Best Known (87−41, 87, s)-Nets in Base 64
(87−41, 87, 513)-Net over F64 — Constructive and digital
Digital (46, 87, 513)-net over F64, using
- t-expansion [i] based on digital (28, 87, 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
(87−41, 87, 546)-Net in Base 64 — Constructive
(46, 87, 546)-net in base 64, using
- (u, u+v)-construction [i] based on
- (8, 28, 258)-net in base 64, using
- base change [i] based on digital (1, 21, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 21, 258)-net over F256, using
- (18, 59, 288)-net in base 64, using
- 4 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
- 4 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- (8, 28, 258)-net in base 64, using
(87−41, 87, 2331)-Net over F64 — Digital
Digital (46, 87, 2331)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6487, 2331, F64, 41) (dual of [2331, 2244, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(6487, 4116, F64, 41) (dual of [4116, 4029, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(33) [i] based on
- linear OA(6481, 4096, F64, 41) (dual of [4096, 4015, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(6467, 4096, F64, 34) (dual of [4096, 4029, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(646, 20, F64, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,64)), using
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- Reed–Solomon code RS(58,64) [i]
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- construction X applied to Ce(40) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(6487, 4116, F64, 41) (dual of [4116, 4029, 42]-code), using
(87−41, 87, 7700856)-Net in Base 64 — Upper bound on s
There is no (46, 87, 7700857)-net in base 64, because
- 1 times m-reduction [i] would yield (46, 86, 7700857)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 214525 352029 515750 306334 064202 395792 645180 449184 990111 221941 208756 994234 938107 985089 610110 316026 328866 656694 157014 095914 334040 359919 224948 244004 352887 738894 > 6486 [i]