Best Known (51, 51+11, s)-Nets in Base 9
(51, 51+11, 212577)-Net over F9 — Constructive and digital
Digital (51, 62, 212577)-net over F9, using
- net defined by OOA [i] based on linear OOA(962, 212577, F9, 11, 11) (dual of [(212577, 11), 2338285, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(962, 1062886, F9, 11) (dual of [1062886, 1062824, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(962, 1062888, F9, 11) (dual of [1062888, 1062826, 12]-code), using
- trace code [i] based on linear OA(8131, 531444, F81, 11) (dual of [531444, 531413, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(8131, 531441, F81, 11) (dual of [531441, 531410, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(8131, 531444, F81, 11) (dual of [531444, 531413, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(962, 1062888, F9, 11) (dual of [1062888, 1062826, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(962, 1062886, F9, 11) (dual of [1062886, 1062824, 12]-code), using
(51, 51+11, 1062888)-Net over F9 — Digital
Digital (51, 62, 1062888)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(962, 1062888, F9, 11) (dual of [1062888, 1062826, 12]-code), using
- trace code [i] based on linear OA(8131, 531444, F81, 11) (dual of [531444, 531413, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(8131, 531441, F81, 11) (dual of [531441, 531410, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(8131, 531444, F81, 11) (dual of [531444, 531413, 12]-code), using
(51, 51+11, large)-Net in Base 9 — Upper bound on s
There is no (51, 62, large)-net in base 9, because
- 9 times m-reduction [i] would yield (51, 53, large)-net in base 9, but