Best Known (27, 27+14, s)-Nets in Base 16
(27, 27+14, 771)-Net over F16 — Constructive and digital
Digital (27, 41, 771)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) for nets [i] based on digital (0, 7, 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
- base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) for nets [i] based on digital (0, 7, 257)-net over F256, using
- digital (14, 28, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 257)-net over F256, using
- digital (6, 13, 257)-net over F16, using
(27, 27+14, 3633)-Net over F16 — Digital
Digital (27, 41, 3633)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1641, 3633, F16, 14) (dual of [3633, 3592, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1641, 4103, F16, 14) (dual of [4103, 4062, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(1640, 4096, F16, 14) (dual of [4096, 4056, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(161, 7, F16, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(1641, 4103, F16, 14) (dual of [4103, 4062, 15]-code), using
(27, 27+14, 2544072)-Net in Base 16 — Upper bound on s
There is no (27, 41, 2544073)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 23 384032 261058 092373 977877 232173 754650 604366 993016 > 1641 [i]