Best Known (40, 72, s)-Nets in Base 64
(40, 72, 513)-Net over F64 — Constructive and digital
Digital (40, 72, 513)-net over F64, using
- t-expansion [i] based on digital (28, 72, 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
(40, 72, 547)-Net in Base 64 — Constructive
(40, 72, 547)-net in base 64, using
- (u, u+v)-construction [i] based on
- (8, 24, 259)-net in base 64, using
- base change [i] based on digital (2, 18, 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, 18, 259)-net over F256, using
- (16, 48, 288)-net in base 64, using
- 1 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- base change [i] based on digital (9, 42, 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, 42, 288)-net over F128, using
- 1 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- (8, 24, 259)-net in base 64, using
(40, 72, 3584)-Net over F64 — Digital
Digital (40, 72, 3584)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6472, 3584, F64, 32) (dual of [3584, 3512, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(6472, 4125, F64, 32) (dual of [4125, 4053, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(21) [i] based on
- linear OA(6463, 4096, F64, 32) (dual of [4096, 4033, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(6443, 4096, F64, 22) (dual of [4096, 4053, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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 Ce(31) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(6472, 4125, F64, 32) (dual of [4125, 4053, 33]-code), using
(40, 72, large)-Net in Base 64 — Upper bound on s
There is no (40, 72, large)-net in base 64, because
- 30 times m-reduction [i] would yield (40, 42, large)-net in base 64, but