Best Known (112−26, 112, s)-Nets in Base 3
(112−26, 112, 464)-Net over F3 — Constructive and digital
Digital (86, 112, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 28, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
(112−26, 112, 767)-Net over F3 — Digital
Digital (86, 112, 767)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3112, 767, F3, 26) (dual of [767, 655, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3112, 772, F3, 26) (dual of [772, 660, 27]-code), using
- 28 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0) [i] based on linear OA(3103, 735, F3, 26) (dual of [735, 632, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3103, 729, F3, 26) (dual of [729, 626, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(30, 6, F3, 0) (dual of [6, 6, 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
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- 28 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0) [i] based on linear OA(3103, 735, F3, 26) (dual of [735, 632, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3112, 772, F3, 26) (dual of [772, 660, 27]-code), using
(112−26, 112, 36543)-Net in Base 3 — Upper bound on s
There is no (86, 112, 36544)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 273936 097304 966692 630606 038768 402295 317488 499307 827585 > 3112 [i]