Best Known (81−22, 81, s)-Nets in Base 4
(81−22, 81, 384)-Net over F4 — Constructive and digital
Digital (59, 81, 384)-net over F4, using
- trace code for nets [i] based on digital (5, 27, 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
(81−22, 81, 450)-Net in Base 4 — Constructive
(59, 81, 450)-net in base 4, using
- trace code for nets [i] based on (5, 27, 150)-net in base 64, using
- 1 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
- base change [i] based on digital (1, 24, 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, 24, 150)-net over F128, using
- 1 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
(81−22, 81, 694)-Net over F4 — Digital
Digital (59, 81, 694)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(481, 694, F4, 22) (dual of [694, 613, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(481, 1023, F4, 22) (dual of [1023, 942, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(481, 1023, F4, 22) (dual of [1023, 942, 23]-code), using
(81−22, 81, 44377)-Net in Base 4 — Upper bound on s
There is no (59, 81, 44378)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 5 847306 539445 012325 006505 038035 762313 836620 416600 > 481 [i]