Best Known (118, 118+28, s)-Nets in Base 3
(118, 118+28, 688)-Net over F3 — Constructive and digital
Digital (118, 146, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (118, 148, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 37, 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, 37, 172)-net over F81, using
(118, 118+28, 2826)-Net over F3 — Digital
Digital (118, 146, 2826)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3146, 2826, F3, 2, 28) (dual of [(2826, 2), 5506, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3146, 3285, F3, 2, 28) (dual of [(3285, 2), 6424, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3146, 6570, F3, 28) (dual of [6570, 6424, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 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(31, 9, F3, 1) (dual of [9, 8, 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
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(3146, 6570, F3, 28) (dual of [6570, 6424, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(3146, 3285, F3, 2, 28) (dual of [(3285, 2), 6424, 29]-NRT-code), using
(118, 118+28, 285825)-Net in Base 3 — Upper bound on s
There is no (118, 146, 285826)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4567 967346 166581 609123 594071 505232 581767 106475 880586 681040 107822 467469 > 3146 [i]