Best Known (58, 58+28, s)-Nets in Base 16
(58, 58+28, 1030)-Net over F16 — Constructive and digital
Digital (58, 86, 1030)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (14, 28, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 257)-net over F256, using
- digital (30, 58, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 29, 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, 29, 258)-net over F256, using
- digital (14, 28, 514)-net over F16, using
(58, 58+28, 5004)-Net over F16 — Digital
Digital (58, 86, 5004)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1686, 5004, F16, 28) (dual of [5004, 4918, 29]-code), using
- 898 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 7 times 0, 1, 33 times 0, 1, 117 times 0, 1, 288 times 0, 1, 446 times 0) [i] based on linear OA(1679, 4099, F16, 28) (dual of [4099, 4020, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(26) [i] based on
- linear OA(1679, 4096, F16, 28) (dual of [4096, 4017, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(1676, 4096, F16, 27) (dual of [4096, 4020, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(27) ⊂ Ce(26) [i] based on
- 898 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 7 times 0, 1, 33 times 0, 1, 117 times 0, 1, 288 times 0, 1, 446 times 0) [i] based on linear OA(1679, 4099, F16, 28) (dual of [4099, 4020, 29]-code), using
(58, 58+28, large)-Net in Base 16 — Upper bound on s
There is no (58, 86, large)-net in base 16, because
- 26 times m-reduction [i] would yield (58, 60, large)-net in base 16, but