Best Known (101−27, 101, s)-Nets in Base 4
(101−27, 101, 384)-Net over F4 — Constructive and digital
Digital (74, 101, 384)-net over F4, using
- t-expansion [i] based on digital (73, 101, 384)-net over F4, using
- 1 times m-reduction [i] based on digital (73, 102, 384)-net over F4, using
- trace code for nets [i] based on digital (5, 34, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 34, 128)-net over F64, using
- 1 times m-reduction [i] based on digital (73, 102, 384)-net over F4, using
(101−27, 101, 450)-Net in Base 4 — Constructive
(74, 101, 450)-net in base 4, using
- 1 times m-reduction [i] based on (74, 102, 450)-net in base 4, using
- trace code for nets [i] based on (6, 34, 150)-net in base 64, using
- 1 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
- base change [i] based on digital (1, 30, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 30, 150)-net over F128, using
- 1 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
- trace code for nets [i] based on (6, 34, 150)-net in base 64, using
(101−27, 101, 850)-Net over F4 — Digital
Digital (74, 101, 850)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4101, 850, F4, 27) (dual of [850, 749, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4101, 1023, F4, 27) (dual of [1023, 922, 28]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(4101, 1023, F4, 27) (dual of [1023, 922, 28]-code), using
(101−27, 101, 80809)-Net in Base 4 — Upper bound on s
There is no (74, 101, 80810)-net in base 4, because
- 1 times m-reduction [i] would yield (74, 100, 80810)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 607178 980666 074826 271558 888290 550422 345291 319529 223700 489880 > 4100 [i]