Best Known (75−20, 75, s)-Nets in Base 64
(75−20, 75, 26391)-Net over F64 — Constructive and digital
Digital (55, 75, 26391)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (7, 17, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-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 (7, 17, 177)-net over F64, using
(75−20, 75, 209717)-Net in Base 64 — Constructive
(55, 75, 209717)-net in base 64, using
- 1 times m-reduction [i] based on (55, 76, 209717)-net in base 64, using
- net defined by OOA [i] based on OOA(6476, 209717, S64, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(6476, 2097171, S64, 21), using
- discarding parts of the base [i] based on linear OA(12865, 2097171, F128, 21) (dual of [2097171, 2097106, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- 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(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(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(20) ⊂ Ce(15) [i] based on
- discarding parts of the base [i] based on linear OA(12865, 2097171, F128, 21) (dual of [2097171, 2097106, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on OA(6476, 2097171, S64, 21), using
- net defined by OOA [i] based on OOA(6476, 209717, S64, 21, 21), using
(75−20, 75, 1696427)-Net over F64 — Digital
Digital (55, 75, 1696427)-net over F64, using
(75−20, 75, large)-Net in Base 64 — Upper bound on s
There is no (55, 75, large)-net in base 64, because
- 18 times m-reduction [i] would yield (55, 57, large)-net in base 64, but