Best Known (123, 123+39, s)-Nets in Base 3
(123, 123+39, 400)-Net over F3 — Constructive and digital
Digital (123, 162, 400)-net over F3, using
- 32 times duplication [i] based on digital (121, 160, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 40, 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, 40, 100)-net over F81, using
(123, 123+39, 832)-Net over F3 — Digital
Digital (123, 162, 832)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3162, 832, F3, 39) (dual of [832, 670, 40]-code), using
- 95 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 16 times 0, 1, 19 times 0, 1, 20 times 0) [i] based on linear OA(3153, 728, F3, 39) (dual of [728, 575, 40]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- 95 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 9 times 0, 1, 13 times 0, 1, 16 times 0, 1, 19 times 0, 1, 20 times 0) [i] based on linear OA(3153, 728, F3, 39) (dual of [728, 575, 40]-code), using
(123, 123+39, 43749)-Net in Base 3 — Upper bound on s
There is no (123, 162, 43750)-net in base 3, because
- 1 times m-reduction [i] would yield (123, 161, 43750)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 65546 012693 578387 409215 097891 674577 322142 379735 246412 416735 600236 533017 720001 > 3161 [i]