Best Known (85, 85+29, s)-Nets in Base 4
(85, 85+29, 531)-Net over F4 — Constructive and digital
Digital (85, 114, 531)-net over F4, using
- 3 times m-reduction [i] based on digital (85, 117, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 39, 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, 39, 177)-net over F64, using
(85, 85+29, 576)-Net in Base 4 — Constructive
(85, 114, 576)-net in base 4, using
- trace code for nets [i] based on (9, 38, 192)-net in base 64, using
- 4 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
- 4 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
(85, 85+29, 1103)-Net over F4 — Digital
Digital (85, 114, 1103)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4114, 1103, F4, 29) (dual of [1103, 989, 30]-code), using
- 71 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 19 times 0, 1, 29 times 0) [i] based on linear OA(4106, 1024, F4, 29) (dual of [1024, 918, 30]-code), using
- an extension Ce(28) of 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]
- 71 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 19 times 0, 1, 29 times 0) [i] based on linear OA(4106, 1024, F4, 29) (dual of [1024, 918, 30]-code), using
(85, 85+29, 145810)-Net in Base 4 — Upper bound on s
There is no (85, 114, 145811)-net in base 4, because
- 1 times m-reduction [i] would yield (85, 113, 145811)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 107 849678 684676 839063 788860 313067 617012 350736 525209 832684 408024 506868 > 4113 [i]