Best Known (242−46, 242, s)-Nets in Base 3
(242−46, 242, 896)-Net over F3 — Constructive and digital
Digital (196, 242, 896)-net over F3, using
- 2 times m-reduction [i] based on digital (196, 244, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 61, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 61, 224)-net over F81, using
(242−46, 242, 3500)-Net over F3 — Digital
Digital (196, 242, 3500)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3242, 3500, F3, 46) (dual of [3500, 3258, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3242, 6570, F3, 46) (dual of [6570, 6328, 47]-code), using
- construction X applied to Ce(45) ⊂ Ce(43) [i] based on
- linear OA(3241, 6561, F3, 46) (dual of [6561, 6320, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- 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(31, 9, F3, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(45) ⊂ Ce(43) [i] based on
- discarding factors / shortening the dual code based on linear OA(3242, 6570, F3, 46) (dual of [6570, 6328, 47]-code), using
(242−46, 242, 493800)-Net in Base 3 — Upper bound on s
There is no (196, 242, 493801)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 29 063357 724791 531484 438303 573666 928511 109942 693848 015233 123876 159904 535306 361890 020350 823469 578049 900623 392529 242347 > 3242 [i]