Best Known (83, 83+22, s)-Nets in Base 4
(83, 83+22, 1046)-Net over F4 — Constructive and digital
Digital (83, 105, 1046)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (6, 17, 18)-net over F4, using
- 1 times m-reduction [i] based on digital (6, 18, 18)-net over F4, using
- digital (66, 88, 1028)-net over F4, 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, 18)-net over F4, using
(83, 83+22, 3725)-Net over F4 — Digital
Digital (83, 105, 3725)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4105, 3725, F4, 22) (dual of [3725, 3620, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4105, 4128, F4, 22) (dual of [4128, 4023, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4104, 4127, F4, 22) (dual of [4127, 4023, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(497, 4096, F4, 22) (dual of [4096, 3999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(473, 4096, F4, 17) (dual of [4096, 4023, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(47, 31, F4, 4) (dual of [31, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4104, 4127, F4, 22) (dual of [4127, 4023, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4105, 4128, F4, 22) (dual of [4128, 4023, 23]-code), using
(83, 83+22, 913745)-Net in Base 4 — Upper bound on s
There is no (83, 105, 913746)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1645 522296 965282 607561 668717 825024 948943 500446 084933 313775 571624 > 4105 [i]