Best Known (111, 111+33, s)-Nets in Base 3
(111, 111+33, 464)-Net over F3 — Constructive and digital
Digital (111, 144, 464)-net over F3, using
- t-expansion [i] based on digital (110, 144, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 36, 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, 36, 116)-net over F81, using
(111, 111+33, 914)-Net over F3 — Digital
Digital (111, 144, 914)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3144, 914, F3, 33) (dual of [914, 770, 34]-code), using
- 171 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 18 times 0, 1, 21 times 0, 1, 23 times 0, 1, 26 times 0, 1, 28 times 0) [i] based on linear OA(3129, 728, F3, 33) (dual of [728, 599, 34]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 171 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 14 times 0, 1, 18 times 0, 1, 21 times 0, 1, 23 times 0, 1, 26 times 0, 1, 28 times 0) [i] based on linear OA(3129, 728, F3, 33) (dual of [728, 599, 34]-code), using
(111, 111+33, 62470)-Net in Base 3 — Upper bound on s
There is no (111, 144, 62471)-net in base 3, because
- 1 times m-reduction [i] would yield (111, 143, 62471)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 169 215192 152797 955341 359954 503246 186478 329793 634911 671218 806423 288033 > 3143 [i]