Best Known (48−21, 48, s)-Nets in Base 64
(48−21, 48, 411)-Net over F64 — Constructive and digital
Digital (27, 48, 411)-net over F64, using
- 642 times duplication [i] based on digital (25, 46, 411)-net over F64, using
- net defined by OOA [i] based on linear OOA(6446, 411, F64, 21, 21) (dual of [(411, 21), 8585, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6446, 4111, F64, 21) (dual of [4111, 4065, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6446, 4114, F64, 21) (dual of [4114, 4068, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(6429, 4097, F64, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6446, 4114, F64, 21) (dual of [4114, 4068, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6446, 4111, F64, 21) (dual of [4111, 4065, 22]-code), using
- net defined by OOA [i] based on linear OOA(6446, 411, F64, 21, 21) (dual of [(411, 21), 8585, 22]-NRT-code), using
(48−21, 48, 1638)-Net in Base 64 — Constructive
(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
(48−21, 48, 3688)-Net over F64 — Digital
Digital (27, 48, 3688)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6448, 3688, F64, 21) (dual of [3688, 3640, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6448, 4120, F64, 21) (dual of [4120, 4072, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(6425, 4097, F64, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(647, 23, F64, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,64)), using
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- Reed–Solomon code RS(57,64) [i]
- discarding factors / shortening the dual code based on linear OA(647, 64, F64, 7) (dual of [64, 57, 8]-code or 64-arc in PG(6,64)), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6448, 4120, F64, 21) (dual of [4120, 4072, 22]-code), using
(48−21, 48, large)-Net in Base 64 — Upper bound on s
There is no (27, 48, large)-net in base 64, because
- 19 times m-reduction [i] would yield (27, 29, large)-net in base 64, but