Best Known (70−21, 70, s)-Nets in Base 4
(70−21, 70, 240)-Net over F4 — Constructive and digital
Digital (49, 70, 240)-net over F4, using
- 2 times m-reduction [i] based on digital (49, 72, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 24, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 24, 80)-net over F64, using
(70−21, 70, 365)-Net over F4 — Digital
Digital (49, 70, 365)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(470, 365, F4, 21) (dual of [365, 295, 22]-code), using
- 98 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 16 times 0, 1, 19 times 0, 1, 21 times 0) [i] based on linear OA(459, 256, F4, 21) (dual of [256, 197, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 255 = 44−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 98 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 16 times 0, 1, 19 times 0, 1, 21 times 0) [i] based on linear OA(459, 256, F4, 21) (dual of [256, 197, 22]-code), using
(70−21, 70, 21523)-Net in Base 4 — Upper bound on s
There is no (49, 70, 21524)-net in base 4, because
- 1 times m-reduction [i] would yield (49, 69, 21524)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 348545 422511 743562 763375 272343 453106 062586 > 469 [i]