Best Known (13, 13+10, s)-Nets in Base 64
(13, 13+10, 822)-Net over F64 — Constructive and digital
Digital (13, 23, 822)-net over F64, using
- net defined by OOA [i] based on linear OOA(6423, 822, F64, 10, 10) (dual of [(822, 10), 8197, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6423, 4110, F64, 10) (dual of [4110, 4087, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(644, 14, F64, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- OA 5-folding and stacking [i] based on linear OA(6423, 4110, F64, 10) (dual of [4110, 4087, 11]-code), using
(13, 13+10, 3277)-Net in Base 64 — Constructive
(13, 23, 3277)-net in base 64, using
- net defined by OOA [i] based on OOA(6423, 3277, S64, 10, 10), using
- OA 5-folding and stacking [i] based on OA(6423, 16385, S64, 10), using
- discarding factors based on OA(6423, 16386, S64, 10), using
- discarding parts of the base [i] based on linear OA(12819, 16386, F128, 10) (dual of [16386, 16367, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 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(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(9) ⊂ Ce(8) [i] based on
- discarding parts of the base [i] based on linear OA(12819, 16386, F128, 10) (dual of [16386, 16367, 11]-code), using
- discarding factors based on OA(6423, 16386, S64, 10), using
- OA 5-folding and stacking [i] based on OA(6423, 16385, S64, 10), using
(13, 13+10, 4205)-Net over F64 — Digital
Digital (13, 23, 4205)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6423, 4205, F64, 10) (dual of [4205, 4182, 11]-code), using
- 101 step Varšamov–Edel lengthening with (ri) = (2, 10 times 0, 1, 89 times 0) [i] based on linear OA(6420, 4101, F64, 10) (dual of [4101, 4081, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(641, 5, F64, 1) (dual of [5, 4, 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 Ce(9) ⊂ Ce(7) [i] based on
- 101 step Varšamov–Edel lengthening with (ri) = (2, 10 times 0, 1, 89 times 0) [i] based on linear OA(6420, 4101, F64, 10) (dual of [4101, 4081, 11]-code), using
(13, 13+10, large)-Net in Base 64 — Upper bound on s
There is no (13, 23, large)-net in base 64, because
- 8 times m-reduction [i] would yield (13, 15, large)-net in base 64, but