Best Known (50−11, 50, s)-Nets in Base 8
(50−11, 50, 6557)-Net over F8 — Constructive and digital
Digital (39, 50, 6557)-net over F8, using
- net defined by OOA [i] based on linear OOA(850, 6557, F8, 11, 11) (dual of [(6557, 11), 72077, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(850, 32786, F8, 11) (dual of [32786, 32736, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(850, 32787, F8, 11) (dual of [32787, 32737, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(846, 32768, F8, 11) (dual of [32768, 32722, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(831, 32768, F8, 7) (dual of [32768, 32737, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(84, 19, F8, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,8)), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(850, 32787, F8, 11) (dual of [32787, 32737, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(850, 32786, F8, 11) (dual of [32786, 32736, 12]-code), using
(50−11, 50, 32787)-Net over F8 — Digital
Digital (39, 50, 32787)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(850, 32787, F8, 11) (dual of [32787, 32737, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(846, 32768, F8, 11) (dual of [32768, 32722, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(831, 32768, F8, 7) (dual of [32768, 32737, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(84, 19, F8, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,8)), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
(50−11, 50, large)-Net in Base 8 — Upper bound on s
There is no (39, 50, large)-net in base 8, because
- 9 times m-reduction [i] would yield (39, 41, large)-net in base 8, but