Best Known (42, 42+17, s)-Nets in Base 64
(42, 42+17, 32848)-Net over F64 — Constructive and digital
Digital (42, 59, 32848)-net over F64, using
- 641 times duplication [i] based on digital (41, 58, 32848)-net over F64, using
- (u, u+v)-construction [i] based on
- 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 (32, 49, 32768)-net over F64, using
- net defined by OOA [i] based on linear OOA(6449, 32768, F64, 17, 17) (dual of [(32768, 17), 557007, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6449, 262145, F64, 17) (dual of [262145, 262096, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(6449, 262145, F64, 17) (dual of [262145, 262096, 18]-code), using
- net defined by OOA [i] based on linear OOA(6449, 32768, F64, 17, 17) (dual of [(32768, 17), 557007, 18]-NRT-code), using
- digital (1, 9, 80)-net over F64, using
- (u, u+v)-construction [i] based on
(42, 42+17, 262144)-Net in Base 64 — Constructive
(42, 59, 262144)-net in base 64, using
- 641 times duplication [i] based on (41, 58, 262144)-net in base 64, using
- net defined by OOA [i] based on OOA(6458, 262144, S64, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(6458, 2097153, S64, 17), using
- discarding factors based on OA(6458, 2097155, S64, 17), using
- discarding parts of the base [i] based on linear OA(12849, 2097155, F128, 17) (dual of [2097155, 2097106, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(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(16) ⊂ Ce(15) [i] based on
- discarding parts of the base [i] based on linear OA(12849, 2097155, F128, 17) (dual of [2097155, 2097106, 18]-code), using
- discarding factors based on OA(6458, 2097155, S64, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(6458, 2097153, S64, 17), using
- net defined by OOA [i] based on OOA(6458, 262144, S64, 17, 17), using
(42, 42+17, 493735)-Net over F64 — Digital
Digital (42, 59, 493735)-net over F64, using
(42, 42+17, large)-Net in Base 64 — Upper bound on s
There is no (42, 59, large)-net in base 64, because
- 15 times m-reduction [i] would yield (42, 44, large)-net in base 64, but