Best Known (37, 74, s)-Nets in Base 27
(37, 74, 182)-Net over F27 — Constructive and digital
Digital (37, 74, 182)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (9, 27, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (10, 47, 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 (9, 27, 88)-net over F27, using
(37, 74, 370)-Net in Base 27 — Constructive
(37, 74, 370)-net in base 27, using
- 10 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
(37, 74, 502)-Net over F27 — Digital
Digital (37, 74, 502)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2774, 502, F27, 37) (dual of [502, 428, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2774, 735, F27, 37) (dual of [735, 661, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,17]) [i] based on
- linear OA(2773, 730, F27, 37) (dual of [730, 657, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(2769, 730, F27, 35) (dual of [730, 661, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,18]) ⊂ C([0,17]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2774, 735, F27, 37) (dual of [735, 661, 38]-code), using
(37, 74, 185400)-Net in Base 27 — Upper bound on s
There is no (37, 74, 185401)-net in base 27, because
- 1 times m-reduction [i] would yield (37, 73, 185401)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 308 733756 490679 809377 324877 830835 028056 781269 038913 927209 802792 477784 410949 137601 441543 562532 160926 715209 > 2773 [i]