Best Known (39, 39+10, s)-Nets in Base 8
(39, 39+10, 52430)-Net over F8 — Constructive and digital
Digital (39, 49, 52430)-net over F8, using
- net defined by OOA [i] based on linear OOA(849, 52430, F8, 10, 10) (dual of [(52430, 10), 524251, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(849, 262150, F8, 10) (dual of [262150, 262101, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(849, 262144, F8, 10) (dual of [262144, 262095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(843, 262144, F8, 9) (dual of [262144, 262101, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(80, 6, F8, 0) (dual of [6, 6, 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
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(849, 262150, F8, 10) (dual of [262150, 262101, 11]-code), using
(39, 39+10, 140968)-Net over F8 — Digital
Digital (39, 49, 140968)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(849, 140968, F8, 10) (dual of [140968, 140919, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(849, 262144, F8, 10) (dual of [262144, 262095, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(849, 262144, F8, 10) (dual of [262144, 262095, 11]-code), using
(39, 39+10, large)-Net in Base 8 — Upper bound on s
There is no (39, 49, large)-net in base 8, because
- 8 times m-reduction [i] would yield (39, 41, large)-net in base 8, but