Best Known (105, 138, s)-Nets in Base 3
(105, 138, 400)-Net over F3 — Constructive and digital
Digital (105, 138, 400)-net over F3, using
- 32 times duplication [i] based on digital (103, 136, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 34, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 34, 100)-net over F81, using
(105, 138, 770)-Net over F3 — Digital
Digital (105, 138, 770)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3138, 770, F3, 33) (dual of [770, 632, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3138, 772, F3, 33) (dual of [772, 634, 34]-code), using
- 35 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) [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]
- 35 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) [i] based on linear OA(3129, 728, F3, 33) (dual of [728, 599, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3138, 772, F3, 33) (dual of [772, 634, 34]-code), using
(105, 138, 41371)-Net in Base 3 — Upper bound on s
There is no (105, 138, 41372)-net in base 3, because
- 1 times m-reduction [i] would yield (105, 137, 41372)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 232149 113131 229057 014857 112250 414482 777710 868086 525318 654093 620865 > 3137 [i]