Best Known (15, 15+8, s)-Nets in Base 81
(15, 15+8, 132862)-Net over F81 — Constructive and digital
Digital (15, 23, 132862)-net over F81, using
- net defined by OOA [i] based on linear OOA(8123, 132862, F81, 8, 8) (dual of [(132862, 8), 1062873, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8123, 531448, F81, 8) (dual of [531448, 531425, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(8116, 531441, F81, 6) (dual of [531441, 531425, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- OA 4-folding and stacking [i] based on linear OA(8123, 531448, F81, 8) (dual of [531448, 531425, 9]-code), using
(15, 15+8, 372313)-Net over F81 — Digital
Digital (15, 23, 372313)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8123, 372313, F81, 8) (dual of [372313, 372290, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8123, 531448, F81, 8) (dual of [531448, 531425, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(8116, 531441, F81, 6) (dual of [531441, 531425, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(8123, 531448, F81, 8) (dual of [531448, 531425, 9]-code), using
(15, 15+8, large)-Net in Base 81 — Upper bound on s
There is no (15, 23, large)-net in base 81, because
- 6 times m-reduction [i] would yield (15, 17, large)-net in base 81, but