Best Known (228−49, 228, s)-Nets in Base 3
(228−49, 228, 688)-Net over F3 — Constructive and digital
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
(228−49, 228, 1808)-Net over F3 — Digital
Digital (179, 228, 1808)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3228, 1808, F3, 49) (dual of [1808, 1580, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(3228, 2192, F3, 49) (dual of [2192, 1964, 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(33, 4, F3, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,3) or 4-cap in PG(2,3)), using
- dual of repetition code with length 4 [i]
- oval in PG(2, 3) [i]
- construction X applied to C([0,24]) ⊂ C([0,22]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3228, 2192, F3, 49) (dual of [2192, 1964, 50]-code), using
(228−49, 228, 159602)-Net in Base 3 — Upper bound on s
There is no (179, 228, 159603)-net in base 3, because
- 1 times m-reduction [i] would yield (179, 227, 159603)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 025507 815098 942939 002865 881803 847163 668711 612865 785696 684056 922845 937977 151276 552486 847271 211187 525634 548689 > 3227 [i]