Best Known (41, 41+17, s)-Nets in Base 3
(41, 41+17, 144)-Net over F3 — Constructive and digital
Digital (41, 58, 144)-net over F3, using
- 31 times duplication [i] based on digital (40, 57, 144)-net over F3, using
- trace code for nets [i] based on digital (2, 19, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- trace code for nets [i] based on digital (2, 19, 48)-net over F27, using
(41, 41+17, 197)-Net over F3 — Digital
Digital (41, 58, 197)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(358, 197, F3, 17) (dual of [197, 139, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(358, 251, F3, 17) (dual of [251, 193, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(356, 243, F3, 17) (dual of [243, 187, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(351, 243, F3, 16) (dual of [243, 192, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(346, 243, F3, 14) (dual of [243, 197, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(358, 251, F3, 17) (dual of [251, 193, 18]-code), using
(41, 41+17, 4714)-Net in Base 3 — Upper bound on s
There is no (41, 58, 4715)-net in base 3, because
- 1 times m-reduction [i] would yield (41, 57, 4715)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1570 693682 372791 578017 473137 > 357 [i]