Best Known (128−29, 128, s)-Nets in Base 16
(128−29, 128, 9379)-Net over F16 — Constructive and digital
Digital (99, 128, 9379)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 14, 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 (85, 114, 9362)-net over F16, using
- net defined by OOA [i] based on linear OOA(16114, 9362, F16, 29, 29) (dual of [(9362, 29), 271384, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16114, 131069, F16, 29) (dual of [131069, 130955, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16114, 131074, F16, 29) (dual of [131074, 130960, 30]-code), using
- trace code [i] based on linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- trace code [i] based on linear OA(25657, 65537, F256, 29) (dual of [65537, 65480, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(16114, 131074, F16, 29) (dual of [131074, 130960, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(16114, 131069, F16, 29) (dual of [131069, 130955, 30]-code), using
- net defined by OOA [i] based on linear OOA(16114, 9362, F16, 29, 29) (dual of [(9362, 29), 271384, 30]-NRT-code), using
- digital (0, 14, 17)-net over F16, using
(128−29, 128, 18724)-Net in Base 16 — Constructive
(99, 128, 18724)-net in base 16, using
- net defined by OOA [i] based on OOA(16128, 18724, S16, 29, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(16128, 262137, S16, 29), using
- discarding factors based on OA(16128, 262147, S16, 29), using
- discarding parts of the base [i] based on linear OA(6485, 262147, F64, 29) (dual of [262147, 262062, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(6485, 262144, F64, 29) (dual of [262144, 262059, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(6482, 262144, F64, 28) (dual of [262144, 262062, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(28) ⊂ Ce(27) [i] based on
- discarding parts of the base [i] based on linear OA(6485, 262147, F64, 29) (dual of [262147, 262062, 30]-code), using
- discarding factors based on OA(16128, 262147, S16, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(16128, 262137, S16, 29), using
(128−29, 128, 240706)-Net over F16 — Digital
Digital (99, 128, 240706)-net over F16, using
(128−29, 128, large)-Net in Base 16 — Upper bound on s
There is no (99, 128, large)-net in base 16, because
- 27 times m-reduction [i] would yield (99, 101, large)-net in base 16, but