Best Known (81−36, 81, s)-Nets in Base 64
(81−36, 81, 513)-Net over F64 — Constructive and digital
Digital (45, 81, 513)-net over F64, using
- t-expansion [i] based on digital (28, 81, 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
(81−36, 81, 548)-Net in Base 64 — Constructive
(45, 81, 548)-net in base 64, using
- (u, u+v)-construction [i] based on
- (10, 28, 260)-net in base 64, using
- base change [i] based on digital (3, 21, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- base change [i] based on digital (3, 21, 260)-net over F256, using
- (17, 53, 288)-net in base 64, using
- 3 times m-reduction [i] based on (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
- 3 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- (10, 28, 260)-net in base 64, using
(81−36, 81, 3803)-Net over F64 — Digital
Digital (45, 81, 3803)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6481, 3803, F64, 36) (dual of [3803, 3722, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(6481, 4128, F64, 36) (dual of [4128, 4047, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(24) [i] based on
- linear OA(6471, 4096, F64, 36) (dual of [4096, 4025, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(6449, 4096, F64, 25) (dual of [4096, 4047, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(6410, 32, F64, 10) (dual of [32, 22, 11]-code or 32-arc in PG(9,64)), using
- discarding factors / shortening the dual code based on linear OA(6410, 64, F64, 10) (dual of [64, 54, 11]-code or 64-arc in PG(9,64)), using
- Reed–Solomon code RS(54,64) [i]
- discarding factors / shortening the dual code based on linear OA(6410, 64, F64, 10) (dual of [64, 54, 11]-code or 64-arc in PG(9,64)), using
- construction X applied to Ce(35) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(6481, 4128, F64, 36) (dual of [4128, 4047, 37]-code), using
(81−36, 81, large)-Net in Base 64 — Upper bound on s
There is no (45, 81, large)-net in base 64, because
- 34 times m-reduction [i] would yield (45, 47, large)-net in base 64, but