Best Known (70−33, 70, s)-Nets in Base 27
(70−33, 70, 190)-Net over F27 — Constructive and digital
Digital (37, 70, 190)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (10, 26, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (11, 44, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 11 and N(F) ≥ 96, using
- net from sequence [i] based on digital (11, 95)-sequence over F27, using
- digital (10, 26, 94)-net over F27, using
(70−33, 70, 370)-Net in Base 27 — Constructive
(37, 70, 370)-net in base 27, using
- 14 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
- base change [i] based on digital (16, 63, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 63, 370)-net over F81, using
(70−33, 70, 718)-Net over F27 — Digital
Digital (37, 70, 718)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2770, 718, F27, 33) (dual of [718, 648, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2770, 747, F27, 33) (dual of [747, 677, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- linear OA(2765, 730, F27, 33) (dual of [730, 665, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(2753, 730, F27, 27) (dual of [730, 677, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(275, 17, F27, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,27)), using
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- Reed–Solomon code RS(22,27) [i]
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- construction X applied to C([0,16]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2770, 747, F27, 33) (dual of [747, 677, 34]-code), using
(70−33, 70, 389325)-Net in Base 27 — Upper bound on s
There is no (37, 70, 389326)-net in base 27, because
- 1 times m-reduction [i] would yield (37, 69, 389326)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 580 916001 985415 132417 920881 912177 862353 426064 699196 966263 935770 260286 744338 078003 027766 237260 507041 > 2769 [i]