Best Known (161−39, 161, s)-Nets in Base 3
(161−39, 161, 400)-Net over F3 — Constructive and digital
Digital (122, 161, 400)-net over F3, using
- 31 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
(161−39, 161, 810)-Net over F3 — Digital
Digital (122, 161, 810)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3161, 810, F3, 39) (dual of [810, 649, 40]-code), using
- 74 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) [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]
- 74 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) [i] based on linear OA(3153, 728, F3, 39) (dual of [728, 575, 40]-code), using
(161−39, 161, 41290)-Net in Base 3 — Upper bound on s
There is no (122, 161, 41291)-net in base 3, because
- 1 times m-reduction [i] would yield (122, 160, 41291)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 21848 590945 144266 910592 552582 584398 293899 504049 332147 916045 439623 489988 933675 > 3160 [i]