Best Known (49, 49+17, s)-Nets in Base 3
(49, 49+17, 204)-Net over F3 — Constructive and digital
Digital (49, 66, 204)-net over F3, using
- trace code for nets [i] based on digital (5, 22, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
(49, 49+17, 362)-Net over F3 — Digital
Digital (49, 66, 362)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(366, 362, F3, 17) (dual of [362, 296, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(366, 374, F3, 17) (dual of [374, 308, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(364, 365, F3, 17) (dual of [365, 301, 18]-code), using an extension Ce(16) of the narrow-sense BCH-code C(I) with length 364 | 36−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(358, 365, F3, 16) (dual of [365, 307, 17]-code), using an extension Ce(15) of the narrow-sense BCH-code C(I) with length 364 | 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(355, 365, F3, 14) (dual of [365, 310, 15]-code), using an extension Ce(13) of the narrow-sense BCH-code C(I) with length 364 | 36−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(30, 7, F3, 0) (dual of [7, 7, 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(366, 374, F3, 17) (dual of [374, 308, 18]-code), using
(49, 49+17, 14159)-Net in Base 3 — Upper bound on s
There is no (49, 66, 14160)-net in base 3, because
- 1 times m-reduction [i] would yield (49, 65, 14160)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 10 305398 816755 514233 361608 392705 > 365 [i]