Best Known (204−29, 204, s)-Nets in Base 4
(204−29, 204, 18729)-Net over F4 — Constructive and digital
Digital (175, 204, 18729)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 14, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (161, 190, 18724)-net over F4, using
- net defined by OOA [i] based on linear OOA(4190, 18724, F4, 29, 29) (dual of [(18724, 29), 542806, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4190, 262137, F4, 29) (dual of [262137, 261947, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4190, 262137, F4, 29) (dual of [262137, 261947, 30]-code), using
- net defined by OOA [i] based on linear OOA(4190, 18724, F4, 29, 29) (dual of [(18724, 29), 542806, 30]-NRT-code), using
- digital (0, 14, 5)-net over F4, using
(204−29, 204, 131104)-Net over F4 — Digital
Digital (175, 204, 131104)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4204, 131104, F4, 2, 29) (dual of [(131104, 2), 262004, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4204, 262208, F4, 29) (dual of [262208, 262004, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4204, 262209, F4, 29) (dual of [262209, 262005, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4204, 262209, F4, 29) (dual of [262209, 262005, 30]-code), using
- OOA 2-folding [i] based on linear OA(4204, 262208, F4, 29) (dual of [262208, 262004, 30]-code), using
(204−29, 204, large)-Net in Base 4 — Upper bound on s
There is no (175, 204, large)-net in base 4, because
- 27 times m-reduction [i] would yield (175, 177, large)-net in base 4, but