Best Known (73, 73+36, s)-Nets in Base 16
(73, 73+36, 1028)-Net over F16 — Constructive and digital
Digital (73, 109, 1028)-net over F16, using
- 1 times m-reduction [i] based on digital (73, 110, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (18, 36, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 18, 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, 18, 257)-net over F256, using
- digital (37, 74, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 37, 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, 37, 257)-net over F256, using
- digital (18, 36, 514)-net over F16, using
- (u, u+v)-construction [i] based on
(73, 73+36, 5232)-Net over F16 — Digital
Digital (73, 109, 5232)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16109, 5232, F16, 36) (dual of [5232, 5123, 37]-code), using
- 1124 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 14 times 0, 1, 46 times 0, 1, 117 times 0, 1, 224 times 0, 1, 326 times 0, 1, 387 times 0) [i] based on linear OA(16100, 4099, F16, 36) (dual of [4099, 3999, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(34) [i] based on
- linear OA(16100, 4096, F16, 36) (dual of [4096, 3996, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- 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(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(35) ⊂ Ce(34) [i] based on
- 1124 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 14 times 0, 1, 46 times 0, 1, 117 times 0, 1, 224 times 0, 1, 326 times 0, 1, 387 times 0) [i] based on linear OA(16100, 4099, F16, 36) (dual of [4099, 3999, 37]-code), using
(73, 73+36, large)-Net in Base 16 — Upper bound on s
There is no (73, 109, large)-net in base 16, because
- 34 times m-reduction [i] would yield (73, 75, large)-net in base 16, but