Best Known (43, 43+35, s)-Nets in Base 64
(43, 43+35, 513)-Net over F64 — Constructive and digital
Digital (43, 78, 513)-net over F64, using
- t-expansion [i] based on digital (28, 78, 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
(43, 43+35, 547)-Net in Base 64 — Constructive
(43, 78, 547)-net in base 64, using
- (u, u+v)-construction [i] based on
- (9, 26, 259)-net in base 64, using
- 2 times m-reduction [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
- 2 times m-reduction [i] based on (9, 28, 259)-net in base 64, using
- (17, 52, 288)-net in base 64, using
- 4 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
- 4 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- (9, 26, 259)-net in base 64, using
(43, 43+35, 3407)-Net over F64 — Digital
Digital (43, 78, 3407)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6478, 3407, F64, 35) (dual of [3407, 3329, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(6478, 4126, F64, 35) (dual of [4126, 4048, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,12]) [i] based on
- linear OA(6469, 4097, F64, 35) (dual of [4097, 4028, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(6449, 4097, F64, 25) (dual of [4097, 4048, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,17]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6478, 4126, F64, 35) (dual of [4126, 4048, 36]-code), using
(43, 43+35, large)-Net in Base 64 — Upper bound on s
There is no (43, 78, large)-net in base 64, because
- 33 times m-reduction [i] would yield (43, 45, large)-net in base 64, but