Best Known (233−35, 233, s)-Nets in Base 4
(233−35, 233, 3876)-Net over F4 — Constructive and digital
Digital (198, 233, 3876)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 24, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (174, 209, 3855)-net over F4, using
- net defined by OOA [i] based on linear OOA(4209, 3855, F4, 35, 35) (dual of [(3855, 35), 134716, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using
- an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- OOA 17-folding and stacking with additional row [i] based on linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using
- net defined by OOA [i] based on linear OOA(4209, 3855, F4, 35, 35) (dual of [(3855, 35), 134716, 36]-NRT-code), using
- digital (7, 24, 21)-net over F4, using
(233−35, 233, 65628)-Net over F4 — Digital
Digital (198, 233, 65628)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4233, 65628, F4, 35) (dual of [65628, 65395, 36]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4230, 65622, F4, 35) (dual of [65622, 65392, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,12]) [i] based on
- linear OA(4209, 65537, F4, 35) (dual of [65537, 65328, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(4145, 65537, F4, 25) (dual of [65537, 65392, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(421, 85, F4, 9) (dual of [85, 64, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(421, 86, F4, 9) (dual of [86, 65, 10]-code), using
- construction X applied to C([0,17]) ⊂ C([0,12]) [i] based on
- linear OA(4230, 65625, F4, 34) (dual of [65625, 65395, 35]-code), using Gilbert–Varšamov bound and bm = 4230 > Vbs−1(k−1) = 583395 089841 095362 866780 074638 214136 189443 508807 998895 387236 658972 805752 578659 334263 577407 611002 600366 190154 910420 896437 406763 219358 271858 [i]
- linear OA(40, 3, F4, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(4230, 65622, F4, 35) (dual of [65622, 65392, 36]-code), using
- construction X with Varšamov bound [i] based on
(233−35, 233, large)-Net in Base 4 — Upper bound on s
There is no (198, 233, large)-net in base 4, because
- 33 times m-reduction [i] would yield (198, 200, large)-net in base 4, but