Best Known (110−25, 110, s)-Nets in Base 3
(110−25, 110, 464)-Net over F3 — Constructive and digital
Digital (85, 110, 464)-net over F3, using
- 32 times duplication [i] based on digital (83, 108, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 27, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 27, 116)-net over F81, using
(110−25, 110, 806)-Net over F3 — Digital
Digital (85, 110, 806)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3110, 806, F3, 25) (dual of [806, 696, 26]-code), using
- 63 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 16 times 0) [i] based on linear OA(397, 730, F3, 25) (dual of [730, 633, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 63 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 16 times 0) [i] based on linear OA(397, 730, F3, 25) (dual of [730, 633, 26]-code), using
(110−25, 110, 57028)-Net in Base 3 — Upper bound on s
There is no (85, 110, 57029)-net in base 3, because
- 1 times m-reduction [i] would yield (85, 109, 57029)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 10144 316255 125981 307558 872316 837679 188255 129547 870513 > 3109 [i]