Best Known (126, 126+27, s)-Nets in Base 3
(126, 126+27, 700)-Net over F3 — Constructive and digital
Digital (126, 153, 700)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 17, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- 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
- digital (4, 17, 12)-net over F3, using
(126, 126+27, 4027)-Net over F3 — Digital
Digital (126, 153, 4027)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3153, 4027, F3, 27) (dual of [4027, 3874, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3153, 6590, F3, 27) (dual of [6590, 6437, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(3145, 6562, F3, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3113, 6562, F3, 21) (dual of [6562, 6449, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3153, 6590, F3, 27) (dual of [6590, 6437, 28]-code), using
(126, 126+27, 1074037)-Net in Base 3 — Upper bound on s
There is no (126, 153, 1074038)-net in base 3, because
- 1 times m-reduction [i] would yield (126, 152, 1074038)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3 329896 599910 793808 078605 618738 675517 506134 072311 987233 344851 385951 454517 > 3152 [i]