Best Known (61−17, 61, s)-Nets in Base 4
(61−17, 61, 312)-Net over F4 — Constructive and digital
Digital (44, 61, 312)-net over F4, using
- 41 times duplication [i] based on digital (43, 60, 312)-net over F4, using
- trace code for nets [i] based on digital (3, 20, 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, 20, 104)-net over F64, using
(61−17, 61, 387)-Net in Base 4 — Constructive
(44, 61, 387)-net in base 4, using
- 41 times duplication [i] based on (43, 60, 387)-net in base 4, using
- trace code for nets [i] based on (3, 20, 129)-net in base 64, using
- 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- base change [i] based on digital (0, 18, 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, 18, 129)-net over F128, using
- 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- trace code for nets [i] based on (3, 20, 129)-net in base 64, using
(61−17, 61, 537)-Net over F4 — Digital
Digital (44, 61, 537)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(461, 537, F4, 17) (dual of [537, 476, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(461, 1023, F4, 17) (dual of [1023, 962, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(461, 1023, F4, 17) (dual of [1023, 962, 18]-code), using
(61−17, 61, 41110)-Net in Base 4 — Upper bound on s
There is no (44, 61, 41111)-net in base 4, because
- 1 times m-reduction [i] would yield (44, 60, 41111)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 329336 489112 293883 713544 430965 535530 > 460 [i]