Best Known (47−23, 47, s)-Nets in Base 64
(47−23, 47, 373)-Net over F64 — Constructive and digital
Digital (24, 47, 373)-net over F64, using
- net defined by OOA [i] based on linear OOA(6447, 373, F64, 23, 23) (dual of [(373, 23), 8532, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6447, 4104, F64, 23) (dual of [4104, 4057, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(22) ⊂ Ce(19) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(6447, 4104, F64, 23) (dual of [4104, 4057, 24]-code), using
(47−23, 47, 515)-Net in Base 64 — Constructive
(24, 47, 515)-net in base 64, using
- (u, u+v)-construction [i] based on
- (4, 15, 257)-net in base 64, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on (4, 16, 257)-net in base 64, using
- (9, 32, 258)-net in base 64, using
- base change [i] based on digital (1, 24, 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, 24, 258)-net over F256, using
- (4, 15, 257)-net in base 64, using
(47−23, 47, 1518)-Net over F64 — Digital
Digital (24, 47, 1518)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6447, 1518, F64, 2, 23) (dual of [(1518, 2), 2989, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6447, 2052, F64, 2, 23) (dual of [(2052, 2), 4057, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6447, 4104, F64, 23) (dual of [4104, 4057, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(22) ⊂ Ce(19) [i] based on
- OOA 2-folding [i] based on linear OA(6447, 4104, F64, 23) (dual of [4104, 4057, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(6447, 2052, F64, 2, 23) (dual of [(2052, 2), 4057, 24]-NRT-code), using
(47−23, 47, 2784768)-Net in Base 64 — Upper bound on s
There is no (24, 47, 2784769)-net in base 64, because
- 1 times m-reduction [i] would yield (24, 46, 2784769)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 121417 178391 803690 458407 725004 306023 709386 158908 663920 904577 118539 896391 005503 587840 > 6446 [i]