Best Known (48−13, 48, s)-Nets in Base 64
(48−13, 48, 45057)-Net over F64 — Constructive and digital
Digital (35, 48, 45057)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (5, 11, 1366)-net over F64, using
- net defined by OOA [i] based on linear OOA(6411, 1366, F64, 6, 6) (dual of [(1366, 6), 8185, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(6411, 4098, F64, 6) (dual of [4098, 4087, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(649, 4096, F64, 5) (dual of [4096, 4087, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- OA 3-folding and stacking [i] based on linear OA(6411, 4098, F64, 6) (dual of [4098, 4087, 7]-code), using
- net defined by OOA [i] based on linear OOA(6411, 1366, F64, 6, 6) (dual of [(1366, 6), 8185, 7]-NRT-code), using
- digital (24, 37, 43691)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 43691, F64, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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 (see above)
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- net defined by OOA [i] based on linear OOA(6437, 43691, F64, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
- digital (5, 11, 1366)-net over F64, using
(48−13, 48, 349528)-Net in Base 64 — Constructive
(35, 48, 349528)-net in base 64, using
- net defined by OOA [i] based on OOA(6448, 349528, S64, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(6448, 2097169, S64, 13), using
- discarding factors based on OA(6448, 2097171, S64, 13), using
- discarding parts of the base [i] based on linear OA(12841, 2097171, F128, 13) (dual of [2097171, 2097130, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(1284, 19, F128, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- Reed–Solomon code RS(124,128) [i]
- discarding factors / shortening the dual code based on linear OA(1284, 128, F128, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- discarding parts of the base [i] based on linear OA(12841, 2097171, F128, 13) (dual of [2097171, 2097130, 14]-code), using
- discarding factors based on OA(6448, 2097171, S64, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(6448, 2097169, S64, 13), using
(48−13, 48, 1408454)-Net over F64 — Digital
Digital (35, 48, 1408454)-net over F64, using
(48−13, 48, large)-Net in Base 64 — Upper bound on s
There is no (35, 48, large)-net in base 64, because
- 11 times m-reduction [i] would yield (35, 37, large)-net in base 64, but