Best Known (28, 28+27, s)-Nets in Base 64
(28, 28+27, 513)-Net over F64 — Constructive and digital
Digital (28, 55, 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
(28, 28+27, 514)-Net in Base 64 — Constructive
(28, 55, 514)-net in base 64, using
- 1 times m-reduction [i] based on (28, 56, 514)-net in base 64, using
- base change [i] based on digital (14, 42, 514)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (0, 14, 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
- digital (0, 28, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- digital (0, 14, 257)-net over F256, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (14, 42, 514)-net over F256, using
(28, 28+27, 1505)-Net over F64 — Digital
Digital (28, 55, 1505)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6455, 1505, F64, 2, 27) (dual of [(1505, 2), 2955, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6455, 2052, F64, 2, 27) (dual of [(2052, 2), 4049, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6455, 4104, F64, 27) (dual of [4104, 4049, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- linear OA(6453, 4096, F64, 27) (dual of [4096, 4043, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(6447, 4096, F64, 24) (dual of [4096, 4049, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- OOA 2-folding [i] based on linear OA(6455, 4104, F64, 27) (dual of [4104, 4049, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(6455, 2052, F64, 2, 27) (dual of [(2052, 2), 4049, 28]-NRT-code), using
(28, 28+27, 2861926)-Net in Base 64 — Upper bound on s
There is no (28, 55, 2861927)-net in base 64, because
- 1 times m-reduction [i] would yield (28, 54, 2861927)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 34 175922 372338 550417 616505 671463 493583 116164 426570 998692 589270 565524 351639 264032 481984 129732 686328 > 6454 [i]