Best Known (42, 42+16, s)-Nets in Base 64
(42, 42+16, 32898)-Net over F64 — Constructive and digital
Digital (42, 58, 32898)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 130)-net over F64, using
- (u, u+v)-construction [i] based on
- 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)
- digital (0, 4, 65)-net over F64, using
- (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 (4, 12, 130)-net over F64, using
(42, 42+16, 262145)-Net in Base 64 — Constructive
(42, 58, 262145)-net in base 64, using
- 642 times duplication [i] based on (40, 56, 262145)-net in base 64, using
- base change [i] based on digital (32, 48, 262145)-net over F128, using
- net defined by OOA [i] based on linear OOA(12848, 262145, F128, 16, 16) (dual of [(262145, 16), 4194272, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(12848, 2097160, F128, 16) (dual of [2097160, 2097112, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(12848, 2097163, F128, 16) (dual of [2097163, 2097115, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [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(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(12848, 2097163, F128, 16) (dual of [2097163, 2097115, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(12848, 2097160, F128, 16) (dual of [2097160, 2097112, 17]-code), using
- net defined by OOA [i] based on linear OOA(12848, 262145, F128, 16, 16) (dual of [(262145, 16), 4194272, 17]-NRT-code), using
- base change [i] based on digital (32, 48, 262145)-net over F128, using
(42, 42+16, 982484)-Net over F64 — Digital
Digital (42, 58, 982484)-net over F64, using
(42, 42+16, 1048581)-Net in Base 64
(42, 58, 1048581)-net in base 64, using
- 642 times duplication [i] based on (40, 56, 1048581)-net in base 64, using
- base change [i] based on digital (32, 48, 1048581)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12848, 1048581, F128, 2, 16) (dual of [(1048581, 2), 2097114, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12848, 2097162, F128, 16) (dual of [2097162, 2097114, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(12848, 2097163, F128, 16) (dual of [2097163, 2097115, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [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(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(12848, 2097163, F128, 16) (dual of [2097163, 2097115, 17]-code), using
- OOA 2-folding [i] based on linear OA(12848, 2097162, F128, 16) (dual of [2097162, 2097114, 17]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12848, 1048581, F128, 2, 16) (dual of [(1048581, 2), 2097114, 17]-NRT-code), using
- base change [i] based on digital (32, 48, 1048581)-net over F128, using
(42, 42+16, large)-Net in Base 64 — Upper bound on s
There is no (42, 58, large)-net in base 64, because
- 14 times m-reduction [i] would yield (42, 44, large)-net in base 64, but