Best Known (199, 199+41, s)-Nets in Base 3
(199, 199+41, 1480)-Net over F3 — Constructive and digital
Digital (199, 240, 1480)-net over F3, using
- 4 times m-reduction [i] based on digital (199, 244, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 61, 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, 61, 370)-net over F81, using
(199, 199+41, 6423)-Net over F3 — Digital
Digital (199, 240, 6423)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3240, 6423, F3, 41) (dual of [6423, 6183, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3240, 6632, F3, 41) (dual of [6632, 6392, 42]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3238, 6630, F3, 41) (dual of [6630, 6392, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(31) [i] based on
- linear OA(3217, 6561, F3, 41) (dual of [6561, 6344, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(321, 69, F3, 8) (dual of [69, 48, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- construction X applied to Ce(40) ⊂ Ce(31) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3238, 6630, F3, 41) (dual of [6630, 6392, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3240, 6632, F3, 41) (dual of [6632, 6392, 42]-code), using
(199, 199+41, 2088676)-Net in Base 3 — Upper bound on s
There is no (199, 240, 2088677)-net in base 3, because
- 1 times m-reduction [i] would yield (199, 239, 2088677)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 076420 787309 665020 497988 775984 406888 401359 095255 327023 942094 736241 342378 076268 194460 222562 821760 321641 598217 447441 > 3239 [i]