Best Known (42, 42+41, s)-Nets in Base 27
(42, 42+41, 192)-Net over F27 — Constructive and digital
Digital (42, 83, 192)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (11, 31, 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 (11, 52, 96)-net over F27, using
- net from sequence [i] based on digital (11, 95)-sequence over F27 (see above)
- digital (11, 31, 96)-net over F27, using
(42, 42+41, 370)-Net in Base 27 — Constructive
(42, 83, 370)-net in base 27, using
- 21 times m-reduction [i] based on (42, 104, 370)-net in base 27, using
- base change [i] based on digital (16, 78, 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, 78, 370)-net over F81, using
(42, 42+41, 588)-Net over F27 — Digital
Digital (42, 83, 588)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2783, 588, F27, 41) (dual of [588, 505, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(2783, 746, F27, 41) (dual of [746, 663, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- linear OA(2778, 729, F27, 41) (dual of [729, 651, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(2766, 729, F27, 35) (dual of [729, 663, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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 Ce(40) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(2783, 746, F27, 41) (dual of [746, 663, 42]-code), using
(42, 42+41, 235996)-Net in Base 27 — Upper bound on s
There is no (42, 83, 235997)-net in base 27, because
- 1 times m-reduction [i] would yield (42, 82, 235997)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 2354 235668 414658 629399 382748 811005 422910 313899 193545 371274 715952 153195 550213 199389 685998 477630 246523 355294 220988 199921 > 2782 [i]