Best Known (69, 89, s)-Nets in Base 16
(69, 89, 13131)-Net over F16 — Constructive and digital
Digital (69, 89, 13131)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- digital (58, 78, 13107)-net over F16, using
- net defined by OOA [i] based on linear OOA(1678, 13107, F16, 20, 20) (dual of [(13107, 20), 262062, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(1678, 131070, F16, 20) (dual of [131070, 130992, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1678, 131072, F16, 20) (dual of [131072, 130994, 21]-code), using
- trace code [i] based on linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- trace code [i] based on linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1678, 131072, F16, 20) (dual of [131072, 130994, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(1678, 131070, F16, 20) (dual of [131070, 130992, 21]-code), using
- net defined by OOA [i] based on linear OOA(1678, 13107, F16, 20, 20) (dual of [(13107, 20), 262062, 21]-NRT-code), using
- digital (1, 11, 24)-net over F16, using
(69, 89, 26215)-Net in Base 16 — Constructive
(69, 89, 26215)-net in base 16, using
- net defined by OOA [i] based on OOA(1689, 26215, S16, 20, 20), using
- OA 10-folding and stacking [i] based on OA(1689, 262150, S16, 20), using
- discarding factors based on OA(1689, 262151, S16, 20), using
- discarding parts of the base [i] based on linear OA(6459, 262151, F64, 20) (dual of [262151, 262092, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(6452, 262144, F64, 18) (dual of [262144, 262092, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- discarding parts of the base [i] based on linear OA(6459, 262151, F64, 20) (dual of [262151, 262092, 21]-code), using
- discarding factors based on OA(1689, 262151, S16, 20), using
- OA 10-folding and stacking [i] based on OA(1689, 262150, S16, 20), using
(69, 89, 230937)-Net over F16 — Digital
Digital (69, 89, 230937)-net over F16, using
(69, 89, large)-Net in Base 16 — Upper bound on s
There is no (69, 89, large)-net in base 16, because
- 18 times m-reduction [i] would yield (69, 71, large)-net in base 16, but