Best Known (39, 76, s)-Nets in Base 64
(39, 76, 513)-Net over F64 — Constructive and digital
Digital (39, 76, 513)-net over F64, using
- t-expansion [i] based on digital (28, 76, 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
(39, 76, 516)-Net in Base 64 — Constructive
(39, 76, 516)-net in base 64, using
- base change [i] based on digital (20, 57, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 19, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 38, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 19, 258)-net over F256, using
- (u, u+v)-construction [i] based on
(39, 76, 1817)-Net over F64 — Digital
Digital (39, 76, 1817)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6476, 1817, F64, 2, 37) (dual of [(1817, 2), 3558, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6476, 2054, F64, 2, 37) (dual of [(2054, 2), 4032, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6476, 4108, F64, 37) (dual of [4108, 4032, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- 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(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(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- OOA 2-folding [i] based on linear OA(6476, 4108, F64, 37) (dual of [4108, 4032, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(6476, 2054, F64, 2, 37) (dual of [(2054, 2), 4032, 38]-NRT-code), using
(39, 76, 4022892)-Net in Base 64 — Upper bound on s
There is no (39, 76, 4022893)-net in base 64, because
- 1 times m-reduction [i] would yield (39, 75, 4022893)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 2907 358644 509425 407228 382308 452829 790610 864832 609624 438146 353684 736947 519158 824925 044471 853284 006127 070347 620985 660640 093317 564580 644460 > 6475 [i]