Best Known (61−22, 61, s)-Nets in Base 16
(61−22, 61, 579)-Net over F16 — Constructive and digital
Digital (39, 61, 579)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 17, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- digital (22, 44, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 22, 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, 22, 257)-net over F256, using
- digital (6, 17, 65)-net over F16, using
(61−22, 61, 2257)-Net over F16 — Digital
Digital (39, 61, 2257)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1661, 2257, F16, 22) (dual of [2257, 2196, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(1661, 4096, F16, 22) (dual of [4096, 4035, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(1661, 4096, F16, 22) (dual of [4096, 4035, 23]-code), using
(61−22, 61, 1557090)-Net in Base 16 — Upper bound on s
There is no (39, 61, 1557091)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 28 269599 839453 558312 136760 676561 814762 308454 568726 457651 738334 991128 538016 > 1661 [i]