Best Known (198, 198+37, s)-Nets in Base 3
(198, 198+37, 1493)-Net over F3 — Constructive and digital
Digital (198, 235, 1493)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 23, 13)-net over F3, using
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 4, N(F) = 12, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- digital (175, 212, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 53, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 53, 370)-net over F81, using
- digital (5, 23, 13)-net over F3, using
(198, 198+37, 10734)-Net over F3 — Digital
Digital (198, 235, 10734)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3235, 10734, F3, 37) (dual of [10734, 10499, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3235, 19703, F3, 37) (dual of [19703, 19468, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([1,18]) [i] based on
- linear OA(3217, 19684, F3, 37) (dual of [19684, 19467, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3216, 19684, F3, 18) (dual of [19684, 19468, 19]-code), using the narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(318, 19, F3, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,3)), using
- dual of repetition code with length 19 [i]
- construction X applied to C([0,18]) ⊂ C([1,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3235, 19703, F3, 37) (dual of [19703, 19468, 38]-code), using
(198, 198+37, 6021089)-Net in Base 3 — Upper bound on s
There is no (198, 235, 6021090)-net in base 3, because
- 1 times m-reduction [i] would yield (198, 234, 6021090)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4429 694358 010127 971973 454479 041523 421810 473806 736112 254495 670915 488937 131401 145986 318736 050781 850553 292695 461621 > 3234 [i]