Best Known (35−9, 35, s)-Nets in Base 4
(35−9, 35, 514)-Net over F4 — Constructive and digital
Digital (26, 35, 514)-net over F4, using
- base reduction for projective spaces (embedding PG(17,16) in PG(34,4)) for nets [i] based on digital (9, 18, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 9, 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, 9, 257)-net over F256, using
(35−9, 35, 942)-Net over F4 — Digital
Digital (26, 35, 942)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(435, 942, F4, 9) (dual of [942, 907, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(435, 1023, F4, 9) (dual of [1023, 988, 10]-code), using
- the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(435, 1023, F4, 9) (dual of [1023, 988, 10]-code), using
(35−9, 35, 96700)-Net in Base 4 — Upper bound on s
There is no (26, 35, 96701)-net in base 4, because
- 1 times m-reduction [i] would yield (26, 34, 96701)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 295 153906 622836 239331 > 434 [i]