Best Known (88, 109, s)-Nets in Base 9
(88, 109, 53144)-Net over F9 — Constructive and digital
Digital (88, 109, 53144)-net over F9, using
- net defined by OOA [i] based on linear OOA(9109, 53144, F9, 21, 21) (dual of [(53144, 21), 1115915, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9109, 531441, F9, 21) (dual of [531441, 531332, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- OOA 10-folding and stacking with additional row [i] based on linear OA(9109, 531441, F9, 21) (dual of [531441, 531332, 22]-code), using
(88, 109, 265723)-Net over F9 — Digital
Digital (88, 109, 265723)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(9109, 265723, F9, 2, 21) (dual of [(265723, 2), 531337, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(9109, 531446, F9, 21) (dual of [531446, 531337, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9109, 531447, F9, 21) (dual of [531447, 531338, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(9109, 531441, F9, 21) (dual of [531441, 531332, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(90, 6, F9, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(9109, 531447, F9, 21) (dual of [531447, 531338, 22]-code), using
- OOA 2-folding [i] based on linear OA(9109, 531446, F9, 21) (dual of [531446, 531337, 22]-code), using
(88, 109, large)-Net in Base 9 — Upper bound on s
There is no (88, 109, large)-net in base 9, because
- 19 times m-reduction [i] would yield (88, 90, large)-net in base 9, but