), using
(74−31, 74, large)-Net in Base 64 — Upper bound on s
There is no (43, 74, large)-net in base 64, because
- 29 times m-reduction [i] would yield (43, 45, large)-net in base 64, but