Best Known (54, 54+22, s)-Nets in Base 64
(54, 54+22, 23911)-Net over F64 — Constructive and digital
Digital (54, 76, 23911)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 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 (42, 64, 23831)-net over F64, using
- net defined by OOA [i] based on linear OOA(6464, 23831, F64, 22, 22) (dual of [(23831, 22), 524218, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(6464, 262141, F64, 22) (dual of [262141, 262077, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(6464, 262141, F64, 22) (dual of [262141, 262077, 23]-code), using
- net defined by OOA [i] based on linear OOA(6464, 23831, F64, 22, 22) (dual of [(23831, 22), 524218, 23]-NRT-code), using
- digital (1, 12, 80)-net over F64, using
(54, 54+22, 190650)-Net in Base 64 — Constructive
(54, 76, 190650)-net in base 64, using
- 641 times duplication [i] based on (53, 75, 190650)-net in base 64, using
- net defined by OOA [i] based on OOA(6475, 190650, S64, 22, 22), using
- OA 11-folding and stacking [i] based on OA(6475, 2097150, S64, 22), using
- discarding factors based on OA(6475, 2097155, S64, 22), using
- discarding parts of the base [i] based on linear OA(12864, 2097155, F128, 22) (dual of [2097155, 2097091, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(12864, 2097152, F128, 22) (dual of [2097152, 2097088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- discarding parts of the base [i] based on linear OA(12864, 2097155, F128, 22) (dual of [2097155, 2097091, 23]-code), using
- discarding factors based on OA(6475, 2097155, S64, 22), using
- OA 11-folding and stacking [i] based on OA(6475, 2097150, S64, 22), using
- net defined by OOA [i] based on OOA(6475, 190650, S64, 22, 22), using
(54, 54+22, 474038)-Net over F64 — Digital
Digital (54, 76, 474038)-net over F64, using
(54, 54+22, large)-Net in Base 64 — Upper bound on s
There is no (54, 76, large)-net in base 64, because
- 20 times m-reduction [i] would yield (54, 56, large)-net in base 64, but