Best Known (68−23, 68, s)-Nets in Base 16
(68−23, 68, 1028)-Net over F16 — Constructive and digital
Digital (45, 68, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (11, 22, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 11, 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, 11, 257)-net over F256, using
- digital (23, 46, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 23, 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, 23, 257)-net over F256, using
- digital (11, 22, 514)-net over F16, using
(68−23, 68, 4008)-Net over F16 — Digital
Digital (45, 68, 4008)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1668, 4008, F16, 23) (dual of [4008, 3940, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(1668, 4104, F16, 23) (dual of [4104, 4036, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(1667, 4097, F16, 23) (dual of [4097, 4030, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(1661, 4097, F16, 21) (dual of [4097, 4036, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(161, 7, F16, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(1668, 4104, F16, 23) (dual of [4104, 4036, 24]-code), using
(68−23, 68, 7064930)-Net in Base 16 — Upper bound on s
There is no (45, 68, 7064931)-net in base 16, because
- 1 times m-reduction [i] would yield (45, 67, 7064931)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 474 284770 534347 134351 261011 553087 457240 326859 922650 970359 560847 029749 461454 611616 > 1667 [i]