Best Known (180, 229, s)-Nets in Base 3
(180, 229, 688)-Net over F3 — Constructive and digital
Digital (180, 229, 688)-net over F3, using
- 31 times duplication [i] based on digital (179, 228, 688)-net over F3, using
- t-expansion [i] based on digital (178, 228, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 57, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 57, 172)-net over F81, using
- t-expansion [i] based on digital (178, 228, 688)-net over F3, using
(180, 229, 1852)-Net over F3 — Digital
Digital (180, 229, 1852)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3229, 1852, F3, 49) (dual of [1852, 1623, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(3229, 2198, F3, 49) (dual of [2198, 1969, 50]-code), using
- construction X applied to C([0,24]) ⊂ C([0,22]) [i] based on
- linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- linear OA(3211, 2188, F3, 45) (dual of [2188, 1977, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- linear OA(34, 10, F3, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), using
- construction X applied to C([0,24]) ⊂ C([0,22]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3229, 2198, F3, 49) (dual of [2198, 1969, 50]-code), using
(180, 229, 167079)-Net in Base 3 — Upper bound on s
There is no (180, 229, 167080)-net in base 3, because
- 1 times m-reduction [i] would yield (180, 228, 167080)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 076692 378275 674701 940084 540595 036328 409170 435731 476725 675773 257218 795671 414881 189565 781107 839484 565549 251841 > 3228 [i]