Best Known (89−27, 89, s)-Nets in Base 4
(89−27, 89, 240)-Net over F4 — Constructive and digital
Digital (62, 89, 240)-net over F4, using
- t-expansion [i] based on digital (61, 89, 240)-net over F4, using
- 1 times m-reduction [i] based on digital (61, 90, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 30, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 30, 80)-net over F64, using
- 1 times m-reduction [i] based on digital (61, 90, 240)-net over F4, using
(89−27, 89, 418)-Net over F4 — Digital
Digital (62, 89, 418)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(489, 418, F4, 27) (dual of [418, 329, 28]-code), using
- 328 step Varšamov–Edel lengthening with (ri) = (7, 4, 2, 2, 1, 2, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0) [i] based on linear OA(427, 28, F4, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,4)), using
- dual of repetition code with length 28 [i]
- 328 step Varšamov–Edel lengthening with (ri) = (7, 4, 2, 2, 1, 2, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0) [i] based on linear OA(427, 28, F4, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,4)), using
(89−27, 89, 22468)-Net in Base 4 — Upper bound on s
There is no (62, 89, 22469)-net in base 4, because
- 1 times m-reduction [i] would yield (62, 88, 22469)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 95825 277166 544562 155010 508208 288393 907738 083061 782560 > 488 [i]