Best Known (59, 59+29, s)-Nets in Base 16
(59, 59+29, 1030)-Net over F16 — Constructive and digital
Digital (59, 88, 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 (31, 60, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 30, 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, 30, 258)-net over F256, using
- digital (14, 28, 514)-net over F16, using
(59, 59+29, 4626)-Net over F16 — Digital
Digital (59, 88, 4626)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1688, 4626, F16, 29) (dual of [4626, 4538, 30]-code), using
- 521 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 43 times 0, 1, 144 times 0, 1, 317 times 0) [i] based on linear OA(1682, 4099, F16, 29) (dual of [4099, 4017, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(1682, 4096, F16, 29) (dual of [4096, 4014, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(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(28) ⊂ Ce(27) [i] based on
- 521 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 43 times 0, 1, 144 times 0, 1, 317 times 0) [i] based on linear OA(1682, 4099, F16, 29) (dual of [4099, 4017, 30]-code), using
(59, 59+29, large)-Net in Base 16 — Upper bound on s
There is no (59, 88, large)-net in base 16, because
- 27 times m-reduction [i] would yield (59, 61, large)-net in base 16, but