Best Known (112, 138, s)-Nets in Base 3
(112, 138, 688)-Net over F3 — Constructive and digital
Digital (112, 138, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (112, 140, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 35, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 35, 172)-net over F81, using
(112, 138, 3102)-Net over F3 — Digital
Digital (112, 138, 3102)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3138, 3102, F3, 2, 26) (dual of [(3102, 2), 6066, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3138, 3285, F3, 2, 26) (dual of [(3285, 2), 6432, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3138, 6570, F3, 26) (dual of [6570, 6432, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3137, 6569, F3, 26) (dual of [6569, 6432, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3137, 6561, F3, 26) (dual of [6561, 6424, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 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
- 1 times code embedding in larger space [i] based on linear OA(3137, 6569, F3, 26) (dual of [6569, 6432, 27]-code), using
- OOA 2-folding [i] based on linear OA(3138, 6570, F3, 26) (dual of [6570, 6432, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(3138, 3285, F3, 2, 26) (dual of [(3285, 2), 6432, 27]-NRT-code), using
(112, 138, 328991)-Net in Base 3 — Upper bound on s
There is no (112, 138, 328992)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 696199 198910 314007 826707 242904 206800 029375 603033 102373 082380 705857 > 3138 [i]