Best Known (111−28, 111, s)-Nets in Base 4
(111−28, 111, 531)-Net over F4 — Constructive and digital
Digital (83, 111, 531)-net over F4, using
- 3 times m-reduction [i] based on digital (83, 114, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 38, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 38, 177)-net over F64, using
(111−28, 111, 576)-Net in Base 4 — Constructive
(83, 111, 576)-net in base 4, using
- trace code for nets [i] based on (9, 37, 192)-net in base 64, using
- 5 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- 5 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
(111−28, 111, 1115)-Net over F4 — Digital
Digital (83, 111, 1115)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4111, 1115, F4, 28) (dual of [1115, 1004, 29]-code), using
- 86 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 24 times 0, 1, 36 times 0) [i] based on linear OA(4105, 1023, F4, 28) (dual of [1023, 918, 29]-code), using
- the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 86 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 24 times 0, 1, 36 times 0) [i] based on linear OA(4105, 1023, F4, 28) (dual of [1023, 918, 29]-code), using
(111−28, 111, 119611)-Net in Base 4 — Upper bound on s
There is no (83, 111, 119612)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 6 740673 967035 956119 465042 230310 631568 499453 904949 686732 941151 145412 > 4111 [i]