Best Known (28, 39, s)-Nets in Base 64
(28, 39, 54446)-Net over F64 — Constructive and digital
Digital (28, 39, 54446)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 2016)-net over F64, using
- net defined by OOA [i] based on linear OOA(647, 2016, F64, 5, 5) (dual of [(2016, 5), 10073, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- net defined by OOA [i] based on linear OOA(647, 2016, F64, 5, 5) (dual of [(2016, 5), 10073, 6]-NRT-code), using
- digital (21, 32, 52430)-net over F64, using
- net defined by OOA [i] based on linear OOA(6432, 52430, F64, 11, 11) (dual of [(52430, 11), 576698, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6432, 262151, F64, 11) (dual of [262151, 262119, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6432, 262152, F64, 11) (dual of [262152, 262120, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(6431, 262145, F64, 11) (dual of [262145, 262114, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(6425, 262145, F64, 9) (dual of [262145, 262120, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6432, 262152, F64, 11) (dual of [262152, 262120, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6432, 262151, F64, 11) (dual of [262151, 262119, 12]-code), using
- net defined by OOA [i] based on linear OOA(6432, 52430, F64, 11, 11) (dual of [(52430, 11), 576698, 12]-NRT-code), using
- digital (2, 7, 2016)-net over F64, using
(28, 39, 419432)-Net in Base 64 — Constructive
(28, 39, 419432)-net in base 64, using
- net defined by OOA [i] based on OOA(6439, 419432, S64, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(6439, 2097161, S64, 11), using
- discarding factors based on OA(6439, 2097163, S64, 11), using
- discarding parts of the base [i] based on linear OA(12833, 2097163, F128, 11) (dual of [2097163, 2097130, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(12831, 2097152, F128, 11) (dual of [2097152, 2097121, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- discarding parts of the base [i] based on linear OA(12833, 2097163, F128, 11) (dual of [2097163, 2097130, 12]-code), using
- discarding factors based on OA(6439, 2097163, S64, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(6439, 2097161, S64, 11), using
(28, 39, 795684)-Net over F64 — Digital
Digital (28, 39, 795684)-net over F64, using
(28, 39, large)-Net in Base 64 — Upper bound on s
There is no (28, 39, large)-net in base 64, because
- 9 times m-reduction [i] would yield (28, 30, large)-net in base 64, but