Best Known (117−41, 117, s)-Nets in Base 16
(117−41, 117, 587)-Net over F16 — Constructive and digital
Digital (76, 117, 587)-net over F16, using
- 161 times duplication [i] based on digital (75, 116, 587)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 26, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- digital (49, 90, 522)-net over F16, using
- trace code for nets [i] based on digital (4, 45, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- trace code for nets [i] based on digital (4, 45, 261)-net over F256, using
- digital (6, 26, 65)-net over F16, using
- (u, u+v)-construction [i] based on
(117−41, 117, 643)-Net in Base 16 — Constructive
(76, 117, 643)-net in base 16, using
- (u, u+v)-construction [i] based on
- (15, 35, 129)-net in base 16, using
- base change [i] based on digital (0, 20, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 20, 129)-net over F128, using
- digital (41, 82, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 41, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 41, 257)-net over F256, using
- (15, 35, 129)-net in base 16, using
(117−41, 117, 3895)-Net over F16 — Digital
Digital (76, 117, 3895)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16117, 3895, F16, 41) (dual of [3895, 3778, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(16117, 4107, F16, 41) (dual of [4107, 3990, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(37) [i] based on
- linear OA(16115, 4096, F16, 41) (dual of [4096, 3981, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(16106, 4096, F16, 38) (dual of [4096, 3990, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(162, 11, F16, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,16)), using
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- Reed–Solomon code RS(14,16) [i]
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- construction X applied to Ce(40) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(16117, 4107, F16, 41) (dual of [4107, 3990, 42]-code), using
(117−41, 117, 5334699)-Net in Base 16 — Upper bound on s
There is no (76, 117, 5334700)-net in base 16, because
- 1 times m-reduction [i] would yield (76, 116, 5334700)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 47 634129 796249 379030 520188 235163 812845 922422 629763 425000 639916 342272 118149 902354 488855 056266 753605 148105 368694 778585 010472 138889 011999 106876 > 16116 [i]