Best Known (90−43, 90, s)-Nets in Base 64
(90−43, 90, 513)-Net over F64 — Constructive and digital
Digital (47, 90, 513)-net over F64, using
- t-expansion [i] based on digital (28, 90, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(90−43, 90, 545)-Net in Base 64 — Constructive
(47, 90, 545)-net in base 64, using
- 641 times duplication [i] based on (46, 89, 545)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 28, 257)-net in base 64, using
- base change [i] based on digital (0, 21, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 21, 257)-net over F256, using
- (18, 61, 288)-net in base 64, using
- 2 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- base change [i] based on digital (9, 54, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 54, 288)-net over F128, using
- 2 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- (7, 28, 257)-net in base 64, using
- (u, u+v)-construction [i] based on
(90−43, 90, 2115)-Net over F64 — Digital
Digital (47, 90, 2115)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6490, 2115, F64, 43) (dual of [2115, 2025, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(6490, 4114, F64, 43) (dual of [4114, 4024, 44]-code), using
- construction X applied to C([0,21]) ⊂ C([0,18]) [i] based on
- linear OA(6485, 4097, F64, 43) (dual of [4097, 4012, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(6473, 4097, F64, 37) (dual of [4097, 4024, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to C([0,21]) ⊂ C([0,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6490, 4114, F64, 43) (dual of [4114, 4024, 44]-code), using
(90−43, 90, 6221771)-Net in Base 64 — Upper bound on s
There is no (47, 90, 6221772)-net in base 64, because
- 1 times m-reduction [i] would yield (47, 89, 6221772)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 56236 555030 044703 626305 605350 695550 174823 983686 525788 062388 490762 163009 127928 961636 676529 973382 470113 721624 501714 782786 826393 657466 676477 347430 579631 146376 007432 > 6489 [i]