Best Known (86, 86+29, s)-Nets in Base 4
(86, 86+29, 531)-Net over F4 — Constructive and digital
Digital (86, 115, 531)-net over F4, using
- t-expansion [i] based on digital (85, 115, 531)-net over F4, using
- 2 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
- 2 times m-reduction [i] based on digital (85, 117, 531)-net over F4, using
(86, 86+29, 576)-Net in Base 4 — Constructive
(86, 115, 576)-net in base 4, using
- 41 times duplication [i] based on (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
- trace code for nets [i] based on (9, 38, 192)-net in base 64, using
(86, 86+29, 1144)-Net over F4 — Digital
Digital (86, 115, 1144)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4115, 1144, F4, 29) (dual of [1144, 1029, 30]-code), using
- 111 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, 1, 39 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]
- 111 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, 1, 39 times 0) [i] based on linear OA(4106, 1024, F4, 29) (dual of [1024, 918, 30]-code), using
(86, 86+29, 160988)-Net in Base 4 — Upper bound on s
There is no (86, 115, 160989)-net in base 4, because
- 1 times m-reduction [i] would yield (86, 114, 160989)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 431 377770 142412 296412 197875 140168 134090 849666 711220 493389 149427 275432 > 4114 [i]