Best Known (27, 40, s)-Nets in Base 16
(27, 40, 1030)-Net over F16 — Constructive and digital
Digital (27, 40, 1030)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 12, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 6, 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, 6, 257)-net over F256, using
- digital (15, 28, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 14, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 14, 258)-net over F256, using
- digital (6, 12, 514)-net over F16, using
(27, 40, 4254)-Net over F16 — Digital
Digital (27, 40, 4254)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1640, 4254, F16, 13) (dual of [4254, 4214, 14]-code), using
- 152 step Varšamov–Edel lengthening with (ri) = (2, 13 times 0, 1, 137 times 0) [i] based on linear OA(1637, 4099, F16, 13) (dual of [4099, 4062, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(1637, 4096, F16, 13) (dual of [4096, 4059, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1634, 4096, F16, 12) (dual of [4096, 4062, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- 152 step Varšamov–Edel lengthening with (ri) = (2, 13 times 0, 1, 137 times 0) [i] based on linear OA(1637, 4099, F16, 13) (dual of [4099, 4062, 14]-code), using
(27, 40, large)-Net in Base 16 — Upper bound on s
There is no (27, 40, large)-net in base 16, because
- 11 times m-reduction [i] would yield (27, 29, large)-net in base 16, but