Best Known (86, 86+27, s)-Nets in Base 3
(86, 86+27, 400)-Net over F3 — Constructive and digital
Digital (86, 113, 400)-net over F3, using
- 31 times duplication [i] based on digital (85, 112, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 28, 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, 28, 100)-net over F81, using
(86, 86+27, 676)-Net over F3 — Digital
Digital (86, 113, 676)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3113, 676, F3, 27) (dual of [676, 563, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3113, 747, F3, 27) (dual of [747, 634, 28]-code), using
- construction XX applied to Ce(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3109, 729, F3, 28) (dual of [729, 620, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(397, 729, F3, 25) (dual of [729, 632, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(391, 729, F3, 23) (dual of [729, 638, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3113, 747, F3, 27) (dual of [747, 634, 28]-code), using
(86, 86+27, 36543)-Net in Base 3 — Upper bound on s
There is no (86, 113, 36544)-net in base 3, because
- 1 times m-reduction [i] would yield (86, 112, 36544)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 273936 097304 966692 630606 038768 402295 317488 499307 827585 > 3112 [i]