Best Known (35, 35+18, s)-Nets in Base 16
(35, 35+18, 771)-Net over F16 — Constructive and digital
Digital (35, 53, 771)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (8, 17, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(8,256) in PG(16,16)) for nets [i] based on digital (0, 9, 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(8,256) in PG(16,16)) for nets [i] based on digital (0, 9, 257)-net over F256, using
- digital (18, 36, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 18, 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, 18, 257)-net over F256, using
- digital (8, 17, 257)-net over F16, using
(35, 35+18, 3706)-Net over F16 — Digital
Digital (35, 53, 3706)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1653, 3706, F16, 18) (dual of [3706, 3653, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(1653, 4112, F16, 18) (dual of [4112, 4059, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(1649, 4096, F16, 18) (dual of [4096, 4047, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(164, 16, F16, 4) (dual of [16, 12, 5]-code or 16-arc in PG(3,16)), using
- Reed–Solomon code RS(12,16) [i]
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(1653, 4112, F16, 18) (dual of [4112, 4059, 19]-code), using
(35, 35+18, 3408696)-Net in Base 16 — Upper bound on s
There is no (35, 53, 3408697)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 6582 019412 183930 114399 151550 666661 316404 848443 724535 825600 564896 > 1653 [i]