Best Known (242−44, 242, s)-Nets in Base 3
(242−44, 242, 1480)-Net over F3 — Constructive and digital
Digital (198, 242, 1480)-net over F3, using
- 32 times duplication [i] based on digital (196, 240, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 60, 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, 60, 370)-net over F81, using
(242−44, 242, 4474)-Net over F3 — Digital
Digital (198, 242, 4474)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3242, 4474, F3, 44) (dual of [4474, 4232, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3242, 6596, F3, 44) (dual of [6596, 6354, 45]-code), using
- construction XX applied to Ce(43) ⊂ Ce(39) ⊂ Ce(37) [i] based on
- linear OA(3233, 6561, F3, 44) (dual of [6561, 6328, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(36, 32, F3, 3) (dual of [32, 26, 4]-code or 32-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
- linear OA(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(43) ⊂ Ce(39) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(3242, 6596, F3, 44) (dual of [6596, 6354, 45]-code), using
(242−44, 242, 801946)-Net in Base 3 — Upper bound on s
There is no (198, 242, 801947)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 29 063916 340962 199114 944320 011979 861910 199145 774848 108628 587956 420293 121710 797864 122626 512317 982798 090743 808109 637741 > 3242 [i]