Best Known (71−24, 71, s)-Nets in Base 16
(71−24, 71, 771)-Net over F16 — Constructive and digital
Digital (47, 71, 771)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (11, 23, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(11,256) in PG(22,16)) for nets [i] based on digital (0, 12, 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(11,256) in PG(22,16)) for nets [i] based on digital (0, 12, 257)-net over F256, using
- digital (24, 48, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- digital (11, 23, 257)-net over F16, using
(71−24, 71, 4082)-Net over F16 — Digital
Digital (47, 71, 4082)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1671, 4082, F16, 24) (dual of [4082, 4011, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(1671, 4113, F16, 24) (dual of [4113, 4042, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- linear OA(1667, 4096, F16, 24) (dual of [4096, 4029, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(1652, 4096, F16, 19) (dual of [4096, 4044, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(164, 17, F16, 4) (dual of [17, 13, 5]-code or 17-arc in PG(3,16)), using
- extended Reed–Solomon code RSe(13,16) [i]
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(1671, 4113, F16, 24) (dual of [4113, 4042, 25]-code), using
(71−24, 71, 4695113)-Net in Base 16 — Upper bound on s
There is no (47, 71, 4695114)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 31 082707 190643 568897 300909 733240 551283 630071 375529 187591 422977 672945 580235 814612 802896 > 1671 [i]