Best Known (44, 44+16, s)-Nets in Base 64
(44, 44+16, 32963)-Net over F64 — Constructive and digital
Digital (44, 60, 32963)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (6, 14, 195)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 65)-net over F64, using
- digital (0, 4, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (0, 8, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- generalized (u, u+v)-construction [i] based on
- digital (30, 46, 32768)-net over F64, using
- net defined by OOA [i] based on linear OOA(6446, 32768, F64, 16, 16) (dual of [(32768, 16), 524242, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(6446, 262144, F64, 16) (dual of [262144, 262098, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OA 8-folding and stacking [i] based on linear OA(6446, 262144, F64, 16) (dual of [262144, 262098, 17]-code), using
- net defined by OOA [i] based on linear OOA(6446, 32768, F64, 16, 16) (dual of [(32768, 16), 524242, 17]-NRT-code), using
- digital (6, 14, 195)-net over F64, using
(44, 44+16, 262146)-Net in Base 64 — Constructive
(44, 60, 262146)-net in base 64, using
- 641 times duplication [i] based on (43, 59, 262146)-net in base 64, using
- net defined by OOA [i] based on OOA(6459, 262146, S64, 16, 16), using
- OA 8-folding and stacking [i] based on OA(6459, 2097168, S64, 16), using
- discarding factors based on OA(6459, 2097171, S64, 16), using
- discarding parts of the base [i] based on linear OA(12850, 2097171, F128, 16) (dual of [2097171, 2097121, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(12846, 2097152, F128, 16) (dual of [2097152, 2097106, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- discarding parts of the base [i] based on linear OA(12850, 2097171, F128, 16) (dual of [2097171, 2097121, 17]-code), using
- discarding factors based on OA(6459, 2097171, S64, 16), using
- OA 8-folding and stacking [i] based on OA(6459, 2097168, S64, 16), using
- net defined by OOA [i] based on OOA(6459, 262146, S64, 16, 16), using
(44, 44+16, 1710598)-Net over F64 — Digital
Digital (44, 60, 1710598)-net over F64, using
(44, 44+16, large)-Net in Base 64 — Upper bound on s
There is no (44, 60, large)-net in base 64, because
- 14 times m-reduction [i] would yield (44, 46, large)-net in base 64, but