Best Known (83−38, 83, s)-Nets in Base 64
(83−38, 83, 513)-Net over F64 — Constructive and digital
Digital (45, 83, 513)-net over F64, using
- t-expansion [i] based on digital (28, 83, 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
(83−38, 83, 547)-Net in Base 64 — Constructive
(45, 83, 547)-net in base 64, using
- (u, u+v)-construction [i] based on
- (9, 28, 259)-net in base 64, using
- base change [i] based on digital (2, 21, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- base change [i] based on digital (2, 21, 259)-net over F256, using
- (17, 55, 288)-net in base 64, using
- 1 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
- 1 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- (9, 28, 259)-net in base 64, using
(83−38, 83, 2930)-Net over F64 — Digital
Digital (45, 83, 2930)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6483, 2930, F64, 38) (dual of [2930, 2847, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(6483, 4122, F64, 38) (dual of [4122, 4039, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(28) [i] based on
- linear OA(6475, 4096, F64, 38) (dual of [4096, 4021, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(6457, 4096, F64, 29) (dual of [4096, 4039, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(648, 26, F64, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,64)), using
- discarding factors / shortening the dual code based on linear OA(648, 64, F64, 8) (dual of [64, 56, 9]-code or 64-arc in PG(7,64)), using
- Reed–Solomon code RS(56,64) [i]
- discarding factors / shortening the dual code based on linear OA(648, 64, F64, 8) (dual of [64, 56, 9]-code or 64-arc in PG(7,64)), using
- construction X applied to Ce(37) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(6483, 4122, F64, 38) (dual of [4122, 4039, 39]-code), using
(83−38, 83, large)-Net in Base 64 — Upper bound on s
There is no (45, 83, large)-net in base 64, because
- 36 times m-reduction [i] would yield (45, 47, large)-net in base 64, but