Best Known (68, 68+21, s)-Nets in Base 3
(68, 68+21, 400)-Net over F3 — Constructive and digital
Digital (68, 89, 400)-net over F3, using
- 31 times duplication [i] based on digital (67, 88, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 22, 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
- trace code for nets [i] based on digital (1, 22, 100)-net over F81, using
(68, 68+21, 626)-Net over F3 — Digital
Digital (68, 89, 626)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(389, 626, F3, 21) (dual of [626, 537, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(389, 747, F3, 21) (dual of [747, 658, 22]-code), using
- construction XX applied to Ce(21) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- linear OA(385, 729, F3, 22) (dual of [729, 644, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(367, 729, F3, 17) (dual of [729, 662, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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
- linear OA(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(21) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(389, 747, F3, 21) (dual of [747, 658, 22]-code), using
(68, 68+21, 35768)-Net in Base 3 — Upper bound on s
There is no (68, 89, 35769)-net in base 3, because
- 1 times m-reduction [i] would yield (68, 88, 35769)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 969969 514972 579321 362681 698000 349604 865497 > 388 [i]