Best Known (116−40, 116, s)-Nets in Base 16
(116−40, 116, 596)-Net over F16 — Constructive and digital
Digital (76, 116, 596)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (16, 36, 82)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 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 (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 (0, 10, 17)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (40, 80, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 40, 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, 40, 257)-net over F256, using
- digital (16, 36, 82)-net over F16, using
(116−40, 116, 643)-Net in Base 16 — Constructive
(76, 116, 643)-net in base 16, using
- 1 times m-reduction [i] based on (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
- (u, u+v)-construction [i] based on
(116−40, 116, 4161)-Net over F16 — Digital
Digital (76, 116, 4161)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16116, 4161, F16, 40) (dual of [4161, 4045, 41]-code), using
- 58 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 43 times 0) [i] based on linear OA(16112, 4099, F16, 40) (dual of [4099, 3987, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(38) [i] based on
- linear OA(16112, 4096, F16, 40) (dual of [4096, 3984, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(16109, 4096, F16, 39) (dual of [4096, 3987, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(39) ⊂ Ce(38) [i] based on
- 58 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 43 times 0) [i] based on linear OA(16112, 4099, F16, 40) (dual of [4099, 3987, 41]-code), using
(116−40, 116, 5334699)-Net in Base 16 — Upper bound on s
There is no (76, 116, 5334700)-net in base 16, because
- 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]