Best Known (45−21, 45, s)-Nets in Base 27
(45−21, 45, 164)-Net over F27 — Constructive and digital
Digital (24, 45, 164)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 17, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (7, 28, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27 (see above)
- digital (7, 17, 82)-net over F27, using
(45−21, 45, 200)-Net in Base 27 — Constructive
(24, 45, 200)-net in base 27, using
- 271 times duplication [i] based on (23, 44, 200)-net in base 27, using
- base change [i] based on digital (12, 33, 200)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (1, 22, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81 (see above)
- digital (1, 11, 100)-net over F81, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (12, 33, 200)-net over F81, using
(45−21, 45, 621)-Net over F27 — Digital
Digital (24, 45, 621)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2745, 621, F27, 21) (dual of [621, 576, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2745, 743, F27, 21) (dual of [743, 698, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(2741, 729, F27, 21) (dual of [729, 688, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2731, 729, F27, 16) (dual of [729, 698, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(274, 14, F27, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2745, 743, F27, 21) (dual of [743, 698, 22]-code), using
(45−21, 45, 345937)-Net in Base 27 — Upper bound on s
There is no (24, 45, 345938)-net in base 27, because
- 1 times m-reduction [i] would yield (24, 44, 345938)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 955 024407 246032 368960 637777 434347 186821 585833 775695 688746 082053 > 2744 [i]