Best Known (109−20, 109, s)-Nets in Base 8
(109−20, 109, 26217)-Net over F8 — Constructive and digital
Digital (89, 109, 26217)-net over F8, using
- net defined by OOA [i] based on linear OOA(8109, 26217, F8, 20, 20) (dual of [(26217, 20), 524231, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8109, 262170, F8, 20) (dual of [262170, 262061, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8109, 262174, F8, 20) (dual of [262174, 262065, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(8103, 262144, F8, 20) (dual of [262144, 262041, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(879, 262144, F8, 15) (dual of [262144, 262065, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(86, 30, F8, 4) (dual of [30, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8109, 262174, F8, 20) (dual of [262174, 262065, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(8109, 262170, F8, 20) (dual of [262170, 262061, 21]-code), using
(109−20, 109, 262174)-Net over F8 — Digital
Digital (89, 109, 262174)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8109, 262174, F8, 20) (dual of [262174, 262065, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(8103, 262144, F8, 20) (dual of [262144, 262041, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(879, 262144, F8, 15) (dual of [262144, 262065, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(86, 30, F8, 4) (dual of [30, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
(109−20, 109, large)-Net in Base 8 — Upper bound on s
There is no (89, 109, large)-net in base 8, because
- 18 times m-reduction [i] would yield (89, 91, large)-net in base 8, but