Best Known (106−35, 106, s)-Nets in Base 16
(106−35, 106, 1030)-Net over F16 — Constructive and digital
Digital (71, 106, 1030)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (17, 34, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 17, 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, 17, 257)-net over F256, using
- digital (37, 72, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 36, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 36, 258)-net over F256, using
- digital (17, 34, 514)-net over F16, using
(106−35, 106, 5141)-Net over F16 — Digital
Digital (71, 106, 5141)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16106, 5141, F16, 35) (dual of [5141, 5035, 36]-code), using
- 1033 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 12 times 0, 1, 38 times 0, 1, 94 times 0, 1, 191 times 0, 1, 305 times 0, 1, 383 times 0) [i] based on linear OA(1697, 4099, F16, 35) (dual of [4099, 4002, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- linear OA(1697, 4096, F16, 35) (dual of [4096, 3999, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(1694, 4096, F16, 34) (dual of [4096, 4002, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- 1033 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 12 times 0, 1, 38 times 0, 1, 94 times 0, 1, 191 times 0, 1, 305 times 0, 1, 383 times 0) [i] based on linear OA(1697, 4099, F16, 35) (dual of [4099, 4002, 36]-code), using
(106−35, 106, large)-Net in Base 16 — Upper bound on s
There is no (71, 106, large)-net in base 16, because
- 33 times m-reduction [i] would yield (71, 73, large)-net in base 16, but