Best Known (42, 42+34, s)-Nets in Base 64
(42, 42+34, 513)-Net over F64 — Constructive and digital
Digital (42, 76, 513)-net over F64, using
- t-expansion [i] based on digital (28, 76, 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
(42, 42+34, 547)-Net in Base 64 — Constructive
(42, 76, 547)-net in base 64, using
- 1 times m-reduction [i] based on (42, 77, 547)-net in base 64, using
- base change [i] based on (31, 66, 547)-net in base 128, using
- (u, u+v)-construction [i] based on
- (5, 22, 259)-net in base 128, using
- 2 times m-reduction [i] based on (5, 24, 259)-net in base 128, 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
- 2 times m-reduction [i] based on (5, 24, 259)-net in base 128, using
- digital (9, 44, 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
- (5, 22, 259)-net in base 128, using
- (u, u+v)-construction [i] based on
- base change [i] based on (31, 66, 547)-net in base 128, using
(42, 42+34, 3458)-Net over F64 — Digital
Digital (42, 76, 3458)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6476, 3458, F64, 34) (dual of [3458, 3382, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(6476, 4125, F64, 34) (dual of [4125, 4049, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(23) [i] based on
- 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(6447, 4096, F64, 24) (dual of [4096, 4049, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(33) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(6476, 4125, F64, 34) (dual of [4125, 4049, 35]-code), using
(42, 42+34, large)-Net in Base 64 — Upper bound on s
There is no (42, 76, large)-net in base 64, because
- 32 times m-reduction [i] would yield (42, 44, large)-net in base 64, but