Best Known (69, 69+34, s)-Nets in Base 16
(69, 69+34, 1028)-Net over F16 — Constructive and digital
Digital (69, 103, 1028)-net over F16, using
- 1 times m-reduction [i] based on digital (69, 104, 1028)-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 (35, 70, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 35, 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, 35, 257)-net over F256, using
- digital (17, 34, 514)-net over F16, using
- (u, u+v)-construction [i] based on
(69, 69+34, 5052)-Net over F16 — Digital
Digital (69, 103, 5052)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16103, 5052, F16, 34) (dual of [5052, 4949, 35]-code), using
- 944 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 11 times 0, 1, 30 times 0, 1, 76 times 0, 1, 161 times 0, 1, 280 times 0, 1, 377 times 0) [i] based on linear OA(1694, 4099, F16, 34) (dual of [4099, 4005, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- 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(1691, 4096, F16, 33) (dual of [4096, 4005, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(33) ⊂ Ce(32) [i] based on
- 944 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 11 times 0, 1, 30 times 0, 1, 76 times 0, 1, 161 times 0, 1, 280 times 0, 1, 377 times 0) [i] based on linear OA(1694, 4099, F16, 34) (dual of [4099, 4005, 35]-code), using
(69, 69+34, large)-Net in Base 16 — Upper bound on s
There is no (69, 103, large)-net in base 16, because
- 32 times m-reduction [i] would yield (69, 71, large)-net in base 16, but