Best Known (68, 68+20, s)-Nets in Base 9
(68, 68+20, 5905)-Net over F9 — Constructive and digital
Digital (68, 88, 5905)-net over F9, using
- 92 times duplication [i] based on digital (66, 86, 5905)-net over F9, using
- net defined by OOA [i] based on linear OOA(986, 5905, F9, 20, 20) (dual of [(5905, 20), 118014, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(986, 59050, F9, 20) (dual of [59050, 58964, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(986, 59054, F9, 20) (dual of [59054, 58968, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(986, 59049, F9, 20) (dual of [59049, 58963, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(981, 59049, F9, 19) (dual of [59049, 58968, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(986, 59054, F9, 20) (dual of [59054, 58968, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(986, 59050, F9, 20) (dual of [59050, 58964, 21]-code), using
- net defined by OOA [i] based on linear OOA(986, 5905, F9, 20, 20) (dual of [(5905, 20), 118014, 21]-NRT-code), using
(68, 68+20, 38645)-Net over F9 — Digital
Digital (68, 88, 38645)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(988, 38645, F9, 20) (dual of [38645, 38557, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(988, 59059, F9, 20) (dual of [59059, 58971, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(986, 59049, F9, 20) (dual of [59049, 58963, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(976, 59049, F9, 17) (dual of [59049, 58973, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(92, 10, F9, 2) (dual of [10, 8, 3]-code or 10-arc in PG(1,9)), using
- extended Reed–Solomon code RSe(8,9) [i]
- Hamming code H(2,9) [i]
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(988, 59059, F9, 20) (dual of [59059, 58971, 21]-code), using
(68, 68+20, large)-Net in Base 9 — Upper bound on s
There is no (68, 88, large)-net in base 9, because
- 18 times m-reduction [i] would yield (68, 70, large)-net in base 9, but