Best Known (19, 33, s)-Nets in Base 64
(19, 33, 588)-Net over F64 — Constructive and digital
Digital (19, 33, 588)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 588, F64, 14, 14) (dual of [(588, 14), 8199, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6433, 4116, F64, 14) (dual of [4116, 4083, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(6) [i] based on
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(646, 20, F64, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,64)), using
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- Reed–Solomon code RS(58,64) [i]
- discarding factors / shortening the dual code based on linear OA(646, 64, F64, 6) (dual of [64, 58, 7]-code or 64-arc in PG(5,64)), using
- construction X applied to Ce(13) ⊂ Ce(6) [i] based on
- OA 7-folding and stacking [i] based on linear OA(6433, 4116, F64, 14) (dual of [4116, 4083, 15]-code), using
(19, 33, 2341)-Net in Base 64 — Constructive
(19, 33, 2341)-net in base 64, using
- net defined by OOA [i] based on OOA(6433, 2341, S64, 14, 14), using
- OA 7-folding and stacking [i] based on OA(6433, 16387, S64, 14), using
- discarding factors based on OA(6433, 16389, S64, 14), using
- discarding parts of the base [i] based on linear OA(12828, 16389, F128, 14) (dual of [16389, 16361, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12828, 16389, F128, 14) (dual of [16389, 16361, 15]-code), using
- discarding factors based on OA(6433, 16389, S64, 14), using
- OA 7-folding and stacking [i] based on OA(6433, 16387, S64, 14), using
(19, 33, 4382)-Net over F64 — Digital
Digital (19, 33, 4382)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6433, 4382, F64, 14) (dual of [4382, 4349, 15]-code), using
- 278 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 12 times 0, 1, 53 times 0, 1, 207 times 0) [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(13) ⊂ Ce(12) [i] based on
- 278 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 12 times 0, 1, 53 times 0, 1, 207 times 0) [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
(19, 33, large)-Net in Base 64 — Upper bound on s
There is no (19, 33, large)-net in base 64, because
- 12 times m-reduction [i] would yield (19, 21, large)-net in base 64, but