Best Known (45, 45+37, s)-Nets in Base 64
(45, 45+37, 513)-Net over F64 — Constructive and digital
Digital (45, 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
(45, 45+37, 548)-Net in Base 64 — Constructive
(45, 82, 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, 54, 288)-net in base 64, using
- 2 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
- 2 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- (10, 28, 260)-net in base 64, using
(45, 45+37, 3325)-Net over F64 — Digital
Digital (45, 82, 3325)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6482, 3325, F64, 37) (dual of [3325, 3243, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(6482, 4126, F64, 37) (dual of [4126, 4044, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,13]) [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(6453, 4097, F64, 27) (dual of [4097, 4044, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(649, 29, F64, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,64)), using
- discarding factors / shortening the dual code based on linear OA(649, 64, F64, 9) (dual of [64, 55, 10]-code or 64-arc in PG(8,64)), using
- Reed–Solomon code RS(55,64) [i]
- discarding factors / shortening the dual code based on linear OA(649, 64, F64, 9) (dual of [64, 55, 10]-code or 64-arc in PG(8,64)), using
- construction X applied to C([0,18]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6482, 4126, F64, 37) (dual of [4126, 4044, 38]-code), using
(45, 45+37, large)-Net in Base 64 — Upper bound on s
There is no (45, 82, large)-net in base 64, because
- 35 times m-reduction [i] would yield (45, 47, large)-net in base 64, but