Best Known (59, 77, s)-Nets in Base 3
(59, 77, 400)-Net over F3 — Constructive and digital
Digital (59, 77, 400)-net over F3, using
- 31 times duplication [i] based on digital (58, 76, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 19, 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, 19, 100)-net over F81, using
(59, 77, 614)-Net over F3 — Digital
Digital (59, 77, 614)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(377, 614, F3, 18) (dual of [614, 537, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(377, 747, F3, 18) (dual of [747, 670, 19]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ 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(361, 729, F3, 16) (dual of [729, 668, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(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(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(377, 747, F3, 18) (dual of [747, 670, 19]-code), using
(59, 77, 25038)-Net in Base 3 — Upper bound on s
There is no (59, 77, 25039)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5 475026 261436 077480 725654 317775 773071 > 377 [i]