Best Known (61, 61+19, s)-Nets in Base 3
(61, 61+19, 400)-Net over F3 — Constructive and digital
Digital (61, 80, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 20, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
(61, 61+19, 577)-Net over F3 — Digital
Digital (61, 80, 577)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(380, 577, F3, 19) (dual of [577, 497, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(380, 754, F3, 19) (dual of [754, 674, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(373, 729, F3, 19) (dual of [729, 656, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(355, 729, F3, 14) (dual of [729, 674, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(37, 25, F3, 4) (dual of [25, 18, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(37, 26, F3, 4) (dual of [26, 19, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(37, 26, F3, 4) (dual of [26, 19, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(380, 754, F3, 19) (dual of [754, 674, 20]-code), using
(61, 61+19, 31964)-Net in Base 3 — Upper bound on s
There is no (61, 80, 31965)-net in base 3, because
- 1 times m-reduction [i] would yield (61, 79, 31965)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 49 274068 730862 218989 134500 224082 442907 > 379 [i]