Best Known (199−37, 199, s)-Nets in Base 3
(199−37, 199, 701)-Net over F3 — Constructive and digital
Digital (162, 199, 701)-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 (139, 176, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 44, 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, 44, 172)-net over F81, using
- digital (5, 23, 13)-net over F3, using
(199−37, 199, 3445)-Net over F3 — Digital
Digital (162, 199, 3445)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3199, 3445, F3, 37) (dual of [3445, 3246, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3199, 6584, F3, 37) (dual of [6584, 6385, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(3193, 6562, F3, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3177, 6562, F3, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(36, 22, F3, 3) (dual of [22, 16, 4]-code or 22-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3199, 6584, F3, 37) (dual of [6584, 6385, 38]-code), using
(199−37, 199, 668994)-Net in Base 3 — Upper bound on s
There is no (162, 199, 668995)-net in base 3, because
- 1 times m-reduction [i] would yield (162, 198, 668995)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 29513 117105 348607 246333 934110 155498 194653 457947 693736 282666 989380 195405 078058 986975 450301 498885 > 3198 [i]