Best Known (41, 41+33, s)-Nets in Base 64
(41, 41+33, 513)-Net over F64 — Constructive and digital
Digital (41, 74, 513)-net over F64, using
- t-expansion [i] based on digital (28, 74, 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
(41, 41+33, 547)-Net in Base 64 — Constructive
(41, 74, 547)-net in base 64, using
- 641 times duplication [i] based on (40, 73, 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, 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
- (8, 24, 259)-net in base 64, using
- (u, u+v)-construction [i] based on
(41, 41+33, 3517)-Net over F64 — Digital
Digital (41, 74, 3517)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6474, 3517, F64, 33) (dual of [3517, 3443, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(6474, 4126, F64, 33) (dual of [4126, 4052, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,11]) [i] based on
- linear OA(6465, 4097, F64, 33) (dual of [4097, 4032, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(6445, 4097, F64, 23) (dual of [4097, 4052, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,16]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6474, 4126, F64, 33) (dual of [4126, 4052, 34]-code), using
(41, 41+33, large)-Net in Base 64 — Upper bound on s
There is no (41, 74, large)-net in base 64, because
- 31 times m-reduction [i] would yield (41, 43, large)-net in base 64, but