Best Known (43, 43+39, s)-Nets in Base 64
(43, 43+39, 513)-Net over F64 — Constructive and digital
Digital (43, 82, 513)-net over F64, using
- t-expansion [i] based on digital (28, 82, 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
(43, 43+39, 545)-Net in Base 64 — Constructive
(43, 82, 545)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 26, 257)-net in base 64, using
- 2 times m-reduction [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
- 2 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
- (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 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, 48, 288)-net over F128, using
- (7, 26, 257)-net in base 64, using
(43, 43+39, 2075)-Net over F64 — Digital
Digital (43, 82, 2075)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6482, 2075, F64, 39) (dual of [2075, 1993, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(6482, 4114, F64, 39) (dual of [4114, 4032, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- linear OA(6477, 4097, F64, 39) (dual of [4097, 4020, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(6465, 4097, F64, 33) (dual of [4097, 4032, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (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([0,19]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6482, 4114, F64, 39) (dual of [4114, 4032, 40]-code), using
(43, 43+39, 6308189)-Net in Base 64 — Upper bound on s
There is no (43, 82, 6308190)-net in base 64, because
- 1 times m-reduction [i] would yield (43, 81, 6308190)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 199 792201 481289 854790 583142 566942 359195 840234 535018 936075 312869 134001 816987 335103 982713 179710 929845 636190 016298 880549 839360 707475 297412 877054 476910 > 6481 [i]