Best Known (78, 78+23, s)-Nets in Base 16
(78, 78+23, 11932)-Net over F16 — Constructive and digital
Digital (78, 101, 11932)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- digital (67, 90, 11915)-net over F16, using
- net defined by OOA [i] based on linear OOA(1690, 11915, F16, 23, 23) (dual of [(11915, 23), 273955, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(1690, 131066, F16, 23) (dual of [131066, 130976, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(1690, 131074, F16, 23) (dual of [131074, 130984, 24]-code), using
- trace code [i] based on linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- trace code [i] based on linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(1690, 131074, F16, 23) (dual of [131074, 130984, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(1690, 131066, F16, 23) (dual of [131066, 130976, 24]-code), using
- net defined by OOA [i] based on linear OOA(1690, 11915, F16, 23, 23) (dual of [(11915, 23), 273955, 24]-NRT-code), using
- digital (0, 11, 17)-net over F16, using
(78, 78+23, 23831)-Net in Base 16 — Constructive
(78, 101, 23831)-net in base 16, using
- net defined by OOA [i] based on OOA(16101, 23831, S16, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(16101, 262142, S16, 23), using
- discarding factors based on OA(16101, 262147, S16, 23), using
- discarding parts of the base [i] based on linear OA(6467, 262147, F64, 23) (dual of [262147, 262080, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(6467, 262144, F64, 23) (dual of [262144, 262077, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 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(22) ⊂ Ce(21) [i] based on
- discarding parts of the base [i] based on linear OA(6467, 262147, F64, 23) (dual of [262147, 262080, 24]-code), using
- discarding factors based on OA(16101, 262147, S16, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(16101, 262142, S16, 23), using
(78, 78+23, 203606)-Net over F16 — Digital
Digital (78, 101, 203606)-net over F16, using
(78, 78+23, large)-Net in Base 16 — Upper bound on s
There is no (78, 101, large)-net in base 16, because
- 21 times m-reduction [i] would yield (78, 80, large)-net in base 16, but