Best Known (99−18, 99, s)-Nets in Base 8
(99−18, 99, 29131)-Net over F8 — Constructive and digital
Digital (81, 99, 29131)-net over F8, using
- 81 times duplication [i] based on digital (80, 98, 29131)-net over F8, using
- net defined by OOA [i] based on linear OOA(898, 29131, F8, 18, 18) (dual of [(29131, 18), 524260, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(898, 262179, F8, 18) (dual of [262179, 262081, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(898, 262181, F8, 18) (dual of [262181, 262083, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(891, 262144, F8, 18) (dual of [262144, 262053, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(861, 262144, F8, 12) (dual of [262144, 262083, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(87, 37, F8, 5) (dual of [37, 30, 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(898, 262181, F8, 18) (dual of [262181, 262083, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(898, 262179, F8, 18) (dual of [262179, 262081, 19]-code), using
- net defined by OOA [i] based on linear OOA(898, 29131, F8, 18, 18) (dual of [(29131, 18), 524260, 19]-NRT-code), using
(99−18, 99, 262183)-Net over F8 — Digital
Digital (81, 99, 262183)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(899, 262183, F8, 18) (dual of [262183, 262084, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(898, 262181, F8, 18) (dual of [262181, 262083, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(891, 262144, F8, 18) (dual of [262144, 262053, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(861, 262144, F8, 12) (dual of [262144, 262083, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(87, 37, F8, 5) (dual of [37, 30, 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(898, 262182, F8, 17) (dual of [262182, 262084, 18]-code), using Gilbert–Varšamov bound and bm = 898 > Vbs−1(k−1) = 791 358920 727010 713090 495044 118138 667381 106473 184905 462452 856577 717517 541734 557402 086574 [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(898, 262181, F8, 18) (dual of [262181, 262083, 19]-code), using
- construction X with Varšamov bound [i] based on
(99−18, 99, large)-Net in Base 8 — Upper bound on s
There is no (81, 99, large)-net in base 8, because
- 16 times m-reduction [i] would yield (81, 83, large)-net in base 8, but