Best Known (28, 28+16, s)-Nets in Base 64
(28, 28+16, 657)-Net over F64 — Constructive and digital
Digital (28, 44, 657)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (5, 13, 145)-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 (1, 9, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (0, 4, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (15, 31, 512)-net over F64, using
- net defined by OOA [i] based on linear OOA(6431, 512, F64, 16, 16) (dual of [(512, 16), 8161, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OA 8-folding and stacking [i] based on linear OA(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using
- net defined by OOA [i] based on linear OOA(6431, 512, F64, 16, 16) (dual of [(512, 16), 8161, 17]-NRT-code), using
- digital (5, 13, 145)-net over F64, using
(28, 28+16, 8193)-Net in Base 64 — Constructive
(28, 44, 8193)-net in base 64, using
- base change [i] based on digital (17, 33, 8193)-net over F256, using
- net defined by OOA [i] based on linear OOA(25633, 8193, F256, 16, 16) (dual of [(8193, 16), 131055, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(25633, 65544, F256, 16) (dual of [65544, 65511, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- OA 8-folding and stacking [i] based on linear OA(25633, 65544, F256, 16) (dual of [65544, 65511, 17]-code), using
- net defined by OOA [i] based on linear OOA(25633, 8193, F256, 16, 16) (dual of [(8193, 16), 131055, 17]-NRT-code), using
(28, 28+16, 20263)-Net over F64 — Digital
Digital (28, 44, 20263)-net over F64, using
(28, 28+16, 21743)-Net in Base 64
(28, 44, 21743)-net in base 64, using
- base change [i] based on digital (17, 33, 21743)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25633, 21743, F256, 3, 16) (dual of [(21743, 3), 65196, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25633, 21848, F256, 3, 16) (dual of [(21848, 3), 65511, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25633, 65544, F256, 16) (dual of [65544, 65511, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- OOA 3-folding [i] based on linear OA(25633, 65544, F256, 16) (dual of [65544, 65511, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(25633, 21848, F256, 3, 16) (dual of [(21848, 3), 65511, 17]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25633, 21743, F256, 3, 16) (dual of [(21743, 3), 65196, 17]-NRT-code), using
(28, 28+16, large)-Net in Base 64 — Upper bound on s
There is no (28, 44, large)-net in base 64, because
- 14 times m-reduction [i] would yield (28, 30, large)-net in base 64, but