Best Known (240−38, 240, s)-Nets in Base 3
(240−38, 240, 1493)-Net over F3 — Constructive and digital
Digital (202, 240, 1493)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 24, 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 (178, 216, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 54, 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, 54, 370)-net over F81, using
- digital (5, 24, 13)-net over F3, using
(240−38, 240, 10467)-Net over F3 — Digital
Digital (202, 240, 10467)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3240, 10467, F3, 38) (dual of [10467, 10227, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3240, 19736, F3, 38) (dual of [19736, 19496, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- linear OA(3226, 19683, F3, 38) (dual of [19683, 19457, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(314, 53, F3, 6) (dual of [53, 39, 7]-code), using
- construction X applied to Ce(37) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(3240, 19736, F3, 38) (dual of [19736, 19496, 39]-code), using
(240−38, 240, 4216755)-Net in Base 3 — Upper bound on s
There is no (202, 240, 4216756)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 229255 870377 201826 963750 223552 653845 755013 259512 890400 525603 136110 376584 552669 998708 301709 098198 145051 960522 041201 > 3240 [i]