Best Known (44−9, 44, s)-Nets in Base 8
(44−9, 44, 65537)-Net over F8 — Constructive and digital
Digital (35, 44, 65537)-net over F8, using
- net defined by OOA [i] based on linear OOA(844, 65537, F8, 9, 9) (dual of [(65537, 9), 589789, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(844, 262149, F8, 9) (dual of [262149, 262105, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(844, 262151, F8, 9) (dual of [262151, 262107, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- 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(837, 262144, F8, 7) (dual of [262144, 262107, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(81, 7, F8, 1) (dual of [7, 6, 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(844, 262151, F8, 9) (dual of [262151, 262107, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(844, 262149, F8, 9) (dual of [262149, 262105, 10]-code), using
(44−9, 44, 170359)-Net over F8 — Digital
Digital (35, 44, 170359)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(844, 170359, F8, 9) (dual of [170359, 170315, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(844, 262151, F8, 9) (dual of [262151, 262107, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- 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(837, 262144, F8, 7) (dual of [262144, 262107, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(81, 7, F8, 1) (dual of [7, 6, 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(844, 262151, F8, 9) (dual of [262151, 262107, 10]-code), using
(44−9, 44, large)-Net in Base 8 — Upper bound on s
There is no (35, 44, large)-net in base 8, because
- 7 times m-reduction [i] would yield (35, 37, large)-net in base 8, but