Best Known (35−10, 35, s)-Nets in Base 64
(35−10, 35, 54445)-Net over F64 — Constructive and digital
Digital (25, 35, 54445)-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 (18, 28, 52429)-net over F64, using
- net defined by OOA [i] based on linear OOA(6428, 52429, F64, 10, 10) (dual of [(52429, 10), 524262, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6428, 262145, F64, 10) (dual of [262145, 262117, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6428, 262147, F64, 10) (dual of [262147, 262119, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(6425, 262144, F64, 9) (dual of [262144, 262119, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(6428, 262147, F64, 10) (dual of [262147, 262119, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6428, 262145, F64, 10) (dual of [262145, 262117, 11]-code), using
- net defined by OOA [i] based on linear OOA(6428, 52429, F64, 10, 10) (dual of [(52429, 10), 524262, 11]-NRT-code), using
- digital (2, 7, 2016)-net over F64, using
(35−10, 35, 419432)-Net in Base 64 — Constructive
(25, 35, 419432)-net in base 64, using
- base change [i] based on digital (20, 30, 419432)-net over F128, using
- net defined by OOA [i] based on linear OOA(12830, 419432, F128, 10, 10) (dual of [(419432, 10), 4194290, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(12830, 2097160, F128, 10) (dual of [2097160, 2097130, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 2097163, F128, 10) (dual of [2097163, 2097133, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 2097163, F128, 10) (dual of [2097163, 2097133, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(12830, 2097160, F128, 10) (dual of [2097160, 2097130, 11]-code), using
- net defined by OOA [i] based on linear OOA(12830, 419432, F128, 10, 10) (dual of [(419432, 10), 4194290, 11]-NRT-code), using
(35−10, 35, 695740)-Net over F64 — Digital
Digital (25, 35, 695740)-net over F64, using
(35−10, 35, 1289797)-Net in Base 64
(25, 35, 1289797)-net in base 64, using
- base change [i] based on digital (20, 30, 1289797)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12830, 1289797, F128, 10) (dual of [1289797, 1289767, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 2097163, F128, 10) (dual of [2097163, 2097133, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 2097163, F128, 10) (dual of [2097163, 2097133, 11]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12830, 1289797, F128, 10) (dual of [1289797, 1289767, 11]-code), using
(35−10, 35, large)-Net in Base 64 — Upper bound on s
There is no (25, 35, large)-net in base 64, because
- 8 times m-reduction [i] would yield (25, 27, large)-net in base 64, but