Best Known (55, 55+20, s)-Nets in Base 4
(55, 55+20, 384)-Net over F4 — Constructive and digital
Digital (55, 75, 384)-net over F4, using
- trace code for nets [i] based on digital (5, 25, 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
(55, 55+20, 450)-Net in Base 4 — Constructive
(55, 75, 450)-net in base 4, using
- trace code for nets [i] based on (5, 25, 150)-net in base 64, using
- 3 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
- 3 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
(55, 55+20, 739)-Net over F4 — Digital
Digital (55, 75, 739)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(475, 739, F4, 20) (dual of [739, 664, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(475, 1023, F4, 20) (dual of [1023, 948, 21]-code), using
- the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(475, 1023, F4, 20) (dual of [1023, 948, 21]-code), using
(55, 55+20, 49457)-Net in Base 4 — Upper bound on s
There is no (55, 75, 49458)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1427 258934 420903 366900 282242 019082 907697 968920 > 475 [i]