Best Known (45, 87, s)-Nets in Base 64
(45, 87, 513)-Net over F64 — Constructive and digital
Digital (45, 87, 513)-net over F64, using
- t-expansion [i] based on digital (28, 87, 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
(45, 87, 516)-Net in Base 64 — Constructive
(45, 87, 516)-net in base 64, using
- 1 times m-reduction [i] based on (45, 88, 516)-net in base 64, using
- base change [i] based on digital (23, 66, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 22, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 44, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 22, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (23, 66, 516)-net over F256, using
(45, 87, 2055)-Net over F64 — Digital
Digital (45, 87, 2055)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6487, 2055, F64, 2, 42) (dual of [(2055, 2), 4023, 43]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6487, 4110, F64, 42) (dual of [4110, 4023, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(36) [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(6473, 4096, F64, 37) (dual of [4096, 4023, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(644, 14, F64, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(41) ⊂ Ce(36) [i] based on
- OOA 2-folding [i] based on linear OA(6487, 4110, F64, 42) (dual of [4110, 4023, 43]-code), using
(45, 87, 4186938)-Net in Base 64 — Upper bound on s
There is no (45, 87, 4186939)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 13 729649 236793 059675 428712 634311 912539 883262 693571 854893 099697 218125 742844 067001 563449 748332 564095 298280 110738 823806 487364 888369 067505 888380 056622 023551 287908 > 6487 [i]