Best Known (37−9, 37, s)-Nets in Base 8
(37−9, 37, 8193)-Net over F8 — Constructive and digital
Digital (28, 37, 8193)-net over F8, using
- net defined by OOA [i] based on linear OOA(837, 8193, F8, 9, 9) (dual of [(8193, 9), 73700, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(837, 32773, F8, 9) (dual of [32773, 32736, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(837, 32774, F8, 9) (dual of [32774, 32737, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(81, 6, F8, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(837, 32774, F8, 9) (dual of [32774, 32737, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(837, 32773, F8, 9) (dual of [32773, 32736, 10]-code), using
(37−9, 37, 21292)-Net over F8 — Digital
Digital (28, 37, 21292)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(837, 21292, F8, 9) (dual of [21292, 21255, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(837, 32774, F8, 9) (dual of [32774, 32737, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(81, 6, F8, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(837, 32774, F8, 9) (dual of [32774, 32737, 10]-code), using
(37−9, 37, large)-Net in Base 8 — Upper bound on s
There is no (28, 37, large)-net in base 8, because
- 7 times m-reduction [i] would yield (28, 30, large)-net in base 8, but