Best Known (66, 66+18, s)-Nets in Base 8
(66, 66+18, 3644)-Net over F8 — Constructive and digital
Digital (66, 84, 3644)-net over F8, using
- 81 times duplication [i] based on digital (65, 83, 3644)-net over F8, using
- net defined by OOA [i] based on linear OOA(883, 3644, F8, 18, 18) (dual of [(3644, 18), 65509, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(883, 32796, F8, 18) (dual of [32796, 32713, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(883, 32800, F8, 18) (dual of [32800, 32717, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(876, 32768, F8, 18) (dual of [32768, 32692, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(87, 32, F8, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(883, 32800, F8, 18) (dual of [32800, 32717, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(883, 32796, F8, 18) (dual of [32796, 32713, 19]-code), using
- net defined by OOA [i] based on linear OOA(883, 3644, F8, 18, 18) (dual of [(3644, 18), 65509, 19]-NRT-code), using
(66, 66+18, 32802)-Net over F8 — Digital
Digital (66, 84, 32802)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(884, 32802, F8, 18) (dual of [32802, 32718, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(883, 32800, F8, 18) (dual of [32800, 32717, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(876, 32768, F8, 18) (dual of [32768, 32692, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(851, 32768, F8, 12) (dual of [32768, 32717, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(87, 32, F8, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(883, 32801, F8, 17) (dual of [32801, 32718, 18]-code), using Gilbert–Varšamov bound and bm = 883 > Vbs−1(k−1) = 2 840348 910947 848899 792909 175307 121440 304483 185745 164138 405175 242440 120611 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(883, 32800, F8, 18) (dual of [32800, 32717, 19]-code), using
- construction X with Varšamov bound [i] based on
(66, 66+18, large)-Net in Base 8 — Upper bound on s
There is no (66, 84, large)-net in base 8, because
- 16 times m-reduction [i] would yield (66, 68, large)-net in base 8, but