Best Known (26, 26+25, s)-Nets in Base 64
(26, 26+25, 354)-Net over F64 — Constructive and digital
Digital (26, 51, 354)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (7, 19, 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 (7, 32, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64 (see above)
- digital (7, 19, 177)-net over F64, using
(26, 26+25, 515)-Net in Base 64 — Constructive
(26, 51, 515)-net in base 64, using
- (u, u+v)-construction [i] based on
- (4, 16, 257)-net in base 64, using
- base change [i] based on digital (0, 12, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 12, 257)-net over F256, using
- (10, 35, 258)-net in base 64, using
- 1 times m-reduction [i] based on (10, 36, 258)-net in base 64, using
- base change [i] based on digital (1, 27, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 27, 258)-net over F256, using
- 1 times m-reduction [i] based on (10, 36, 258)-net in base 64, using
- (4, 16, 257)-net in base 64, using
(26, 26+25, 1505)-Net over F64 — Digital
Digital (26, 51, 1505)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6451, 1505, F64, 2, 25) (dual of [(1505, 2), 2959, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6451, 2052, F64, 2, 25) (dual of [(2052, 2), 4053, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6451, 4104, F64, 25) (dual of [4104, 4053, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(6449, 4096, F64, 25) (dual of [4096, 4047, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(6443, 4096, F64, 22) (dual of [4096, 4053, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(6451, 4104, F64, 25) (dual of [4104, 4053, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(6451, 2052, F64, 2, 25) (dual of [(2052, 2), 4053, 26]-NRT-code), using
(26, 26+25, 2816889)-Net in Base 64 — Upper bound on s
There is no (26, 51, 2816890)-net in base 64, because
- 1 times m-reduction [i] would yield (26, 50, 2816890)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 2 037036 514311 860010 595279 159049 615091 961114 628201 680141 095409 708193 831396 045834 506819 074148 > 6450 [i]