Best Known (20, 20+10, s)-Nets in Base 16
(20, 20+10, 1028)-Net over F16 — Constructive and digital
Digital (20, 30, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (5, 10, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 5, 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, 5, 257)-net over F256, using
- digital (10, 20, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 10, 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, 10, 257)-net over F256, using
- digital (5, 10, 514)-net over F16, using
(20, 20+10, 4120)-Net over F16 — Digital
Digital (20, 30, 4120)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1630, 4120, F16, 10) (dual of [4120, 4090, 11]-code), using
- 16 step Varšamov–Edel lengthening with (ri) = (1, 15 times 0) [i] based on linear OA(1629, 4103, F16, 10) (dual of [4103, 4074, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(1628, 4096, F16, 10) (dual of [4096, 4068, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1622, 4096, F16, 8) (dual of [4096, 4074, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- 16 step Varšamov–Edel lengthening with (ri) = (1, 15 times 0) [i] based on linear OA(1629, 4103, F16, 10) (dual of [4103, 4074, 11]-code), using
(20, 20+10, 2913832)-Net in Base 16 — Upper bound on s
There is no (20, 30, 2913833)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 1 329229 819749 255321 824241 899120 355976 > 1630 [i]