Best Known (47−20, 47, s)-Nets in Base 64
(47−20, 47, 412)-Net over F64 — Constructive and digital
Digital (27, 47, 412)-net over F64, using
- net defined by OOA [i] based on linear OOA(6447, 412, F64, 20, 20) (dual of [(412, 20), 8193, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(6447, 4120, F64, 20) (dual of [4120, 4073, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(6447, 4122, F64, 20) (dual of [4122, 4075, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(10) [i] based on
- 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(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(648, 26, F64, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,64)), using
- discarding factors / shortening the dual code based on linear OA(648, 64, F64, 8) (dual of [64, 56, 9]-code or 64-arc in PG(7,64)), using
- Reed–Solomon code RS(56,64) [i]
- discarding factors / shortening the dual code based on linear OA(648, 64, F64, 8) (dual of [64, 56, 9]-code or 64-arc in PG(7,64)), using
- construction X applied to Ce(19) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(6447, 4122, F64, 20) (dual of [4122, 4075, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(6447, 4120, F64, 20) (dual of [4120, 4073, 21]-code), using
(47−20, 47, 1638)-Net in Base 64 — Constructive
(27, 47, 1638)-net in base 64, using
- 1 times m-reduction [i] based on (27, 48, 1638)-net in base 64, using
- net defined by OOA [i] based on OOA(6448, 1638, S64, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(6448, 16381, S64, 21), using
- discarding factors based on OA(6448, 16386, S64, 21), using
- discarding parts of the base [i] based on linear OA(12841, 16386, F128, 21) (dual of [16386, 16345, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(12841, 16384, F128, 21) (dual of [16384, 16343, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12839, 16384, F128, 20) (dual of [16384, 16345, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(12841, 16386, F128, 21) (dual of [16386, 16345, 22]-code), using
- discarding factors based on OA(6448, 16386, S64, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(6448, 16381, S64, 21), using
- net defined by OOA [i] based on OOA(6448, 1638, S64, 21, 21), using
(47−20, 47, 4402)-Net over F64 — Digital
Digital (27, 47, 4402)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6447, 4402, F64, 20) (dual of [4402, 4355, 21]-code), using
- 296 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 7 times 0, 1, 23 times 0, 1, 70 times 0, 1, 189 times 0) [i] based on linear OA(6439, 4098, F64, 20) (dual of [4098, 4059, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- 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(6437, 4096, F64, 19) (dual of [4096, 4059, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- 296 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 7 times 0, 1, 23 times 0, 1, 70 times 0, 1, 189 times 0) [i] based on linear OA(6439, 4098, F64, 20) (dual of [4098, 4059, 21]-code), using
(47−20, 47, large)-Net in Base 64 — Upper bound on s
There is no (27, 47, large)-net in base 64, because
- 18 times m-reduction [i] would yield (27, 29, large)-net in base 64, but