Best Known (136−25, 136, s)-Nets in Base 3
(136−25, 136, 688)-Net over F3 — Constructive and digital
Digital (111, 136, 688)-net over F3, using
- t-expansion [i] based on digital (109, 136, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 34, 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, 34, 172)-net over F81, using
(136−25, 136, 3294)-Net over F3 — Digital
Digital (111, 136, 3294)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3136, 3294, F3, 2, 25) (dual of [(3294, 2), 6452, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3136, 6588, F3, 25) (dual of [6588, 6452, 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(37, 27, F3, 4) (dual of [27, 20, 5]-code), using
- an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- OOA 2-folding [i] based on linear OA(3136, 6588, F3, 25) (dual of [6588, 6452, 26]-code), using
(136−25, 136, 616506)-Net in Base 3 — Upper bound on s
There is no (111, 136, 616507)-net in base 3, because
- 1 times m-reduction [i] would yield (111, 135, 616507)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 25785 546496 640029 935025 406015 211480 566584 109971 655072 870151 388361 > 3135 [i]