Best Known (40, 57, s)-Nets in Base 4
(40, 57, 240)-Net over F4 — Constructive and digital
Digital (40, 57, 240)-net over F4, using
- t-expansion [i] based on digital (39, 57, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 19, 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, 19, 80)-net over F64, using
(40, 57, 327)-Net over F4 — Digital
Digital (40, 57, 327)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(457, 327, F4, 17) (dual of [327, 270, 18]-code), using
- 62 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 0, 1, 6 times 0, 1, 10 times 0, 1, 16 times 0, 1, 21 times 0) [i] based on linear OA(449, 257, F4, 17) (dual of [257, 208, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 257 | 48−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 62 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 0, 1, 6 times 0, 1, 10 times 0, 1, 16 times 0, 1, 21 times 0) [i] based on linear OA(449, 257, F4, 17) (dual of [257, 208, 18]-code), using
(40, 57, 20552)-Net in Base 4 — Upper bound on s
There is no (40, 57, 20553)-net in base 4, because
- 1 times m-reduction [i] would yield (40, 56, 20553)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 5193 898381 122807 047476 303368 550585 > 456 [i]