Best Known (53, 53+20, s)-Nets in Base 64
(53, 53+20, 26344)-Net over F64 — Constructive and digital
Digital (53, 73, 26344)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (5, 15, 130)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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 (0, 10, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 5, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- 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 (5, 15, 130)-net over F64, using
(53, 53+20, 209717)-Net in Base 64 — Constructive
(53, 73, 209717)-net in base 64, using
- net defined by OOA [i] based on OOA(6473, 209717, S64, 20, 20), using
- OA 10-folding and stacking [i] based on OA(6473, 2097170, S64, 20), using
- discarding factors based on OA(6473, 2097171, S64, 20), using
- discarding parts of the base [i] based on linear OA(12862, 2097171, F128, 20) (dual of [2097171, 2097109, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [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(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- discarding parts of the base [i] based on linear OA(12862, 2097171, F128, 20) (dual of [2097171, 2097109, 21]-code), using
- discarding factors based on OA(6473, 2097171, S64, 20), using
- OA 10-folding and stacking [i] based on OA(6473, 2097170, S64, 20), using
(53, 53+20, 1094995)-Net over F64 — Digital
Digital (53, 73, 1094995)-net over F64, using
(53, 53+20, large)-Net in Base 64 — Upper bound on s
There is no (53, 73, large)-net in base 64, because
- 18 times m-reduction [i] would yield (53, 55, large)-net in base 64, but