Best Known (64−31, 64, s)-Nets in Base 27
(64−31, 64, 178)-Net over F27 — Constructive and digital
Digital (33, 64, 178)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (8, 23, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- digital (10, 41, 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 (8, 23, 84)-net over F27, using
(64−31, 64, 370)-Net in Base 27 — Constructive
(33, 64, 370)-net in base 27, using
- 4 times m-reduction [i] based on (33, 68, 370)-net in base 27, using
- base change [i] based on digital (16, 51, 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, 51, 370)-net over F81, using
(64−31, 64, 564)-Net over F27 — Digital
Digital (33, 64, 564)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2764, 564, F27, 31) (dual of [564, 500, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2764, 741, F27, 31) (dual of [741, 677, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(2761, 730, F27, 31) (dual of [730, 669, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (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(273, 11, F27, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,27) or 11-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2764, 741, F27, 31) (dual of [741, 677, 32]-code), using
(64−31, 64, 253809)-Net in Base 27 — Upper bound on s
There is no (33, 64, 253810)-net in base 27, because
- 1 times m-reduction [i] would yield (33, 63, 253810)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 1 499429 372016 119350 666793 732552 000934 591372 994872 119129 081332 025011 777882 146800 479845 269385 > 2763 [i]