Best Known (135−23, 135, s)-Nets in Base 9
(135−23, 135, 96626)-Net over F9 — Constructive and digital
Digital (112, 135, 96626)-net over F9, using
- 91 times duplication [i] based on digital (111, 134, 96626)-net over F9, using
- net defined by OOA [i] based on linear OOA(9134, 96626, F9, 23, 23) (dual of [(96626, 23), 2222264, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9134, 1062887, F9, 23) (dual of [1062887, 1062753, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9134, 1062888, F9, 23) (dual of [1062888, 1062754, 24]-code), using
- trace code [i] based on linear OA(8167, 531444, F81, 23) (dual of [531444, 531377, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(8167, 531441, F81, 23) (dual of [531441, 531374, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(8164, 531441, F81, 22) (dual of [531441, 531377, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(8167, 531444, F81, 23) (dual of [531444, 531377, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9134, 1062888, F9, 23) (dual of [1062888, 1062754, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9134, 1062887, F9, 23) (dual of [1062887, 1062753, 24]-code), using
- net defined by OOA [i] based on linear OOA(9134, 96626, F9, 23, 23) (dual of [(96626, 23), 2222264, 24]-NRT-code), using
(135−23, 135, 1062890)-Net over F9 — Digital
Digital (112, 135, 1062890)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9135, 1062890, F9, 23) (dual of [1062890, 1062755, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9134, 1062888, F9, 23) (dual of [1062888, 1062754, 24]-code), using
- trace code [i] based on linear OA(8167, 531444, F81, 23) (dual of [531444, 531377, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(8167, 531441, F81, 23) (dual of [531441, 531374, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(8164, 531441, F81, 22) (dual of [531441, 531377, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(8167, 531444, F81, 23) (dual of [531444, 531377, 24]-code), using
- linear OA(9134, 1062889, F9, 22) (dual of [1062889, 1062755, 23]-code), using Gilbert–Varšamov bound and bm = 9134 > Vbs−1(k−1) = 649678 752301 596562 497170 090905 767171 706738 256180 707716 261276 593950 875842 087626 563334 305694 762745 986859 304522 199582 856754 993729 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 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
- linear OA(9134, 1062888, F9, 23) (dual of [1062888, 1062754, 24]-code), using
- construction X with Varšamov bound [i] based on
(135−23, 135, large)-Net in Base 9 — Upper bound on s
There is no (112, 135, large)-net in base 9, because
- 21 times m-reduction [i] would yield (112, 114, large)-net in base 9, but