Best Known (78−21, 78, s)-Nets in Base 64
(78−21, 78, 26391)-Net over F64 — Constructive and digital
Digital (57, 78, 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 (40, 61, 26214)-net over F64, using
- net defined by OOA [i] based on linear OOA(6461, 26214, F64, 21, 21) (dual of [(26214, 21), 550433, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6461, 262141, F64, 21) (dual of [262141, 262080, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6461, 262141, F64, 21) (dual of [262141, 262080, 22]-code), using
- net defined by OOA [i] based on linear OOA(6461, 26214, F64, 21, 21) (dual of [(26214, 21), 550433, 22]-NRT-code), using
- digital (7, 17, 177)-net over F64, using
(78−21, 78, 209717)-Net in Base 64 — Constructive
(57, 78, 209717)-net in base 64, using
- 641 times duplication [i] based on (56, 77, 209717)-net in base 64, using
- base change [i] based on digital (45, 66, 209717)-net over F128, using
- 1281 times duplication [i] based on digital (44, 65, 209717)-net over F128, using
- net defined by OOA [i] based on linear OOA(12865, 209717, F128, 21, 21) (dual of [(209717, 21), 4403992, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [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
- OOA 10-folding and stacking with additional row [i] based on linear OA(12865, 2097171, F128, 21) (dual of [2097171, 2097106, 22]-code), using
- net defined by OOA [i] based on linear OOA(12865, 209717, F128, 21, 21) (dual of [(209717, 21), 4403992, 22]-NRT-code), using
- 1281 times duplication [i] based on digital (44, 65, 209717)-net over F128, using
- base change [i] based on digital (45, 66, 209717)-net over F128, using
(78−21, 78, 1459051)-Net over F64 — Digital
Digital (57, 78, 1459051)-net over F64, using
(78−21, 78, large)-Net in Base 64 — Upper bound on s
There is no (57, 78, large)-net in base 64, because
- 19 times m-reduction [i] would yield (57, 59, large)-net in base 64, but