Best Known (86−42, 86, s)-Nets in Base 64
(86−42, 86, 513)-Net over F64 — Constructive and digital
Digital (44, 86, 513)-net over F64, using
- t-expansion [i] based on digital (28, 86, 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
(86−42, 86, 515)-Net in Base 64 — Constructive
(44, 86, 515)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 28, 257)-net in base 64, using
- base change [i] based on digital (0, 21, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 21, 257)-net over F256, using
- (16, 58, 258)-net in base 64, using
- 2 times m-reduction [i] based on (16, 60, 258)-net in base 64, using
- base change [i] based on digital (1, 45, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 45, 258)-net over F256, using
- 2 times m-reduction [i] based on (16, 60, 258)-net in base 64, using
- (7, 28, 257)-net in base 64, using
(86−42, 86, 1879)-Net over F64 — Digital
Digital (44, 86, 1879)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6486, 1879, F64, 2, 42) (dual of [(1879, 2), 3672, 43]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6486, 2053, F64, 2, 42) (dual of [(2053, 2), 4020, 43]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6486, 4106, F64, 42) (dual of [4106, 4020, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(6486, 4107, F64, 42) (dual of [4107, 4021, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(37) [i] based on
- linear OA(6483, 4096, F64, 42) (dual of [4096, 4013, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(6475, 4096, F64, 38) (dual of [4096, 4021, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to Ce(41) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(6486, 4107, F64, 42) (dual of [4107, 4021, 43]-code), using
- OOA 2-folding [i] based on linear OA(6486, 4106, F64, 42) (dual of [4106, 4020, 43]-code), using
- discarding factors / shortening the dual code based on linear OOA(6486, 2053, F64, 2, 42) (dual of [(2053, 2), 4020, 43]-NRT-code), using
(86−42, 86, 3434691)-Net in Base 64 — Upper bound on s
There is no (44, 86, 3434692)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 214525 456352 175189 168272 584370 559532 952475 496837 945370 684926 242888 461087 922265 908047 072481 980963 627623 043262 254174 221157 067163 152566 559345 045644 151233 792368 > 6486 [i]