Best Known (20−9, 20, s)-Nets in Base 64
(20−9, 20, 1026)-Net over F64 — Constructive and digital
Digital (11, 20, 1026)-net over F64, using
- net defined by OOA [i] based on linear OOA(6420, 1026, F64, 9, 9) (dual of [(1026, 9), 9214, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6420, 4105, F64, 9) (dual of [4105, 4085, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6420, 4108, F64, 9) (dual of [4108, 4088, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(649, 4097, F64, 5) (dual of [4097, 4088, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6420, 4108, F64, 9) (dual of [4108, 4088, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6420, 4105, F64, 9) (dual of [4105, 4085, 10]-code), using
(20−9, 20, 4096)-Net in Base 64 — Constructive
(11, 20, 4096)-net in base 64, using
- net defined by OOA [i] based on OOA(6420, 4096, S64, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(6420, 16385, S64, 9), using
- discarding factors based on OA(6420, 16386, S64, 9), using
- discarding parts of the base [i] based on linear OA(12817, 16386, F128, 9) (dual of [16386, 16369, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- discarding parts of the base [i] based on linear OA(12817, 16386, F128, 9) (dual of [16386, 16369, 10]-code), using
- discarding factors based on OA(6420, 16386, S64, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(6420, 16385, S64, 9), using
(20−9, 20, 4139)-Net over F64 — Digital
Digital (11, 20, 4139)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6420, 4139, F64, 9) (dual of [4139, 4119, 10]-code), using
- 38 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 33 times 0) [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- 38 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 33 times 0) [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
(20−9, 20, large)-Net in Base 64 — Upper bound on s
There is no (11, 20, large)-net in base 64, because
- 7 times m-reduction [i] would yield (11, 13, large)-net in base 64, but