Best Known (30, 30+13, s)-Nets in Base 64
(30, 30+13, 43756)-Net over F64 — Constructive and digital
Digital (30, 43, 43756)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (24, 37, 43691)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 43691, F64, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- net defined by OOA [i] based on linear OOA(6437, 43691, F64, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
- digital (0, 6, 65)-net over F64, using
(30, 30+13, 262212)-Net over F64 — Digital
Digital (30, 43, 262212)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6443, 262212, F64, 13) (dual of [262212, 262169, 14]-code), using
- (u, u+v)-construction [i] based on
- linear OA(646, 65, F64, 6) (dual of [65, 59, 7]-code or 65-arc in PG(5,64)), using
- extended Reed–Solomon code RSe(59,64) [i]
- linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(646, 65, F64, 6) (dual of [65, 59, 7]-code or 65-arc in PG(5,64)), using
- (u, u+v)-construction [i] based on
(30, 30+13, large)-Net in Base 64 — Upper bound on s
There is no (30, 43, large)-net in base 64, because
- 11 times m-reduction [i] would yield (30, 32, large)-net in base 64, but