Best Known (31, 31+16, s)-Nets in Base 16
(31, 31+16, 771)-Net over F16 — Constructive and digital
Digital (31, 47, 771)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (7, 15, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(7,256) in PG(14,16)) for nets [i] based on digital (0, 8, 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(7,256) in PG(14,16)) for nets [i] based on digital (0, 8, 257)-net over F256, using
- digital (16, 32, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 16, 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, 16, 257)-net over F256, using
- digital (7, 15, 257)-net over F16, using
(31, 31+16, 3639)-Net over F16 — Digital
Digital (31, 47, 3639)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1647, 3639, F16, 16) (dual of [3639, 3592, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(1647, 4103, F16, 16) (dual of [4103, 4056, 17]-code), using
- 1 times truncation [i] based on linear OA(1648, 4104, F16, 17) (dual of [4104, 4056, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(1646, 4096, F16, 17) (dual of [4096, 4050, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(162, 8, F16, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,16)), using
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- Reed–Solomon code RS(14,16) [i]
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- 1 times truncation [i] based on linear OA(1648, 4104, F16, 17) (dual of [4104, 4056, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(1647, 4103, F16, 16) (dual of [4103, 4056, 17]-code), using
(31, 31+16, 2977166)-Net in Base 16 — Upper bound on s
There is no (31, 47, 2977167)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 392 319439 158741 481740 518267 603922 547623 887486 114837 883691 > 1647 [i]