Best Known (137−25, 137, s)-Nets in Base 3
(137−25, 137, 688)-Net over F3 — Constructive and digital
Digital (112, 137, 688)-net over F3, using
- 3 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
(137−25, 137, 3296)-Net over F3 — Digital
Digital (112, 137, 3296)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3137, 3296, F3, 2, 25) (dual of [(3296, 2), 6455, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3137, 6592, F3, 25) (dual of [6592, 6455, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3137, 6593, F3, 25) (dual of [6593, 6456, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- 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(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(38, 32, F3, 4) (dual of [32, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3137, 6593, F3, 25) (dual of [6593, 6456, 26]-code), using
- OOA 2-folding [i] based on linear OA(3137, 6592, F3, 25) (dual of [6592, 6455, 26]-code), using
(137−25, 137, 675613)-Net in Base 3 — Upper bound on s
There is no (112, 137, 675614)-net in base 3, because
- 1 times m-reduction [i] would yield (112, 136, 675614)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 77356 244185 792330 023345 120310 763627 973453 162704 627126 343415 104729 > 3136 [i]