Best Known (31−11, 31, s)-Nets in Base 81
(31−11, 31, 106288)-Net over F81 — Constructive and digital
Digital (20, 31, 106288)-net over F81, using
- net defined by OOA [i] based on linear OOA(8131, 106288, F81, 11, 11) (dual of [(106288, 11), 1169137, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [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]
- OOA 5-folding and stacking with additional row [i] based on linear OA(8131, 531441, F81, 11) (dual of [531441, 531410, 12]-code), using
(31−11, 31, 265722)-Net over F81 — Digital
Digital (20, 31, 265722)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8131, 265722, F81, 2, 11) (dual of [(265722, 2), 531413, 12]-NRT-code), using
- OOA 2-folding [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
- OOA 2-folding [i] based on linear OA(8131, 531444, F81, 11) (dual of [531444, 531413, 12]-code), using
(31−11, 31, large)-Net in Base 81 — Upper bound on s
There is no (20, 31, large)-net in base 81, because
- 9 times m-reduction [i] would yield (20, 22, large)-net in base 81, but