Best Known (147−29, 147, s)-Nets in Base 3
(147−29, 147, 688)-Net over F3 — Constructive and digital
Digital (118, 147, 688)-net over F3, using
- 1 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
(147−29, 147, 2051)-Net over F3 — Digital
Digital (118, 147, 2051)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3147, 2051, F3, 29) (dual of [2051, 1904, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3147, 2222, F3, 29) (dual of [2222, 2075, 30]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3141, 2188, F3, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3113, 2188, F3, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(36, 34, F3, 3) (dual of [34, 28, 4]-code or 34-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3147, 2222, F3, 29) (dual of [2222, 2075, 30]-code), using
(147−29, 147, 285825)-Net in Base 3 — Upper bound on s
There is no (118, 147, 285826)-net in base 3, because
- 1 times m-reduction [i] would yield (118, 146, 285826)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4567 967346 166581 609123 594071 505232 581767 106475 880586 681040 107822 467469 > 3146 [i]