Best Known (61, 61+18, s)-Nets in Base 16
(61, 61+18, 14581)-Net over F16 — Constructive and digital
Digital (61, 79, 14581)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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 (52, 70, 14564)-net over F16, using
- net defined by OOA [i] based on linear OOA(1670, 14564, F16, 18, 18) (dual of [(14564, 18), 262082, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(1670, 131076, F16, 18) (dual of [131076, 131006, 19]-code), using
- trace code [i] based on linear OA(25635, 65538, F256, 18) (dual of [65538, 65503, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(25633, 65536, F256, 17) (dual of [65536, 65503, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- trace code [i] based on linear OA(25635, 65538, F256, 18) (dual of [65538, 65503, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(1670, 131076, F16, 18) (dual of [131076, 131006, 19]-code), using
- net defined by OOA [i] based on linear OOA(1670, 14564, F16, 18, 18) (dual of [(14564, 18), 262082, 19]-NRT-code), using
- digital (0, 9, 17)-net over F16, using
(61, 61+18, 29127)-Net in Base 16 — Constructive
(61, 79, 29127)-net in base 16, using
- 161 times duplication [i] based on (60, 78, 29127)-net in base 16, using
- base change [i] based on digital (34, 52, 29127)-net over F64, using
- net defined by OOA [i] based on linear OOA(6452, 29127, F64, 18, 18) (dual of [(29127, 18), 524234, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6452, 262143, F64, 18) (dual of [262143, 262091, 19]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(6452, 262144, F64, 18) (dual of [262144, 262092, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(6452, 262143, F64, 18) (dual of [262143, 262091, 19]-code), using
- net defined by OOA [i] based on linear OOA(6452, 29127, F64, 18, 18) (dual of [(29127, 18), 524234, 19]-NRT-code), using
- base change [i] based on digital (34, 52, 29127)-net over F64, using
(61, 61+18, 188577)-Net over F16 — Digital
Digital (61, 79, 188577)-net over F16, using
(61, 61+18, large)-Net in Base 16 — Upper bound on s
There is no (61, 79, large)-net in base 16, because
- 16 times m-reduction [i] would yield (61, 63, large)-net in base 16, but