Best Known (83−40, 83, s)-Nets in Base 64
(83−40, 83, 513)-Net over F64 — Constructive and digital
Digital (43, 83, 513)-net over F64, using
- t-expansion [i] based on digital (28, 83, 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
(83−40, 83, 516)-Net in Base 64 — Constructive
(43, 83, 516)-net in base 64, using
- 1 times m-reduction [i] based on (43, 84, 516)-net in base 64, using
- base change [i] based on digital (22, 63, 516)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (1, 21, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- digital (1, 42, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256 (see above)
- digital (1, 21, 258)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (22, 63, 516)-net over F256, using
(83−40, 83, 2055)-Net over F64 — Digital
Digital (43, 83, 2055)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6483, 2055, F64, 2, 40) (dual of [(2055, 2), 4027, 41]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6483, 4110, F64, 40) (dual of [4110, 4027, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(34) [i] based on
- linear OA(6479, 4096, F64, 40) (dual of [4096, 4017, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(6469, 4096, F64, 35) (dual of [4096, 4027, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(39) ⊂ Ce(34) [i] based on
- OOA 2-folding [i] based on linear OA(6483, 4110, F64, 40) (dual of [4110, 4027, 41]-code), using
(83−40, 83, 4126782)-Net in Base 64 — Upper bound on s
There is no (43, 83, 4126783)-net in base 64, because
- the generalized Rao bound for nets shows that 64m ≥ 818351 140673 465293 885803 228065 230969 915385 533664 033119 446935 290363 852050 400899 041003 151297 612407 541945 247751 734774 977230 348087 666279 116473 785702 679049 > 6483 [i]