Best Known (49, 49+20, s)-Nets in Base 64
(49, 49+20, 26294)-Net over F64 — Constructive and digital
Digital (49, 69, 26294)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 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 (38, 58, 26214)-net over F64, using
- net defined by OOA [i] based on linear OOA(6458, 26214, F64, 20, 20) (dual of [(26214, 20), 524222, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(6458, 262140, F64, 20) (dual of [262140, 262082, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(6458, 262140, F64, 20) (dual of [262140, 262082, 21]-code), using
- net defined by OOA [i] based on linear OOA(6458, 26214, F64, 20, 20) (dual of [(26214, 20), 524222, 21]-NRT-code), using
- digital (1, 11, 80)-net over F64, using
(49, 49+20, 209715)-Net in Base 64 — Constructive
(49, 69, 209715)-net in base 64, using
- 641 times duplication [i] based on (48, 68, 209715)-net in base 64, using
- net defined by OOA [i] based on OOA(6468, 209715, S64, 20, 20), using
- OA 10-folding and stacking [i] based on OA(6468, 2097150, S64, 20), using
- discarding factors based on OA(6468, 2097155, S64, 20), using
- discarding parts of the base [i] based on linear OA(12858, 2097155, F128, 20) (dual of [2097155, 2097097, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(12858, 2097152, F128, 20) (dual of [2097152, 2097094, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- discarding parts of the base [i] based on linear OA(12858, 2097155, F128, 20) (dual of [2097155, 2097097, 21]-code), using
- discarding factors based on OA(6468, 2097155, S64, 20), using
- OA 10-folding and stacking [i] based on OA(6468, 2097150, S64, 20), using
- net defined by OOA [i] based on OOA(6468, 209715, S64, 20, 20), using
(49, 49+20, 456215)-Net over F64 — Digital
Digital (49, 69, 456215)-net over F64, using
(49, 49+20, large)-Net in Base 64 — Upper bound on s
There is no (49, 69, large)-net in base 64, because
- 18 times m-reduction [i] would yield (49, 51, large)-net in base 64, but