Best Known (90−30, 90, s)-Nets in Base 16
(90−30, 90, 1028)-Net over F16 — Constructive and digital
Digital (60, 90, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (15, 30, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
- digital (30, 60, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 30, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 30, 257)-net over F256, using
- digital (15, 30, 514)-net over F16, using
(90−30, 90, 4357)-Net over F16 — Digital
Digital (60, 90, 4357)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1690, 4357, F16, 30) (dual of [4357, 4267, 31]-code), using
- 253 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 14 times 0, 1, 57 times 0, 1, 175 times 0) [i] based on linear OA(1685, 4099, F16, 30) (dual of [4099, 4014, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(1685, 4096, F16, 30) (dual of [4096, 4011, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(1682, 4096, F16, 29) (dual of [4096, 4014, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(29) ⊂ Ce(28) [i] based on
- 253 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 14 times 0, 1, 57 times 0, 1, 175 times 0) [i] based on linear OA(1685, 4099, F16, 30) (dual of [4099, 4014, 31]-code), using
(90−30, 90, 7184470)-Net in Base 16 — Upper bound on s
There is no (60, 90, 7184471)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 2 348542 750522 582172 906993 669682 196940 866090 735531 940598 155047 808717 472605 367010 918099 279633 403692 581846 948976 > 1690 [i]