Best Known (21, 21+5, s)-Nets in Base 9
(21, 21+5, 531443)-Net over F9 — Constructive and digital
Digital (21, 26, 531443)-net over F9, using
- net defined by OOA [i] based on linear OOA(926, 531443, F9, 5, 5) (dual of [(531443, 5), 2657189, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(926, 1062887, F9, 5) (dual of [1062887, 1062861, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(926, 1062888, F9, 5) (dual of [1062888, 1062862, 6]-code), using
- trace code [i] based on linear OA(8113, 531444, F81, 5) (dual of [531444, 531431, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(8113, 531441, F81, 5) (dual of [531441, 531428, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(8110, 531441, F81, 4) (dual of [531441, 531431, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(4) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(8113, 531444, F81, 5) (dual of [531444, 531431, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(926, 1062888, F9, 5) (dual of [1062888, 1062862, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(926, 1062887, F9, 5) (dual of [1062887, 1062861, 6]-code), using
(21, 21+5, 1062888)-Net over F9 — Digital
Digital (21, 26, 1062888)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(926, 1062888, F9, 5) (dual of [1062888, 1062862, 6]-code), using
- trace code [i] based on linear OA(8113, 531444, F81, 5) (dual of [531444, 531431, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(8113, 531441, F81, 5) (dual of [531441, 531428, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(8110, 531441, F81, 4) (dual of [531441, 531431, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(4) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(8113, 531444, F81, 5) (dual of [531444, 531431, 6]-code), using
(21, 21+5, large)-Net in Base 9 — Upper bound on s
There is no (21, 26, large)-net in base 9, because
- 3 times m-reduction [i] would yield (21, 23, large)-net in base 9, but