Best Known (55, 76, s)-Nets in Base 4
(55, 76, 312)-Net over F4 — Constructive and digital
Digital (55, 76, 312)-net over F4, using
- 2 times m-reduction [i] based on digital (55, 78, 312)-net over F4, using
- trace code for nets [i] based on digital (3, 26, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 26, 104)-net over F64, using
(55, 76, 387)-Net in Base 4 — Constructive
(55, 76, 387)-net in base 4, using
- 41 times duplication [i] based on (54, 75, 387)-net in base 4, using
- trace code for nets [i] based on (4, 25, 129)-net in base 64, using
- 3 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
- base change [i] based on digital (0, 24, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 24, 129)-net over F128, using
- 3 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
- trace code for nets [i] based on (4, 25, 129)-net in base 64, using
(55, 76, 615)-Net over F4 — Digital
Digital (55, 76, 615)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(476, 615, F4, 21) (dual of [615, 539, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(476, 1023, F4, 21) (dual of [1023, 947, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(476, 1023, F4, 21) (dual of [1023, 947, 22]-code), using
(55, 76, 49457)-Net in Base 4 — Upper bound on s
There is no (55, 76, 49458)-net in base 4, because
- 1 times m-reduction [i] would yield (55, 75, 49458)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1427 258934 420903 366900 282242 019082 907697 968920 > 475 [i]